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For)1 453 2 1080 4288 t (mail research!netlib)1 795 1 1540 4468 t (send only ttgrx1 from port)4 1055 1 1540 4588 t (send only dttgrx6 from port)4 1105 1 1540 4708 t (.)1540 4828 w ( and the sixth example in)5 1032(will cause you to receive in the mail the first example in single)12 2568 2 1080 5008 t (double precision.)1 688 1 1080 5128 t (June 5, 1985)2 508 1 720 5608 t cleartomark showpage saveobj restore end %%EndPage: 0 1 %%Page: 1 2 DpostDict begin /saveobj save def mark 2 pagesetup 12 CW f (TTGR)2117 1230 w 12 B f (- A Package for Solving)4 1207 1 2435 1230 t (Partial Differential Equations in Two Space Variables.)6 2801 1 1479 1380 t 10 I f ( an n)2 50( ma)1 50( fm)1 72( au uf)2 78( Ka)1 50( K)1 92( .)1 0(L L.)1 81 8 2643 1620 t ( r)1 0( ye er)2 83( ry)1 44( hr)1 39( ch)1 50( Sc)1 44( S)1 75( .)1 0( L.)1 25( L)1 81( .)1 0(N N.)1 92 12 2613 1800 t 10 R f (AT&T Bell Laboratories)2 993 1 2383 1980 t (Murray Hill, New Jersey 07974)4 1267 1 2246 2100 t 10 B f (1. Introduction.)1 695 1 720 2460 t 10 R f ( \()1 72(Many physical problems require the solution of partial differential equations)9 3171 2 970 2616 t 10 B f (pde)4213 2616 w 10 R f ('s\) in two space)3 671 1 4369 2616 t ( sufficiently complex that their solution must be carried out numeri-)10 2714( these equations are)3 787(variables. Typically)1 819 3 720 2736 t (cally.)720 2856 w (This paper describes a formulation for solving systems of)8 2321 1 970 3012 t 10 B f (pde)3319 3012 w 10 R f ( on a rectan-)3 507('s in two spatial variables,)4 1058 2 3475 3012 t ( formulation allows for terms of the form)7 1652( The)1 206( time.)1 229(gle, and)1 342 4 720 3132 t 10 B f (u)3175 3132 w 10 R f (,)3231 3132 w 10 B f (u)3281 3132 w 7 I f (x x)1 31 1 3348 3152 t 10 R f (,)3387 3132 w 10 B f (u)3437 3132 w 7 I f (y y)1 31 1 3504 3152 t 10 R f (,)3543 3132 w 10 B f (u)3593 3132 w 7 I f (t t)1 20 1 3660 3152 t 10 R f (,)3688 3132 w 10 B f (u)3738 3132 w 7 I f ( t)1 0(x xt)1 51 2 3805 3152 t 10 R f (,)3864 3132 w 10 B f (u)3914 3132 w 7 I f ( t)1 0(y yt)1 51 2 3981 3152 t 10 R f (,)4040 3132 w 10 B f (u)4090 3132 w 7 I f (x xx x)2 62 1 4157 3152 t 10 R f (,)4227 3132 w 10 B f (u)4277 3132 w 7 I f ( t)1 0(x xx xt)2 82 2 4344 3152 t 10 R f (,)4434 3132 w 10 B f (u)4484 3132 w 7 I f (x xy y)2 62 1 4551 3152 t 10 R f (,)4621 3132 w 10 B f (u)4671 3132 w 7 I f ( t)1 0(x xy yt)2 82 2 4738 3152 t 10 R f (,)4828 3132 w 10 B f (u)4878 3132 w 7 I f (y yy y)2 62 1 4945 3152 t 10 R f (,)5015 3132 w 10 B f (u)720 3252 w 7 I f ( t)1 0(y yy yt)2 82 2 787 3272 t 10 R f (in the)1 228 1 904 3252 t 10 B f (pde)1160 3252 w 10 R f ('s, and)1 269 1 1316 3252 t 10 B f (u)1613 3252 w 10 R f (,)1669 3252 w 10 B f (u)1722 3252 w 7 I f (x x)1 31 1 1789 3272 t 10 R f (,)1828 3252 w 10 B f (u)1881 3252 w 7 I f (y y)1 31 1 1948 3272 t 10 R f (,)1987 3252 w 10 B f (u)2040 3252 w 7 I f (t t)1 20 1 2107 3272 t 10 R f (,)2135 3252 w 10 B f (u)2188 3252 w 7 I f ( t)1 0(x xt)1 51 2 2255 3272 t 10 R f (,)2314 3252 w 10 B f (u)2367 3252 w 7 I f ( t)1 0(y yt)1 51 2 2434 3272 t 10 R f (in the boundary conditions \()4 1139 1 2521 3252 t 10 B f (bc)3688 3252 w 10 R f ('s \), where)2 429 1 3788 3252 t 10 B f (u)4245 3252 w 10 R f (is a vector of)3 527 1 4329 3252 t 10 B f (pde)4884 3252 w 10 R f (variables, and)1 554 1 720 3372 t 10 B f (u)1299 3372 w 7 I f (x x)1 31 1 1366 3392 t 10 R f (denotes)1430 3372 w 10 S f (\266)1760 3372 w 10 B f (u)1817 3372 w 10 I f (/ /)1 28 1 1905 3372 t 10 S f (\266)1941 3372 w 10 I f (x x)1 44 1 1998 3372 t 10 R f ( mathematical formulation is given in section 2.)7 1910( The)1 205(, etc.)1 191 3 2042 3372 t 10 B f (Getting Started.)1 688 1 720 3612 t 10 R f ( to solve a simple)4 752(If you have not read this screed before, want)8 1871 2 970 3768 t 10 B f (pde)3631 3768 w 10 R f (and have no interest in fancy)5 1215 1 3825 3768 t (things having nothing to do with your immediate needs, just do the following)12 3092 1 720 3888 t 10 S f (\267)970 4044 w 10 R f (Read section 2, Statement of the Problem, p 3.)8 1857 1 1041 4044 t 10 S f (\267)970 4200 w 10 R f (Read section 4, Formulation, Example 1, p 6.)7 1813 1 1041 4200 t 10 S f (\267)970 4356 w 10 R f (Skim section 5, Software, pp 11-16.)5 1438 1 1041 4356 t 10 S f (\267)970 4512 w 10 R f (Read Appendix 4, Programs, Example 1, pp 1-9.)7 1945 1 1041 4512 t 10 S f (\267)970 4668 w 10 R f (Copy the example program \( either from a file or the paper \).)12 2430 1 1041 4668 t 10 S f (\267)970 4824 w 10 R f (Run it and make sure it works as advertised.)8 1767 1 1041 4824 t 10 S f (\267)970 4980 w 10 R f (Alter it to solve your problem; see the start of Appendix 4 for the steps involved here.)16 3426 1 1041 4980 t ( above scheme typically involves changing only a couple of)9 2469( The)1 214( solved.)1 320(With luck, your problem is now)5 1317 4 720 5136 t (dozen lines of code in the example to get a problem solved.)11 2374 1 720 5256 t 10 B f (The Examples)1 609 1 720 5496 t 10 R f ( address your problem, there are many exam-)7 1834(If the above "Getting Started" reading does not seem to)9 2236 2 970 5652 t ( examples are)2 548( The)1 205( of them is quite likely to be of help.)9 1455( One)1 216(ples discussed in section 4.)4 1080 5 720 5772 t 10 S f (\267)970 5928 w 10 R f (A simple)1 364 1 1041 5928 t 10 B f (pde)1430 5928 w 10 R f (, the heat equation, see Example 1, p-6.)7 1572 1 1586 5928 t 10 S f (\267)970 6084 w 10 R f (A coupled system of)3 824 1 1041 6084 t 10 B f (pde)1890 6084 w 10 R f (s, see Example 2, p 6.)5 871 1 2046 6084 t 10 S f (\267)970 6240 w 10 R f (A material interface, see Example 3, p 7)7 1604 1 1041 6240 t 10 S f (\267)970 6396 w 10 R f (A non-rectangular domain, see Example 4, p 8)7 1860 1 1041 6396 t 10 S f (\267)970 6552 w 10 R f (A static problem, see Example 5, pp 8-9)7 1606 1 1041 6552 t 10 S f (\267)970 6708 w 10 R f (Error estimation, see Example 6, pp 9-10)6 1642 1 1041 6708 t cleartomark showpage saveobj restore end %%EndPage: 1 2 %%Page: 2 3 DpostDict begin /saveobj save def mark 3 pagesetup 10 R f (- 2 -)2 166 1 2797 480 t 10 B f (A Principle.)1 511 1 720 840 t 10 R f (The guiding principle used during the design of)7 1901 1 720 996 t 10 CW f (TTGR)2646 996 w 10 R f (was)2911 996 w (It is better a user complain the package runs slowly than)10 2246 1 1080 1176 t (complain the package cannot solve the problem at hand.)8 2243 1 1080 1296 t ( people wanting to solve various model equations in a hurry, so they can get on to other models)18 3864(As a result,)2 456 2 720 1512 t (and problems \( that is, people whose time is more valuable than machine time \), should find)16 3834 1 720 1632 t 10 CW f (TTGR)4589 1632 w 10 R f (very)4863 1632 w ( problem many times in a production environment may)8 2236( people wanting to solve the same)6 1372(useful. However,)1 712 3 720 1752 t (find)720 1872 w 10 CW f (TTGR)908 1872 w 10 R f ( for their needs; see section 6 for ways to speed)10 1900( slow)1 240( settings,)1 357(, with the default)3 683 4 1148 1872 t 10 CW f (TTGR)4354 1872 w 10 R f (up consid-)1 420 1 4620 1872 t (erably.)720 1992 w ( Galerkin's method in space, using B-splines, and a vari-)9 2338(The numerical solution technique employs)4 1732 2 970 2148 t ( section 3 for an out-)5 844(able order, variable time-step extrapolated backward difference procedure in time; see)10 3476 2 720 2268 t ( manual for the software called)5 1258( 5 is a user)4 439( Section)1 353( examples are formulated in section 4.)6 1540(line. Many)1 461 5 720 2388 t 10 CW f (TTGR)4800 2388 w 10 R f ( 6 describes)2 481( Section)1 355(for Transient Tensor product Galerkin for partial differential equations on Rectangles.)10 3484 3 720 2508 t (ways of making)2 668 1 720 2628 t 10 CW f (TTGR)1429 2628 w 10 R f (run faster than the default settings allow and also describes alternative ways of)12 3330 1 1710 2628 t (entering and using the package.)4 1261 1 720 2748 t ( 2 gives a brief tutorial on extrapolation.)7 1686( Appendix)1 455( a brief tutorial on B-splines.)5 1204(Appendix 1 gives)2 725 4 970 2904 t (Appendix 3 discusses improvements that could be made in)8 2380 1 720 3024 t 10 CW f (TTGR)3129 3024 w 10 R f ( 4 presents the programs used)5 1199(. Appendix)1 472 2 3369 3024 t (to solve the examples discussed in section 4.)7 1782 1 720 3144 t ( in)1 118(Appendix 5 summarizes the basic procedures available)6 2279 2 970 3300 t 10 CW f (TTGR)3407 3300 w 10 R f (, along with their arguments, and)5 1393 1 3647 3300 t ( using)1 243(gives a list of error states and problems that may arise when)11 2415 2 720 3420 t 10 CW f (TTGR)3404 3420 w 10 R f (, along with the common causes of)6 1396 1 3644 3420 t (such difficulties.)1 666 1 720 3540 t 10 B f (A Warning.)1 506 1 720 3780 t 10 R f ( may be anywhere, and there are oodles of corners in which)11 2431( Bugs)1 261( an infant state.)3 625(This software is in)3 753 4 970 3936 t ( goal is to create a)5 758( The)1 212( using the best tools at hand.)6 1178( code has been created modularly,)5 1397( The)1 213(they can lurk.)2 562 6 720 4056 t (robust and widely applicable package for solving two-dimensional)7 2765 1 720 4176 t 10 B f (pde)3525 4176 w 10 R f ( this modularity has)3 840(s. However,)1 519 2 3681 4176 t ( example, the)2 537( For)1 190(hurt a bit.)2 388 3 720 4296 t 10 B f (ode)1861 4296 w 10 R f (solver)2037 4296 w 10 CW f (IODE)2307 4296 w 10 R f ( to solve the spatially dis-)5 1025(from the Port Library has been used)6 1442 2 2573 4296 t ( the name of the)4 659( the user can change)4 820( Thus,)1 278(cretized equations.)1 751 4 720 4416 t 10 B f (pde)3257 4416 w 10 R f (defining subroutine, but not that for the)6 1598 1 3442 4416 t 10 B f (bc)720 4536 w 10 R f (s:)820 4536 w 10 CW f (IODE)953 4536 w 10 R f ( is a learning experience for that soft-)7 1528( This)1 233( one subroutine name it can pass below it.)8 1713(only has)1 342 4 1224 4536 t ( linear algebra is robust, but)5 1150( The)1 212( is very robust, but slow.)5 1021( spatial discretization scheme)3 1191( The)1 211(ware as well.)2 535 6 720 4656 t ( to quote a user: "Expensive solutions are better)8 1977( However,)1 449( optimal in run-time and space.)5 1288(not necessarily)1 606 4 720 4776 t ( use it any)3 432( Please)1 313( bottom line here is: be careful with and mistrustful of this code.)12 2663( The)1 212(than none at all.")3 700 5 720 4896 t ( know generally how things go with)6 1438(way you see fit, report any bugs or anomalous behavior to us, and let us)14 2882 2 720 5016 t ( plan many improvements to)4 1188( we)1 151(it. Since)1 363 3 720 5136 t 10 CW f (TTGR)2458 5136 w 10 R f (, see Appendix 3, your comments are both welcome and)9 2342 1 2698 5136 t ( electronic mail addresses are)4 1174( Our)1 205(likely to be effective.)3 848 3 720 5256 t (research!lck)2637 5436 w (research!nls)2639 5556 w 10 B f (The examples are available through electronic mail.)6 2207 1 1901 5772 t 10 R f (There are 6 examples and the)5 1196 1 970 5928 t 10 B f (fortran)2196 5928 w 10 R f ( exam-)1 280( For)1 195( precision.)1 422(for each is available in single or double)7 1607 4 2536 5928 t (ple, the command)2 713 1 720 6048 t (mail research!netlib)1 795 1 1180 6228 t (send only ttgrx1 from port)4 1055 1 1180 6348 t (send only dttgrx6 from port)4 1105 1 1180 6468 t (.)1180 6588 w (will cause you to receive in the mail the first example in single and the sixth example in double precision.)19 4223 1 720 6768 t cleartomark showpage saveobj restore end %%EndPage: 2 3 %%Page: 3 4 DpostDict begin /saveobj save def mark 4 pagesetup 10 R f (- 3 -)2 166 1 2797 480 t 10 B f ( of the Problem.)3 682(2. Statement)1 557 2 720 840 t 10 R f ( can be handled by)4 808(The general form of equations that)5 1453 2 970 996 t 10 CW f (TTGR)3271 996 w 10 R f (is an essentially classical, text-book)4 1489 1 3551 996 t (divergence-form)720 1116 w 10 B f (pde)1409 1116 w 10 R f (with a general set of boundary conditions \()7 1711 1 1590 1116 t 10 B f (bc)3326 1116 w 10 R f (s \).)1 122 1 3426 1116 t 10 B f (The pde-bc Formulation.)2 1070 1 720 1356 t 10 R f (The general)1 477 1 970 1512 t 10 B f (pde)1476 1512 w 10 R f (-)1632 1512 w 10 B f (bc)1665 1512 w 10 R f (form that can be solved with the approach used in)9 2024 1 1794 1512 t 10 CW f (TTGR)3847 1512 w 10 R f ( by the follow-)3 606(is given)1 318 2 4116 1512 t (ing equations, where)2 849 1 720 1632 t 10 B f (u)1604 1632 w 10 R f (is a vector of)3 548 1 1695 1632 t 10 B f (pde)2277 1632 w 10 R f (variables of length)2 761 1 2467 1632 t 10 I f (n n)1 50 1 3262 1632 t 7 I f (u u)1 35 1 3323 1652 t 10 R f ( all physical laws that are second)6 1368(. Since)1 306 2 3366 1632 t ( the)1 155(order in space can be written in divergence form,)8 2014 2 720 1752 t 10 B f (pde)2922 1752 w 10 R f ('s are assumed to be in semi-linear, divergence-)7 1962 1 3078 1752 t (form)720 1872 w 10 S f (\266)1245 2162 w 10 I f (x x)1 44 1 1302 2162 t 10 S f (\266)1271 2032 w 10 S1 f (_ __)1 131 1 1230 2062 t 10 B f (a)1403 2092 w 7 R f (\( 1 \))2 91 1 1458 2052 t 10 R f (\()1565 2092 w 10 I f (t t)1 28 1 1606 2092 t 10 R f (,)1642 2092 w 10 I f (x x)1 44 1 1708 2092 t 10 R f (,)1760 2092 w 10 I f (y y)1 44 1 1826 2092 t 10 R f (,)1878 2092 w 10 B f (u)1944 2092 w 10 R f (,)2008 2092 w 10 B f (u)2074 2092 w 7 I f (x x)1 31 1 2141 2112 t 10 R f (,)2188 2092 w 10 B f (u)2254 2092 w 7 I f (y y)1 31 1 2321 2112 t 10 R f (,)2368 2092 w 10 B f (u)2434 2092 w 7 I f (t t)1 20 1 2501 2112 t 10 R f (,)2537 2092 w 10 B f (u)2603 2092 w 7 I f ( t)1 0(x xt)1 51 2 2670 2112 t 10 R f (,)2737 2092 w 10 B f (u)2803 2092 w 7 I f ( t)1 0(y yt)1 51 2 2870 2112 t 10 R f (\))2937 2092 w 10 S f (+ +)1 55 1 3027 2092 t (\266)1245 2432 w 10 I f (y y)1 44 1 1302 2432 t 10 S f (\266)1271 2302 w 10 S1 f (_ __)1 131 1 1230 2332 t 10 B f (a)1403 2362 w 7 R f (\( 2 \))2 91 1 1458 2322 t 10 R f (\()1565 2362 w 10 I f (t t)1 28 1 1606 2362 t 10 R f (,)1642 2362 w 10 I f (x x)1 44 1 1708 2362 t 10 R f (,)1760 2362 w 10 I f (y y)1 44 1 1826 2362 t 10 R f (,)1878 2362 w 10 B f (u)1944 2362 w 10 R f (,)2008 2362 w 10 B f (u)2074 2362 w 7 I f (x x)1 31 1 2141 2382 t 10 R f (,)2188 2362 w 10 B f (u)2254 2362 w 7 I f (y y)1 31 1 2321 2382 t 10 R f (,)2368 2362 w 10 B f (u)2434 2362 w 7 I f (t t)1 20 1 2501 2382 t 10 R f (,)2537 2362 w 10 B f (u)2603 2362 w 7 I f ( t)1 0(x xt)1 51 2 2670 2382 t 10 R f (,)2737 2362 w 10 B f (u)2803 2362 w 7 I f ( t)1 0(y yt)1 51 2 2870 2382 t 10 R f (\))2937 2362 w 10 S f (= =)1 55 1 3027 2362 t 10 R f (\(2.1\))4849 2362 w 10 B f (f)1411 2632 w 10 R f (\()1452 2632 w 10 I f (t t)1 28 1 1493 2632 t 10 R f (,)1529 2632 w 10 I f (x x)1 44 1 1595 2632 t 10 R f (,)1647 2632 w 10 I f (y y)1 44 1 1713 2632 t 10 R f (,)1765 2632 w 10 B f (u)1831 2632 w 10 R f (,)1895 2632 w 10 B f (u)1961 2632 w 7 I f (x x)1 31 1 2028 2652 t 10 R f (,)2075 2632 w 10 B f (u)2141 2632 w 7 I f (y y)1 31 1 2208 2652 t 10 R f (,)2255 2632 w 10 B f (u)2321 2632 w 7 I f (t t)1 20 1 2388 2652 t 10 R f (,)2424 2632 w 10 B f (u)2490 2632 w 7 I f ( t)1 0(x xt)1 51 2 2557 2652 t 10 R f (,)2624 2632 w 10 B f (u)2690 2632 w 7 I f ( t)1 0(y yt)1 51 2 2757 2652 t 10 R f (\) ,)1 98 1 2824 2632 t (where)720 2862 w 10 B f (a)1002 2862 w 10 R f (and)1091 2862 w 10 B f (f)1274 2862 w 10 R f ( of their arguments, for)4 983(are vector-valued functions)2 1119 2 1346 2862 t 10 I f (L L)1 56 1 3488 2862 t 7 I f (x x)1 31 1 3555 2882 t 10 S f (\243)3635 2862 w 10 I f (x x)1 44 1 3731 2862 t 10 S f (\243)3816 2862 w 10 I f (R R)1 61 1 3912 2862 t 7 I f (x x)1 31 1 3984 2882 t 10 R f (and)4063 2862 w 10 I f (L L)1 56 1 4247 2862 t 7 I f (y y)1 31 1 4314 2882 t 10 S f (\243)4394 2862 w 10 I f (y y)1 44 1 4490 2862 t 10 S f (\243)4575 2862 w 10 I f (R R)1 61 1 4671 2862 t 7 I f (y y)1 31 1 4743 2882 t 10 R f ( is)1 107(. It)1 151 2 4782 2862 t (required that the length of)4 1041 1 720 2982 t 10 B f (a)1787 2982 w 10 R f (and)1863 2982 w 10 B f (f)2033 2982 w 10 R f (be equal to)2 440 1 2092 2982 t 10 I f (n n)1 50 1 2558 2982 t 7 I f (u u)1 35 1 2619 3002 t 10 R f (, the number of)3 613 1 2662 2982 t 10 B f (pde)3301 2982 w 10 R f (variables, that is, the length of the vec-)7 1557 1 3483 2982 t (tor)720 3102 w 10 B f (u)856 3102 w 10 R f ( boundary conditions are assumed to have the form)8 2041(. The)1 230 2 912 3102 t 10 B f (b)1220 3282 w 10 R f (\()1317 3282 w 10 I f (t t)1 28 1 1391 3282 t 10 R f (,)1427 3282 w 10 I f (x x)1 44 1 1493 3282 t 10 R f (,)1545 3282 w 10 I f (y y)1 44 1 1611 3282 t 10 R f (,)1663 3282 w 10 B f (u)1729 3282 w 10 R f (,)1793 3282 w 10 B f (u)1859 3282 w 7 I f (x x)1 31 1 1926 3302 t 10 R f (,)1973 3282 w 10 B f (u)2039 3282 w 7 I f (y y)1 31 1 2106 3302 t 10 R f (,)2153 3282 w 10 B f (u)2219 3282 w 7 I f (t t)1 20 1 2286 3302 t 10 R f (,)2322 3282 w 10 B f (u)2388 3282 w 7 I f ( t)1 0(x xt)1 51 2 2455 3302 t 10 R f (,)2522 3282 w 10 B f (u)2588 3282 w 7 I f ( t)1 0(y yt)1 51 2 2655 3302 t 10 R f (\))2755 3282 w 10 S f (= =)1 55 1 2845 3282 t 10 R f (0 \(2.2\))1 2091 1 2949 3282 t (where)720 3462 w 10 B f (b)991 3462 w 10 R f (is a vector-valued function, of length)5 1490 1 1075 3462 t 10 I f (n n)1 50 1 2593 3462 t 7 I f (u u)1 35 1 2654 3482 t 10 R f ( identically zero component of the)5 1387( Any)1 226( its arguments.)2 594(, of)1 136 4 2697 3462 t 10 B f (bc)720 3582 w 10 R f (vector)848 3582 w 10 B f (b)1125 3582 w 10 R f (is treated as an inactive)4 943 1 1209 3582 t 10 B f (bc)2180 3582 w 10 R f ( each of the)3 471(. If)1 144 2 2280 3582 t 10 B f (pde)2923 3582 w 10 R f ('s is second order in space, then each of the)9 1761 1 3079 3582 t 10 B f (bc)4868 3582 w 10 R f ('s)4968 3582 w ( any of the)3 436( If)1 120(will have to be active.)4 895 3 720 3702 t 10 B f (pde)2201 3702 w 10 R f ('s are of order less than 2 in space, some of the)11 1928 1 2357 3702 t 10 B f (bc)4315 3702 w 10 R f ('s must accord-)2 625 1 4415 3702 t ( conditions \()2 500( Initial)1 289(ingly be inactive.)2 691 3 720 3822 t 10 B f (ic)2200 3822 w 10 R f ('s\))2272 3822 w 10 B f (u)2402 3822 w 10 R f (\( 0 ,)2 124 1 2466 3822 t 10 I f (x x)1 44 1 2598 3822 t 10 R f (,)2650 3822 w 10 I f (y y)1 44 1 2683 3822 t 10 R f (\) must be supplied, but need not satisfy the)8 1713 1 2735 3822 t 10 B f (bc)4473 3822 w 10 R f ('s \(2.2\).)1 313 1 4573 3822 t (A classical example of the above form \( with)8 1796 1 970 3978 t 10 I f (n n)1 50 1 2791 3978 t 7 I f (u u)1 35 1 2852 3998 t 10 S f (= =)1 55 1 2935 3978 t 10 R f (1 \) is the heat equation [22])6 1098 1 3030 3978 t 10 I f (u u)1 50 1 1220 4158 t 7 I f (t t)1 20 1 1281 4178 t 10 S f (= =)1 55 1 1358 4158 t 10 I f (u u)1 50 1 1462 4158 t 7 I f (x xx x)2 62 1 1523 4178 t 10 S f (+ +)1 55 1 1642 4158 t 10 I f (u u)1 50 1 1746 4158 t 7 I f (y yy y)2 62 1 1807 4178 t 10 R f ( 0 , 1 ])4 190(on [)1 240 2 2116 4158 t 10 S f (\264)2595 4158 w 10 R f ([ 0 , 1 ])4 223 1 2691 4158 t (subject to boundary conditions)3 1230 1 720 4338 t 10 I f (u u)1 50 1 1220 4518 t 10 R f (\()1278 4518 w 10 I f (t t)1 28 1 1319 4518 t 10 R f (,)1355 4518 w 10 I f (x x)1 44 1 1388 4518 t 10 R f (,)1440 4518 w 10 I f (y y)1 44 1 1473 4518 t 10 R f (\))1525 4518 w 10 S f (= =)1 55 1 1615 4518 t 10 I f (g g)1 50 1 1719 4518 t 10 R f (\()1777 4518 w 10 I f (x x)1 44 1 1818 4518 t 10 R f (,)1870 4518 w 10 I f (y y)1 44 1 1903 4518 t 10 R f (\))1955 4518 w (with initial conditions)2 879 1 720 4698 t 10 I f (u u)1 50 1 1220 4878 t 10 R f (\( 0 ,)2 124 1 1278 4878 t 10 I f (x x)1 44 1 1410 4878 t 10 R f (,)1462 4878 w 10 I f (y y)1 44 1 1495 4878 t 10 R f (\))1547 4878 w 10 S f (= =)1 55 1 1637 4878 t 10 R f (0.)1741 4878 w (Note that these initial conditions do not satisfy the)8 2011 1 720 5058 t 10 B f (bc)2756 5058 w 10 R f ('s, unless)1 372 1 2856 5058 t 10 I f (g g)1 50 1 3253 5058 t 10 R f (\()3311 5058 w 10 I f (x x)1 44 1 3352 5058 t 10 R f (,)3404 5058 w 10 I f (y y)1 44 1 3437 5058 t 10 R f (\))3489 5058 w 10 S f (\272)3571 5058 w 10 R f ( this equation we have)4 893(0. For)1 264 2 3667 5058 t 10 I f (a a)1 50 1 1220 5238 t 7 R f (\( 1 \))2 91 1 1281 5198 t 10 S f (= =)1 55 1 1437 5238 t 10 I f (u u)1 50 1 1541 5238 t 7 I f (x x)1 31 1 1602 5258 t 10 R f (,)1649 5238 w 10 I f (a a)1 50 1 1781 5238 t 7 R f (\( 2 \))2 91 1 1842 5198 t 10 S f (= =)1 55 1 1957 5238 t 10 I f (u u)1 50 1 2028 5238 t 7 I f (y y)1 31 1 2089 5258 t 10 R f (and)2235 5238 w 10 I f (f f)1 28 1 2494 5238 t 10 S f (= =)1 55 1 2587 5238 t 10 I f (u u)1 50 1 2691 5238 t 7 I f (t t)1 20 1 2752 5258 t 10 R f (with)720 5418 w 10 B f (bc)923 5418 w 10 R f (s)1023 5418 w 10 B f (b)1220 5598 w 10 S f (= =)1 55 1 1325 5598 t 10 I f (u u)1 50 1 1429 5598 t 10 S f (- -)1 55 1 1519 5598 t 10 I f ( .)1 0( .)1 33(g g)1 50 3 1614 5598 t 10 R f (Note that the form of \(2.1\)-\(2.2\) encompasses parabolic \()8 2291 1 970 5814 t 10 I f (u u)1 50 1 3288 5814 t 7 I f (t t)1 20 1 3349 5834 t 10 S f (= =)1 55 1 3426 5814 t 10 I f (u u)1 50 1 3530 5814 t 7 I f (x xx x)2 62 1 3591 5834 t 10 S f (+ +)1 55 1 3710 5814 t 10 I f (u u)1 50 1 3814 5814 t 7 I f (y yy y)2 62 1 3875 5834 t 10 R f (\), elliptic \()2 423 1 3972 5814 t 10 I f (u u)1 50 1 4422 5814 t 7 I f (x xx x)2 62 1 4483 5834 t 10 S f (+ +)1 55 1 4602 5814 t 10 I f (u u)1 50 1 4706 5814 t 7 I f (y yy y)2 62 1 4767 5834 t 10 S f (= =)1 55 1 4886 5814 t 10 R f (0)4990 5814 w (\) and hyperbolic \()3 712 1 720 5934 t 10 I f (u u)1 50 1 1457 5934 t 7 I f (t t)1 20 1 1518 5954 t 10 S f (= =)1 55 1 1595 5934 t 10 I f (u u)1 50 1 1699 5934 t 7 I f (x x)1 31 1 1760 5954 t 10 S f (+ +)1 55 1 1848 5934 t 10 I f (u u)1 50 1 1952 5934 t 7 I f (y y)1 31 1 2013 5954 t 10 R f ( also encompasses)2 732( It)1 111(\) problems.)1 455 3 2077 5934 t 10 B f (pde)3400 5934 w 10 R f ('s that have no solution, such as)6 1274 1 3556 5934 t 10 I f (u u)1 50 1 1220 6114 t 7 I f (x x)1 31 1 1275 6133 t 7 R f (2)1275 6074 w 10 S f (+ +)1 55 1 1358 6114 t 10 I f (u u)1 50 1 1453 6114 t 7 I f (y y)1 31 1 1508 6133 t 7 R f (2)1508 6074 w 10 S f ( -)1 0( -)1 112(= =)1 55 3 1600 6114 t 10 R f (1)1783 6114 w (over the real field.)3 731 1 720 6294 t 10 B f (History.)720 6534 w 10 R f (Several other Fortran software packages are available for solving)8 2654 1 970 6690 t 10 B f (pde)3657 6690 w 10 R f ('s in two spatial variables, for)5 1227 1 3813 6690 t ( packages assume that the)4 1031( These)1 288(example [28,29].)1 679 3 720 6810 t 10 B f (pde)2743 6810 w 10 R f (has a form like)3 596 1 2924 6810 t 10 B f (u)1220 6990 w 7 I f (t t)1 20 1 1287 7010 t 10 S f (= =)1 55 1 1364 6990 t 10 B f (f)1468 6990 w 10 R f (\()1509 6990 w 10 I f (t t)1 28 1 1583 6990 t 10 R f (,)1619 6990 w 10 I f (x x)1 44 1 1685 6990 t 10 R f (,)1737 6990 w 10 I f (y y)1 44 1 1803 6990 t 10 R f (,)1855 6990 w 10 B f (u)1921 6990 w 10 R f (,)1985 6990 w 10 B f (u)2051 6990 w 7 I f (x x)1 31 1 2118 7010 t 10 R f (,)2165 6990 w 10 B f (u)2231 6990 w 7 I f (y y)1 31 1 2298 7010 t 10 R f (,)2345 6990 w 10 B f (u)2411 6990 w 7 I f (x xx x)2 62 1 2478 7010 t 10 R f (,)2556 6990 w 10 B f (u)2622 6990 w 7 I f (y yy y)2 62 1 2689 7010 t 10 R f (,)2767 6990 w 10 B f (u)2833 6990 w 7 I f (x xy y)2 62 1 2900 7010 t 10 R f (\) \(2.3\))1 2038 1 3002 6990 t (with boundary conditions of the form)5 1496 1 720 7170 t cleartomark showpage saveobj restore end %%EndPage: 3 4 %%Page: 4 5 DpostDict begin /saveobj save def mark 5 pagesetup 10 R f (- 4 -)2 166 1 2797 480 t 10 B f (u)1220 840 w 10 S f ( a)1 5( a)1 112(= =)1 55 3 1325 840 t 10 R f (\()1505 840 w 10 I f (t t)1 28 1 1546 840 t 10 R f (\))1582 840 w (or)720 1020 w 10 S f (a a)1 68 1 1220 1200 t 10 R f (\()1296 1200 w 10 I f (t t)1 28 1 1337 1200 t 10 R f (,)1373 1200 w 10 I f (x x)1 44 1 1406 1200 t 10 R f (,)1458 1200 w 10 I f (y y)1 44 1 1491 1200 t 10 R f (,)1543 1200 w 10 B f (u)1576 1200 w 10 R f (\))1640 1200 w 10 S f ( b)1 5( b)1 95(+ +)1 55 3 1721 1200 t 10 R f (\()1884 1200 w 10 I f (t t)1 28 1 1925 1200 t 10 R f (,)1961 1200 w 10 I f (x x)1 44 1 1994 1200 t 10 R f (,)2046 1200 w 10 I f (y y)1 44 1 2079 1200 t 10 R f (,)2131 1200 w 10 B f (u)2164 1200 w 10 R f (\))2228 1200 w 10 B f (u)2277 1200 w 7 I f (N N)1 47 1 2344 1220 t 10 S f ( g)1 5( g)1 90(= =)1 55 3 2448 1200 t 10 R f (\()2606 1200 w 10 I f (t t)1 28 1 2647 1200 t 10 R f (,)2683 1200 w 10 I f (x x)1 44 1 2716 1200 t 10 R f (,)2768 1200 w 10 I f (y y)1 44 1 2801 1200 t 10 R f (,)2853 1200 w 10 B f (u)2886 1200 w 10 R f (\). \(2.4\))1 2090 1 2950 1200 t (where)720 1380 w 10 B f (u)992 1380 w 7 I f (N N)1 47 1 1059 1400 t 10 R f ( [29] allows no)3 617( example,)1 392( For)1 193( are slight variations on this theme.)6 1426( There)1 286(is the normal derivative.)3 983 6 1143 1380 t 10 S f (\266)745 1620 w 10 I f (x x)1 44 1 802 1620 t 10 S f (\266)878 1620 w 10 I f (y y)1 44 1 935 1620 t 10 S f (\266)770 1490 w 7 R f (2)824 1450 w 10 B f (u)899 1490 w 10 S1 f (_ _____)1 264 1 731 1520 t 10 R f (terms in the)2 480 1 1034 1550 t 10 B f (pde)1542 1550 w 10 R f ( [28] allows no dependence of)5 1220(\(2.3\). Also,)1 483 2 1726 1550 t 10 B f (u)3457 1550 w 10 R f (in the coefficients of the)4 982 1 3541 1550 t 10 B f (bc)4551 1550 w 10 R f (s in \(2.4\),)2 389 1 4651 1550 t (and only applies to)3 758 1 720 1720 t 10 I f ( r)1 0( ar)1 39( la)1 50( al)1 28( ca)1 50(s sc)1 83 6 1503 1720 t 10 B f (pde)1778 1720 w 10 R f ( interesting)1 448( formulation \(2.3\)-\(2.4\) covers a wide range of physically)8 2300(s. While)1 358 3 1934 1720 t (problems, it does not cover problems with)6 1691 1 720 1840 t 10 B f (u)2437 1840 w 7 I f ( t)1 0(x xt)1 51 2 2504 1860 t 10 R f (or)2589 1840 w 10 B f (u)2698 1840 w 7 I f ( t)1 0(x xx xt)2 82 2 2765 1860 t 10 R f ( problems having tan-)3 874(terms [32], nor does it deal with)6 1285 2 2881 1840 t (gential components in their)3 1091 1 720 1960 t 10 B f (bc)1836 1960 w 10 R f (s [34].)1 255 1 1936 1960 t 10 B f ( Method of Solution.)3 873(3. General)1 469 2 720 2200 t 10 R f (Let the solution)2 658 1 970 2356 t 10 B f (u)1668 2356 w 10 R f (\()1732 2356 w 10 I f (t t)1 28 1 1773 2356 t 10 R f (,)1809 2356 w 10 I f (x x)1 44 1 1842 2356 t 10 R f (,)1894 2356 w 10 I f (y y)1 44 1 1927 2356 t 10 R f ( B-)1 141(\), for a given instant in time, be approximated by a tensor-product of)12 2920 2 1979 2356 t (splines [27,1,2,3] of)2 844 1 720 2476 t 10 B f (order)1610 2476 w 10 I f (k k)1 44 1 1894 2476 t 7 I f (x x)1 31 1 1949 2496 t 10 R f (and)2034 2476 w 10 I f (k k)1 44 1 2223 2476 t 7 I f (y y)1 31 1 2278 2496 t 10 R f (on)2362 2476 w 10 B f (meshes)2507 2476 w 10 I f (X X)1 61 1 2857 2476 t 10 R f (\( 1 \))2 132 1 2926 2476 t 10 S f (\243)3107 2476 w 10 R f (. . .)2 125 1 3195 2451 t 10 S f (\243)3353 2476 w 10 I f (X X)1 61 1 3449 2476 t 10 R f (\()3518 2476 w 10 I f ( X)1 0(N NX)1 128 2 3559 2476 t 10 R f (\), and)1 247 1 3695 2476 t 10 I f (Y Y)1 56 1 3987 2476 t 10 R f (\( 1 \))2 132 1 4051 2476 t 10 S f (\243)4232 2476 w 10 R f (. . .)2 125 1 4320 2451 t 10 S f (\243)4478 2476 w 10 I f (Y Y)1 56 1 4574 2476 t 10 R f (\()4638 2476 w 10 I f ( Y)1 0(N NY)1 123 2 4679 2476 t 10 R f (\), see)1 230 1 4810 2476 t ( is, for any fixed)4 725( That)1 250(Appendix 1.)1 511 3 720 2596 t 10 I f (y y)1 44 1 2248 2596 t 10 R f (, each component of the solution will be approximated in)9 2434 1 2292 2596 t 10 I f (x x)1 44 1 4768 2596 t 10 R f (by a)1 186 1 4854 2596 t (piecewise polynomial function of degree less than)6 2044 1 720 2716 t 10 I f (k k)1 44 1 2796 2716 t 7 I f (x x)1 31 1 2851 2736 t 10 R f (, with)1 235 1 2890 2716 t 10 I f (k k)1 44 1 3157 2716 t 7 I f (x x)1 31 1 3212 2736 t 10 S f (- -)1 55 1 3267 2716 t 10 R f (2 continuous derivatives, where)3 1291 1 3338 2716 t 10 I f (k k)1 44 1 4661 2716 t 7 I f (x x)1 31 1 4716 2736 t 10 S f (\263)4796 2716 w 10 R f (2 is)1 148 1 4892 2716 t ( similar statement holds about the degree in)7 1880(any integer the user desires; a)5 1278 2 720 2836 t 10 I f (y y)1 44 1 3923 2836 t 10 R f (. Let)1 228 1 3967 2836 t 10 I f (B B)1 61 1 4240 2836 t 7 I f (p p)1 35 1 4312 2856 t 10 R f (\()4363 2836 w 10 I f (x x)1 44 1 4404 2836 t 10 R f (\) be the basis)3 584 1 4456 2836 t (functions in)1 475 1 720 2956 t 10 I f (x x)1 44 1 1220 2956 t 10 R f (and)1289 2956 w 10 I f (C C)1 67 1 1458 2956 t 7 I f (q q)1 35 1 1536 2976 t 10 R f (\()1587 2956 w 10 I f (y y)1 44 1 1628 2956 t 10 R f (\) be those in)3 491 1 1680 2956 t 10 I f (y y)1 44 1 2196 2956 t 10 R f (. Then)1 280 1 2240 2956 t 10 I f (u u)1 50 1 1220 3136 t 7 I f (i i)1 20 1 1281 3156 t 10 R f (\()1317 3136 w 10 I f (t t)1 28 1 1358 3136 t 10 R f (,)1394 3136 w 10 I f (x x)1 44 1 1427 3136 t 10 R f (,)1479 3136 w 10 I f (y y)1 44 1 1512 3136 t 10 R f (\))1564 3136 w 10 S f (= =)1 55 1 1654 3136 t 7 I f (p p)1 35 1 1801 3236 t 15 S f (S)1774 3166 w 7 I f (q q)1 35 1 1971 3236 t 15 S f (S)1944 3166 w 10 I f (U U)1 72 1 2098 3136 t 7 I f (q q)1 35 1 2181 3156 t 7 R f (,)2221 3156 w 7 I f (p p)1 35 1 2244 3156 t 7 R f (,)2284 3156 w 7 I f (i i)1 20 1 2307 3156 t 10 R f (\()2343 3136 w 10 I f (t t)1 28 1 2384 3136 t 10 R f (\))2420 3136 w 10 I f (B B)1 61 1 2493 3136 t 7 I f (p p)1 35 1 2565 3156 t 10 R f (\()2616 3136 w 10 I f (x x)1 44 1 2657 3136 t 10 R f (\))2709 3136 w 10 I f (C C)1 67 1 2782 3136 t 7 I f (q q)1 35 1 2860 3156 t 10 R f (\()2911 3136 w 10 I f (y y)1 44 1 2952 3136 t 10 R f (\))3004 3136 w 10 I f (. .)1 25 1 3053 3136 t 10 R f (\(3.1\))4849 3136 w (If we set)2 649 1 720 3396 t 10 I f (h h)1 50 1 1547 3396 t 7 I f (x x)1 31 1 1608 3416 t 10 S f (= =)1 55 1 1696 3396 t 7 I f (i i)1 20 1 1876 3466 t 10 R f (max)1800 3396 w 10 S f (\357 \357)1 49 1 2005 3413 t 10 I f (X X)1 61 1 2086 3396 t 10 R f (\()2155 3396 w 10 I f (i i)1 28 1 2196 3396 t 10 S f (+ +)1 55 1 2248 3396 t 10 R f (1 \))1 91 1 2319 3396 t 10 S f (- -)1 55 1 2467 3396 t 10 I f (X X)1 61 1 2571 3396 t 10 R f (\()2640 3396 w 10 I f (i i)1 28 1 2681 3396 t 10 R f (\))2717 3396 w 10 S f (\357 \357)1 49 1 2791 3413 t 10 R f (, and)1 347 1 2807 3396 t 10 I f (h h)1 50 1 3332 3396 t 7 I f (y y)1 31 1 3393 3416 t 10 R f (similarly, then the error,)3 1430 1 3610 3396 t 10 S f ( \357)1 0( \357)1 24(\357 \357)1 49 3 720 3588 t 10 I f (u u)1 50 1 825 3571 t 7 I f (i i)1 20 1 886 3591 t 10 S f (- -)1 55 1 954 3571 t 10 I f (u u)1 50 1 1049 3571 t 11 R f (\303)1058 3566 w 7 I f (i i)1 20 1 1110 3591 t 10 S f ( \357)1 0( \357)1 24(\357 \357)1 49 3 1171 3588 t 7 S f (\245)1233 3591 w 10 S f (\272)1333 3571 w 7 I f (x x)1 31 1 1470 3641 t 7 R f (,)1506 3641 w 7 I f (y y)1 31 1 1529 3641 t 10 R f (max)1429 3571 w 10 S f (\357 \357)1 49 1 1634 3588 t 10 I f (u u)1 50 1 1715 3571 t 7 I f (i i)1 20 1 1776 3591 t 10 R f (\()1812 3571 w 10 I f (t t)1 28 1 1853 3571 t 10 R f (,)1889 3571 w 10 I f (x x)1 44 1 1922 3571 t 10 R f (,)1974 3571 w 10 I f (y y)1 44 1 2007 3571 t 10 R f (\))2059 3571 w 10 S f (- -)1 55 1 2149 3571 t 10 I f (u u)1 50 1 2253 3571 t 11 R f (\303)2262 3566 w 7 I f (i i)1 20 1 2314 3591 t 10 R f (\()2350 3571 w 10 I f (t t)1 28 1 2391 3571 t 10 R f (,)2427 3571 w 10 I f (x x)1 44 1 2460 3571 t 10 R f (,)2512 3571 w 10 I f (y y)1 44 1 2545 3571 t 10 R f (\))2597 3571 w 10 S f (\357 \357)1 49 1 2671 3588 t 10 R f (, is)1 123 1 2687 3571 t 10 I f (O O)1 72 1 2841 3571 t 10 R f (\()2921 3571 w 10 I f (h h)1 50 1 2962 3571 t 7 I f (x x)1 31 1 3017 3590 t (k k)1 31 1 3017 3517 t 4 I f (x x)1 18 1 3054 3531 t 10 S f (+ +)1 55 1 3125 3571 t 10 I f (h h)1 50 1 3220 3571 t 7 I f (y y)1 31 1 3275 3590 t (k k)1 31 1 3275 3517 t 4 I f (y y)1 18 1 3312 3531 t 10 R f ( B-spline)1 369(\), for some)2 447 2 3351 3571 t 10 I f (u u)1 50 1 4197 3571 t 11 R f (\303)4206 3566 w 7 I f (i i)1 20 1 4258 3591 t 10 R f ( Since)1 277(, see [3].)2 353 2 4286 3571 t 10 I f (k k)1 44 1 4946 3571 t 7 I f (x x)1 31 1 5001 3591 t 10 R f (and)720 3741 w 10 I f (k k)1 44 1 895 3741 t 7 I f (y y)1 31 1 950 3761 t 10 R f ( integer)1 309(may be taken to be any)5 953 2 1020 3741 t 10 S f (\263 \263)1 55 1 2314 3741 t 10 R f (2, this gives a powerful technique for approximating the solution)9 2655 1 2385 3741 t 10 B f (u)720 3861 w 10 R f (\()784 3861 w 10 I f (t t)1 28 1 825 3861 t 10 R f (,)861 3861 w 10 I f (x x)1 44 1 894 3861 t 10 R f (,)946 3861 w 10 I f (y y)1 44 1 979 3861 t 10 R f ( \(R-R-G\) method [31,25] to find essentially the)7 1929( can use the Rayleigh-Ritz-Galerkin)4 1464( We)1 195(\) in space.)2 421 4 1031 3861 t ( of the)2 265(projection of the solution)3 1020 2 720 3981 t 10 B f (pde)2035 3981 w 10 R f ( reduces the)2 486( This)1 233(onto the space of B-splines we have selected.)7 1842 3 2221 3981 t 10 B f (pde)4812 3981 w 10 R f ('s)4968 3981 w (in space and time to)4 799 1 720 4101 t 10 B f (ode)1544 4101 w 10 R f ('s in time [15,31] for the coefficients)6 1472 1 1694 4101 t 10 I f (U U)1 72 1 3191 4101 t 7 I f (q q)1 35 1 3274 4121 t 7 R f (,)3314 4121 w 7 I f (p p)1 35 1 3337 4121 t 7 R f (,)3377 4121 w 7 I f (i i)1 20 1 3400 4121 t 10 R f (\()3436 4101 w 10 I f (t t)1 28 1 3477 4101 t 10 R f (\) in the expansion \(3.1\).)4 954 1 3513 4101 t (Thus, after the spatial discretization, only)5 1696 1 970 4257 t 10 B f (ode)2699 4257 w 10 R f ( these)1 239( Since)1 280('s in time remain to be solved.)6 1261 3 2849 4257 t 10 B f (ode)4663 4257 w 10 R f ('s are)1 227 1 4813 4257 t ( vir-)1 170( This)1 229( general "stiff" [9,10], an implicit differencing scheme must be used to solve them.)13 3318(known to be in)3 603 4 720 4377 t (tually requires that the partial derivatives of the)7 1936 1 720 4497 t 10 B f (a)2687 4497 w 10 R f (and)2768 4497 w 10 B f (f)2943 4497 w 10 R f (in \(2.1\) and of the)4 742 1 3007 4497 t 10 B f (b)3780 4497 w 10 R f ( respect to their)3 639(in \(2.2\), with)2 534 2 3867 4497 t (arguments be known, either analytically or numerically.)6 2235 1 720 4617 t (The next step is the solution of these time-varying)8 2044 1 970 4773 t 10 B f (ode)3044 4773 w 10 R f ( we assume that some basic one-step)6 1500('s. Here)1 346 2 3194 4773 t 10 B f (ode)720 4893 w 10 R f ( [22], or an)3 548( example, a backwards-Euler or Crank-Nicholson scheme)6 2518( For)1 225(solver is available.)2 818 4 931 4893 t ( an explicit method such as Gragg's modified mid-point rule)9 2534(exponentially-fitted technique [18], or even)4 1786 2 720 5013 t ([17,18], could be used.)3 936 1 720 5133 t 10 CW f (TTGR)1783 5133 w 10 R f ( [26])1 197( See)1 200( the default time discretization scheme.)5 1594(uses backwards-Euler as)2 994 4 2055 5133 t (for a description of the method used to solve the nonlinear equations arising at each time-step.)15 3761 1 720 5253 t ( others, have the property that for a given time-step)9 2118(All the above techniques, and many)5 1466 2 970 5409 t 10 S f (d)4587 5409 w 10 R f (they pro-)1 371 1 4669 5409 t ( to)1 105(duce an approximate solution accurate)4 1547 2 720 5529 t 10 I f (O O)1 72 1 2399 5529 t 10 R f (\()2479 5529 w 10 S f (d)2520 5529 w 7 S f (g)2574 5489 w 10 R f (\), where typically)2 705 1 2619 5529 t 10 S f (g)3351 5529 w 10 R f ( if the equations are)4 795( Moreover,)1 470(is 1 or 2.)3 356 3 3419 5529 t (solved using time-steps of)3 1050 1 720 5649 t 10 S f (d)1796 5649 w 10 R f (and)1871 5649 w 10 S f (d)2041 5649 w 10 I f (/ /)1 28 1 2098 5649 t 10 R f ( results of these two computations can be combined using extrapo-)10 2683(2, the)1 223 2 2134 5649 t ( to obtain a result accurate to)6 1159(lation [5,17])1 496 2 720 5769 t 10 I f (O O)1 72 1 2401 5769 t 10 R f (\()2481 5769 w 10 S f (d)2522 5769 w 7 R f (2)2576 5729 w 7 S f (g)2616 5729 w 10 R f ( process can be repeated indefinitely, so a basic pro-)9 2092(\). This)1 287 2 2661 5769 t (cess of accuracy)2 680 1 720 5889 t 10 S f (d)1439 5889 w 7 S f (g)1493 5849 w 10 R f (can be used to generate a sequence of processes of accuracy)10 2530 1 1569 5889 t 10 I f (O O)1 72 1 4139 5889 t 10 R f (\()4219 5889 w 10 S f (d)4260 5889 w 7 S f (g)4314 5849 w 10 R f (\),)4359 5889 w 10 I f (O O)1 72 1 4457 5889 t 10 R f (\()4537 5889 w 10 S f (d)4578 5889 w 7 R f (2)4632 5849 w 7 S f (g)4672 5849 w 10 R f (\) ,)1 74 1 4717 5889 t (. . .)2 125 1 4824 5864 t (,)4982 5889 w 10 I f (O O)1 72 1 720 6009 t 10 R f (\()800 6009 w 10 S f (d)841 6009 w 7 I f (P P)1 43 1 895 5969 t 7 S f (g)943 5969 w 10 R f (\) ,)1 74 1 988 6009 t (. . .)2 125 1 1095 5984 t (, see Appendix 2.)3 696 1 1270 6009 t ( available [23,24] for carrying out this extrapolation process and)9 2711(A step-size and order monitor is)5 1359 2 970 6165 t 10 I f ( y)1 0( ll ly)2 72( al)1 28( ca)1 50( ti ic)2 72( at)1 28( ma)1 50( om)1 72( to)1 50(a au ut)2 128 10 720 6285 t 10 R f (deciding what time-step)2 984 1 1308 6285 t 10 S f (d)2329 6285 w 10 R f (and order)1 391 1 2415 6285 t 10 I f (P P)1 61 1 2843 6285 t 10 S f (g)2912 6285 w 10 R f (should be used, given the accuracy desired in the)8 2050 1 2990 6285 t ( time should be computed, and the time)7 1598( user need only specify how accurately the solution in)9 2167(solution. The)1 555 3 720 6405 t (integration then proceeds automatically, with no need for the user to worry about choosing)13 3622 1 720 6525 t 10 S f (d)4367 6525 w 10 R f (.)4416 6525 w (The algorithm implemented by)3 1241 1 720 6681 t 10 CW f (TTGR)1986 6681 w 10 R f (for solving such)2 644 1 2251 6681 t 10 B f (pde)2920 6681 w 10 R f ('s then consists of 3 steps:)5 1047 1 3076 6681 t ( the equations in space using R-R-G with B-splines.)8 2074(1\) Discretize)1 654 2 970 6873 t ( a one-step method for solving the resulting)7 1740(2\) Produce)1 577 2 970 7029 t 10 B f (ode)3312 7029 w 10 R f ('s.)3462 7029 w ( that one-step process to the extrapolation step-size and order monitor.)10 2809(3\) Feed)1 444 2 970 7185 t cleartomark showpage saveobj restore end %%EndPage: 4 5 %%Page: 5 6 DpostDict begin /saveobj save def mark 6 pagesetup 10 R f (- 5 -)2 166 1 2797 480 t ( and the various)3 692(Section 6 gives a somewhat more detailed outline of the spatial discretization process)12 3628 2 720 840 t (parameters that describe the solution procedure.)5 1912 1 720 960 t 10 B f ( - Formulation.)2 647(4. Examples)1 542 2 720 1200 t 10 R f ( Appendix 4 for pro-)4 830( See)1 194(This section gives the formulation of many examples in terms of \(2.1\)-\(2.2\).)11 3046 3 970 1356 t (grams that solve these problems using)5 1524 1 720 1476 t 10 CW f (TTGR)2269 1476 w 10 R f (.)2509 1476 w 10 B f (Example 1 - A Simple Heat Equation.)6 1604 1 720 1716 t 10 R f (As a simple example of the use of)7 1356 1 970 1872 t 10 CW f (TTGR)2351 1872 w 10 R f (, consider solving the scalar heat equation)6 1672 1 2591 1872 t 10 I f (u u)1 50 1 1220 2052 t 7 I f (t t)1 20 1 1281 2072 t 10 S f (+ +)1 55 1 1349 2052 t 10 I f (u u)1 50 1 1444 2052 t 7 I f (x x)1 31 1 1505 2072 t 10 S f (+ +)1 55 1 1584 2052 t 10 I f (u u)1 50 1 1679 2052 t 7 I f (y y)1 31 1 1740 2072 t 10 S f (= =)1 55 1 1828 2052 t 10 B f (u)1932 2052 w 7 I f (x x)1 31 1 1999 2072 t 10 S f (+ +)1 55 1 2078 2052 t 10 B f (u)2173 2052 w 7 I f (x xx x)2 62 1 2240 2072 t 10 S f (+ +)1 55 1 2350 2052 t 10 I f (. .)1 25 1 2445 2052 t 10 R f (1)2478 2052 w 10 B f (u)2560 2052 w 7 I f (x xy y)2 62 1 2627 2072 t 10 S f (+ +)1 55 1 2737 2052 t 10 B f (u)2832 2052 w 7 I f (y y)1 31 1 2899 2072 t 10 S f (+ +)1 55 1 2978 2052 t 10 B f (u)3073 2052 w 7 I f (y yy y)2 62 1 3140 2072 t 10 S f (+ +)1 55 1 3250 2052 t 10 I f (. .)1 25 1 3345 2052 t 10 R f (1)3378 2052 w 10 B f (u)3460 2052 w 7 I f (x xy y)2 62 1 3527 2072 t 10 S f (+ +)1 55 1 3613 2052 t 10 I f (g g)1 50 1 3684 2052 t 10 R f (\()3742 2052 w 10 I f (t t)1 28 1 3783 2052 t 10 R f (,)3819 2052 w 10 I f (x x)1 44 1 3852 2052 t 10 R f (,)3904 2052 w 10 I f (y y)1 44 1 3937 2052 t 10 R f ( \(4.1\))1 977(\) ,)1 74 2 3989 2052 t ( 0 , 1 ])4 190( unit square [)3 545(on the)1 253 3 720 2232 t 10 S f (\264)1724 2232 w 10 R f ( where the source term)4 936([ 0 , 1 ],)4 248 2 1787 2232 t 10 I f (g g)1 50 1 3003 2232 t 10 R f (\()3061 2232 w 10 I f (t t)1 28 1 3102 2232 t 10 R f (,)3138 2232 w 10 I f (x x)1 44 1 3171 2232 t 10 R f (,)3223 2232 w 10 I f (y y)1 44 1 3256 2232 t 10 R f (\) is chosen so that the solution is a known)9 1732 1 3308 2232 t (function,)720 2352 w 10 I f (u u)1 50 1 1103 2352 t 10 R f (\()1161 2352 w 10 I f (t t)1 28 1 1202 2352 t 10 R f (,)1238 2352 w 10 I f (x x)1 44 1 1271 2352 t 10 R f (\))1323 2352 w 10 S f (= =)1 55 1 1413 2352 t 10 I f (t t)1 28 1 1517 2352 t 10 R f (.)1553 2322 w 10 I f (x x)1 44 1 1586 2352 t 10 R f (.)1638 2322 w 10 I f (y y)1 44 1 1671 2352 t 10 R f ( boundary conditions are then taken to be)7 1650(. The)1 230 2 1715 2352 t 10 I f (u u)1 50 1 1220 2532 t 10 R f (\()1278 2532 w 10 I f (t t)1 28 1 1319 2532 t 10 R f (,)1355 2532 w 10 I f (x x)1 44 1 1388 2532 t 10 R f (,)1440 2532 w 10 I f (y y)1 44 1 1473 2532 t 10 R f (\))1525 2532 w 10 S f (= =)1 55 1 1574 2532 t 10 I f (t t)1 28 1 1645 2532 t 10 R f (.)1681 2502 w 10 I f (x x)1 44 1 1714 2532 t 10 R f (.)1766 2502 w 10 I f (y y)1 44 1 1799 2532 t 10 R f (\(4.2\))4849 2532 w (with initial conditions)2 879 1 720 2748 t 10 I f (u u)1 50 1 1220 2928 t 10 R f (\( 0 ,)2 124 1 1278 2928 t 10 I f (x x)1 44 1 1410 2928 t 10 R f (,)1462 2928 w 10 I f (y y)1 44 1 1495 2928 t 10 R f (\))1547 2928 w 10 S f (= =)1 55 1 1637 2928 t 10 R f (0)1741 2928 w 10 I f (. .)1 25 1 1823 2928 t 10 R f (\(4.3\))4849 2928 w (The)720 3144 w 10 B f (pde)900 3144 w 10 R f (\(4.1\) is equivalent to \(2.1\) with)5 1246 1 1081 3144 t 10 I f (a a)1 50 1 1220 3324 t 7 R f (\( 1 \))2 91 1 1281 3284 t 10 S f (= =)1 55 1 1437 3324 t 10 I f (u u)1 50 1 1541 3324 t 10 S f (+ +)1 55 1 1631 3324 t 10 I f (u u)1 50 1 1726 3324 t 7 I f (x x)1 31 1 1787 3344 t 10 S f (+ +)1 55 1 1866 3324 t 10 I f (. .)1 25 1 1961 3324 t 10 R f (1)1994 3324 w 10 I f (u u)1 50 1 2076 3324 t 7 I f (y y)1 31 1 2137 3344 t 10 R f (,)2184 3324 w 10 I f (a a)1 50 1 1228 3504 t 7 R f (\( 2 \))2 91 1 1289 3464 t 10 S f (= =)1 55 1 1445 3504 t 10 I f (u u)1 50 1 1549 3504 t 10 S f (+ +)1 55 1 1639 3504 t 10 I f (u u)1 50 1 1734 3504 t 7 I f (y y)1 31 1 1795 3524 t 10 S f (+ +)1 55 1 1874 3504 t 10 I f (. .)1 25 1 1969 3504 t 10 R f (1)2002 3504 w 10 I f (u u)1 50 1 2084 3504 t 7 I f (x x)1 31 1 2145 3524 t 10 R f (,)2192 3504 w 10 I f (f f)1 28 1 1236 3684 t 10 S f (= =)1 55 1 1329 3684 t 10 I f (u u)1 50 1 1433 3684 t 7 I f (t t)1 20 1 1494 3704 t 10 S f (+ +)1 55 1 1562 3684 t 10 I f (u u)1 50 1 1657 3684 t 7 I f (x x)1 31 1 1718 3704 t 10 S f (+ +)1 55 1 1797 3684 t 10 I f (u u)1 50 1 1892 3684 t 7 I f (y y)1 31 1 1953 3704 t 10 S f (- -)1 55 1 2032 3684 t 10 I f (g g)1 50 1 2127 3684 t 10 R f (\()2185 3684 w 10 I f (t t)1 28 1 2226 3684 t 10 R f (,)2262 3684 w 10 I f (x x)1 44 1 2295 3684 t 10 R f (,)2347 3684 w 10 I f (y y)1 44 1 2380 3684 t 10 R f (\))2432 3684 w (while the)1 369 1 720 3864 t 10 B f (bc)1114 3864 w 10 R f (s \(4.2\) are equivalent to \(2.2\) with)6 1364 1 1214 3864 t 10 B f (b)1220 4044 w 10 S f (= =)1 55 1 1325 4044 t 10 I f (u u)1 50 1 1429 4044 t 10 R f (\()1487 4044 w 10 I f (t t)1 28 1 1528 4044 t 10 R f (,)1564 4044 w 10 I f (x x)1 44 1 1597 4044 t 10 R f (,)1649 4044 w 10 I f (y y)1 44 1 1682 4044 t 10 R f (\))1734 4044 w 10 S f (- -)1 55 1 1815 4044 t 10 I f ( .)1 0( .)1 33( y)1 0( y)1 76( x)1 0( x)1 76(t t)1 28 7 1910 4044 t 10 R f (See example 1 in Appendix 4 for code solving this problem.)10 2406 1 720 4224 t 10 B f (Example 2 - Two Heat Equations.)5 1440 1 720 4464 t 10 R f (Consider solving the coupled system of heat equations)7 2179 1 970 4620 t 10 I f (u u)1 50 1 1220 4940 t 7 R f (2)1281 4960 w 7 I f (t t)1 20 1 1321 4960 t 10 S f (= =)1 55 1 1398 4940 t 10 I f (u u)1 50 1 1502 4940 t 7 R f (2)1563 4960 w 7 I f (x xx x)2 62 1 1603 4960 t 10 S f (+ +)1 55 1 1713 4940 t 10 I f (u u)1 50 1 1808 4940 t 7 R f (2)1869 4960 w 7 I f (y yy y)2 62 1 1909 4960 t 10 S f (- -)1 55 1 2019 4940 t 10 I f (u u)1 50 1 2114 4940 t 7 R f (1)2175 4960 w 10 I f (u u)1 50 1 2250 4940 t 7 R f (2)2311 4960 w 10 S f (+ +)1 55 1 2394 4940 t 10 I f (g g)1 50 1 2489 4940 t 7 R f (2)2550 4960 w 10 I f (u u)1 50 1 1220 4780 t 7 R f (1)1281 4800 w 7 I f (t t)1 20 1 1321 4800 t 10 S f (= =)1 55 1 1398 4780 t 10 I f (u u)1 50 1 1502 4780 t 7 R f (1)1563 4800 w 7 I f (x xx x)2 62 1 1603 4800 t 10 S f (+ +)1 55 1 1713 4780 t 10 I f (u u)1 50 1 1808 4780 t 7 R f (1)1869 4800 w 7 I f (y yy y)2 62 1 1909 4800 t 10 S f (- -)1 55 1 2019 4780 t 10 I f (u u)1 50 1 2090 4780 t 7 R f (1)2151 4800 w 10 I f (u u)1 50 1 2226 4780 t 7 R f (2)2287 4800 w 10 S f (+ +)1 55 1 2370 4780 t 10 I f (g g)1 50 1 2465 4780 t 7 R f (1)2526 4800 w 10 R f (\(4.4\))4849 4890 w ( 0 , 1 ])4 190(on the unit square [)4 771 2 720 5120 t 10 S f (\264)1697 5120 w 10 R f ( where)1 268([ 0 , 1 ],)4 248 2 1760 5120 t 10 I f (g g)1 50 1 2301 5120 t 7 R f (1)2362 5140 w 10 R f (and)2430 5120 w 10 I f (g g)1 50 1 2599 5120 t 7 R f (2)2660 5140 w 10 R f (are chosen so that the solution is given by)8 1671 1 2728 5120 t 10 I f (u u)1 50 1 1220 5300 t 7 R f (1)1281 5320 w 10 R f (\()1332 5300 w 10 I f (t t)1 28 1 1373 5300 t 10 R f (,)1409 5300 w 10 I f (x x)1 44 1 1442 5300 t 10 R f (,)1494 5300 w 10 I f (y y)1 44 1 1527 5300 t 10 R f (\))1579 5300 w 10 S f (= =)1 55 1 1669 5300 t 10 I f (e e)1 44 1 1773 5300 t 7 I f (t t)1 20 1 1828 5260 t 7 R f (\()1869 5260 w 7 I f (x x)1 31 1 1897 5260 t 7 S f (- -)1 39 1 1944 5260 t 7 I f (y y)1 31 1 1994 5260 t 7 R f (\))2030 5260 w 10 R f (and)2275 5300 w 10 I f (u u)1 50 1 2625 5300 t 7 R f (2)2686 5320 w 10 R f (\()2737 5300 w 10 I f (t t)1 28 1 2778 5300 t 10 R f (,)2814 5300 w 10 I f (x x)1 44 1 2847 5300 t 10 R f (,)2899 5300 w 10 I f (y y)1 44 1 2932 5300 t 10 R f (\))2984 5300 w 10 S f (= =)1 55 1 3074 5300 t 10 I f (e e)1 44 1 3178 5300 t 7 S f (- -)1 39 1 3233 5260 t 7 I f (t t)1 20 1 3283 5260 t 7 R f (\()3308 5260 w 7 I f (x x)1 31 1 3336 5260 t 7 S f (- -)1 39 1 3383 5260 t 7 I f (y y)1 31 1 3433 5260 t 7 R f (\))3469 5260 w 10 I f (. .)1 25 1 3516 5300 t 10 R f (The boundary conditions are then taken to be)7 1805 1 720 5480 t 10 I f (u u)1 50 1 1220 5660 t 7 R f (1)1281 5680 w 10 R f (\()1332 5660 w 10 I f (t t)1 28 1 1373 5660 t 10 R f (,)1409 5660 w 10 I f (x x)1 44 1 1442 5660 t 10 R f (,)1494 5660 w 10 I f (y y)1 44 1 1527 5660 t 10 R f (\))1579 5660 w 10 S f (= =)1 55 1 1628 5660 t 10 I f (e e)1 44 1 1699 5660 t 7 I f (t t)1 20 1 1754 5620 t 7 R f (\()1779 5620 w 7 I f (x x)1 31 1 1807 5620 t 7 S f (- -)1 39 1 1854 5620 t 7 I f (y y)1 31 1 1904 5620 t 7 R f (\))1940 5620 w 10 R f (and)2185 5660 w 10 I f (u u)1 50 1 2535 5660 t 7 R f (2)2596 5680 w 10 R f (\()2647 5660 w 10 I f (t t)1 28 1 2688 5660 t 10 R f (,)2724 5660 w 10 I f (x x)1 44 1 2757 5660 t 10 R f (,)2809 5660 w 10 I f (y y)1 44 1 2842 5660 t 10 R f (\))2894 5660 w 10 S f (= =)1 55 1 2943 5660 t 10 I f (e e)1 44 1 3014 5660 t 7 S f (- -)1 39 1 3069 5620 t 7 I f (t t)1 20 1 3119 5620 t 7 R f (\()3144 5620 w 7 I f (x x)1 31 1 3172 5620 t 7 S f (- -)1 39 1 3219 5620 t 7 I f (y y)1 31 1 3269 5620 t 7 R f (\))3305 5620 w 10 R f (\(4.5\))4849 5660 w (with initial conditions)2 879 1 720 5876 t 10 I f (u u)1 50 1 1220 6056 t 7 R f (1)1281 6076 w 10 R f (\( 0 ,)2 124 1 1332 6056 t 10 I f (x x)1 44 1 1464 6056 t 10 R f (,)1516 6056 w 10 I f (y y)1 44 1 1549 6056 t 10 R f (\))1601 6056 w 10 S f (= =)1 55 1 1691 6056 t 10 R f (1)1795 6056 w 10 S f (= =)1 55 1 1894 6056 t 10 I f (u u)1 50 1 1998 6056 t 7 R f (2)2059 6076 w 10 R f (\( 0 ,)2 124 1 2110 6056 t 10 I f (x x)1 44 1 2242 6056 t 10 R f (,)2294 6056 w 10 I f (y y)1 44 1 2327 6056 t 10 R f (\). \(4.6\))1 2661 1 2379 6056 t (The)720 6272 w 10 B f (pde)900 6272 w 10 R f (\(4.4\) is equivalent to \(2.1\) with)5 1246 1 1081 6272 t 10 I f (a a)1 50 1 1220 6452 t 7 R f (1)1275 6471 w (\( 1 \))2 91 1 1275 6412 t 10 S f (= =)1 55 1 1431 6452 t 10 I f (u u)1 50 1 1535 6452 t 7 R f (1)1596 6472 w 7 I f (x x)1 31 1 1636 6472 t 10 R f (,)1683 6452 w 10 I f (a a)1 50 1 1815 6452 t 7 R f (2)1870 6471 w (\( 1 \))2 91 1 1870 6412 t 10 S f (= =)1 55 1 2026 6452 t 10 I f (u u)1 50 1 2130 6452 t 7 R f (2)2191 6472 w 7 I f (x x)1 31 1 2231 6472 t 10 R f (,)2278 6452 w 10 I f (a a)1 50 1 1228 6632 t 7 R f (1)1283 6651 w (\( 2 \))2 91 1 1283 6592 t 10 S f (= =)1 55 1 1439 6632 t 10 I f (u u)1 50 1 1543 6632 t 7 R f (1)1604 6652 w 7 I f (y y)1 31 1 1644 6652 t 10 R f (,)1691 6632 w 10 I f (a a)1 50 1 1823 6632 t 7 R f (2)1878 6651 w (\( 2 \))2 91 1 1878 6592 t 10 S f (= =)1 55 1 2034 6632 t 10 I f (u u)1 50 1 2138 6632 t 7 R f (2)2199 6652 w 7 I f (y y)1 31 1 2239 6652 t 10 R f (,)2286 6632 w 10 I f (f f)1 28 1 1236 6812 t 7 R f (1)1275 6832 w 10 S f (= =)1 55 1 1367 6812 t 10 I f (u u)1 50 1 1471 6812 t 7 R f (1)1532 6832 w 7 I f (t t)1 20 1 1572 6832 t 10 S f (+ +)1 55 1 1640 6812 t 10 I f (u u)1 50 1 1735 6812 t 7 R f (1)1796 6832 w 10 I f (u u)1 50 1 1871 6812 t 7 R f (2)1932 6832 w 10 S f (- -)1 55 1 2015 6812 t 10 I f (g g)1 50 1 2110 6812 t 7 R f (1)2171 6832 w 10 R f (,)2222 6812 w 10 I f (f f)1 28 1 2362 6812 t 7 R f (2)2401 6832 w 10 S f (= =)1 55 1 2493 6812 t 10 I f (u u)1 50 1 2597 6812 t 7 R f (2)2658 6832 w 7 I f (t t)1 20 1 2698 6832 t 10 S f (+ +)1 55 1 2766 6812 t 10 I f (u u)1 50 1 2861 6812 t 7 R f (1)2922 6832 w 10 I f (u u)1 50 1 2997 6812 t 7 R f (2)3058 6832 w 10 S f (- -)1 55 1 3141 6812 t 10 I f (g g)1 50 1 3236 6812 t 7 R f (2)3297 6832 w 10 R f (while the)1 369 1 720 6992 t 10 B f (bc)1114 6992 w 10 R f (s \(4.5\) are equivalent to \(2.2\) with)6 1364 1 1214 6992 t 10 I f (b b)1 50 1 1220 7172 t 7 R f (1)1281 7192 w 10 S f (= =)1 55 1 1373 7172 t 10 I f (u u)1 50 1 1477 7172 t 7 R f (1)1538 7192 w 10 R f (\()1589 7172 w 10 I f (t t)1 28 1 1630 7172 t 10 R f (,)1666 7172 w 10 I f (x x)1 44 1 1699 7172 t 10 R f (,)1751 7172 w 10 I f (y y)1 44 1 1784 7172 t 10 R f (\))1836 7172 w 10 S f (- -)1 55 1 1917 7172 t 10 I f (e e)1 44 1 2012 7172 t 7 I f (t t)1 20 1 2067 7132 t 7 R f (\()2092 7132 w 7 I f (x x)1 31 1 2120 7132 t 7 S f (- -)1 39 1 2167 7132 t 7 I f (y y)1 31 1 2217 7132 t 7 R f (\))2253 7132 w 10 R f (and)2498 7172 w 10 I f (b b)1 50 1 2848 7172 t 7 R f (2)2909 7192 w 10 S f (= =)1 55 1 3001 7172 t 10 I f (u u)1 50 1 3105 7172 t 7 R f (2)3166 7192 w 10 R f (\()3217 7172 w 10 I f (t t)1 28 1 3258 7172 t 10 R f (,)3294 7172 w 10 I f (x x)1 44 1 3327 7172 t 10 R f (,)3379 7172 w 10 I f (y y)1 44 1 3412 7172 t 10 R f (\))3464 7172 w 10 S f (- -)1 55 1 3545 7172 t 10 I f (e e)1 44 1 3640 7172 t 7 S f (- -)1 39 1 3695 7132 t 7 I f (t t)1 20 1 3745 7132 t 7 R f (\()3770 7132 w 7 I f (x x)1 31 1 3798 7132 t 7 S f (- -)1 39 1 3845 7132 t 7 I f (y y)1 31 1 3895 7132 t 7 R f (\))3931 7132 w 10 I f (. .)1 25 1 3978 7172 t cleartomark showpage saveobj restore end %%EndPage: 5 6 %%Page: 6 7 DpostDict begin /saveobj save def mark 7 pagesetup 10 R f (- 6 -)2 166 1 2797 480 t (See example 2 in Appendix 4 for code solving this problem.)10 2406 1 720 840 t 10 B f (More Meaty Examples)2 970 1 720 1080 t 10 R f ( excellent schemes for spatially discretizing \(2.1\)-\(2.2\), reducing them to a system of)12 3439(There are many)2 631 2 970 1236 t 10 B f (ode)720 1356 w 10 R f ( most elegant and most acceptable to people in the physical sci-)11 2615( The)1 212('s, and thus solving the problem.)5 1343 3 870 1356 t ( method has the property that the term)7 1537( That)1 235( Galerkin's method [25,31].)3 1118(ences is)1 314 4 720 1476 t 10 B f (a \()1 91 1 3951 1476 t 10 R f (.)4050 1446 w (\) in \(2.1\) is forced to be)6 957 1 4083 1476 t ( Since)1 274(continuous everywhere, see [25].)3 1327 2 720 1596 t 10 B f (a \()1 91 1 2348 1596 t 10 R f (.)2447 1566 w ( contin-)1 309(\) is a flux, physically, and physicists expect fluxes to be)10 2251 2 2480 1596 t ( spatial discretization)2 855( Other)1 280(uous, this means that Galerkin's method gives the solution the physicists want.)11 3185 3 720 1716 t ( not automatically make)3 990(methods such as finite-differences, least-squares and collocation do)7 2764 2 720 1836 t 10 B f (a \()1 91 1 4508 1836 t 10 R f (.)4607 1806 w (\) continu-)1 400 1 4640 1836 t ( in this section are stripped down versions of real problems and the)12 2770( of the rest of the examples)6 1123(ous. Most)1 427 3 720 1956 t (use of Galerkin's method in)4 1115 1 720 2076 t 10 CW f (TTGR)1860 2076 w 10 R f (is important in their formulation.)4 1314 1 2125 2076 t 10 B f (Example 3 - Interfaces)3 962 1 720 2316 t 10 R f ( structure with three rect-)4 1012( a layered)2 387( Assume)1 372(This example shows how to deal with material interfaces.)8 2299 4 970 2472 t (angles piled one on another, each with its own material constant,)10 2705 1 720 2592 t 10 S f (k)3462 2592 w 10 R f (\()3549 2592 w 10 I f (x x)1 44 1 3590 2592 t 10 R f (,)3642 2592 w 10 I f (y y)1 44 1 3675 2592 t 10 R f ( heat flow problem, the)4 973( a)1 81(\). For)1 259 3 3727 2592 t (modeling equations might look like)4 1423 1 720 2712 t 10 B f (u)1220 2892 w 7 I f (t t)1 20 1 1287 2912 t 10 S f ( \321)1 120(= =)1 55 2 1364 2892 t 10 R f (.)1571 2862 w (\()1628 2892 w 10 S f (k)1693 2892 w 10 R f (\()1780 2892 w 10 I f (x x)1 44 1 1821 2892 t 10 R f (,)1873 2892 w 10 I f (y y)1 44 1 1906 2892 t 10 R f (\))1958 2892 w 10 S f (\321)2031 2892 w 10 B f (u)2134 2892 w 10 R f (\))2222 2892 w 10 S f (+ +)1 55 1 2303 2892 t 10 I f (g g)1 50 1 2398 2892 t 10 R f (\()2456 2892 w 10 I f (t t)1 28 1 2497 2892 t 10 R f (,)2533 2892 w 10 I f (x x)1 44 1 2566 2892 t 10 R f (,)2618 2892 w 10 I f (y y)1 44 1 2651 2892 t 10 R f (\) \(4.7\))1 2337 1 2703 2892 t ( 0 , 1 ])4 190(on the domain [)3 630 2 720 3072 t 10 S f (\264)1556 3072 w 10 R f ( where)1 268([ 0 , 3 ],)4 248 2 1619 3072 t 10 S f (k)2160 3072 w 10 R f (is piecewise constant on the different rectangles, say,)7 2120 1 2240 3072 t 10 S f (k \272)1 151 1 1220 3437 t (\354)1420 3250 w (\357)1420 3350 w (\355)1420 3450 w (\357)1420 3550 w (\356)1420 3650 w 10 R f (1)1535 3587 w 10 I f (/ /)1 28 1 1593 3587 t 10 R f (32)1629 3587 w 10 S f (< <)1 55 1 1745 3587 t 10 I f (y y)1 44 1 1816 3587 t 10 S f (\243)1868 3587 w 10 R f (3)1931 3587 w (1)1535 3447 w 10 I f (/ /)1 28 1 1593 3447 t 10 R f (21)1629 3447 w 10 S f (< <)1 55 1 1745 3447 t 10 I f (y y)1 44 1 1816 3447 t 10 S f (\243)1868 3447 w 10 R f (2)1931 3447 w (10)1535 3307 w 10 S f (\243)1643 3307 w 10 I f (y y)1 44 1 1706 3307 t 10 S f (\243)1758 3307 w 10 R f (1)1821 3307 w (and)720 3822 w 10 I f (g g)1 50 1 889 3822 t 10 R f (is chosen so that)3 658 1 964 3822 t 10 I f (u u)1 50 1 1220 4187 t 10 S f (\272)1311 4187 w (\354)1415 4000 w (\357)1415 4100 w (\355)1415 4200 w (\357)1415 4300 w (\356)1415 4400 w 10 R f (3)1530 4337 w 10 I f (t t)1 28 1 1588 4337 t 10 R f (\()1648 4337 w 10 I f (y y)1 44 1 1689 4337 t 10 S f (- -)1 55 1 1757 4337 t 10 R f (1 \) 2)2 149 1 1828 4337 t 10 S f (< <)1 55 1 1993 4337 t 10 I f (y y)1 44 1 2064 4337 t 10 S f (\243)2116 4337 w 10 R f (3)2179 4337 w 10 I f (. .)1 25 1 2237 4337 t (t t)1 28 1 1530 4197 t 10 R f (\( 2)1 91 1 1590 4197 t 10 I f (y y)1 44 1 1689 4197 t 10 S f (- -)1 55 1 1757 4197 t 10 R f (1 \) 1)2 149 1 1828 4197 t 10 S f (< <)1 55 1 1993 4197 t 10 I f (y y)1 44 1 2064 4197 t 10 S f (\243)2116 4197 w 10 R f (2)2179 4197 w 10 I f ( y)1 0(t ty)1 72 2 1530 4057 t 10 R f (0)1610 4057 w 10 S f (\243)1668 4057 w 10 I f (y y)1 44 1 1731 4057 t 10 S f (\243)1783 4057 w 10 R f (1)1846 4057 w (The)720 4572 w 10 B f (bc)900 4572 w 10 R f (s on the bottom and top are given by, say, insulation)10 2088 1 1000 4572 t 10 B f (u)1220 4752 w 7 I f (N N)1 47 1 1287 4772 t 10 S f (= =)1 55 1 1391 4752 t 10 R f (0 \(4.8a\))1 3545 1 1495 4752 t (where)720 4932 w 10 B f (u)988 4932 w 7 I f (N N)1 47 1 1055 4952 t 10 R f (is the normal derivative, and on the sides Dirichlet data)9 2208 1 1135 4932 t 10 B f (u)1220 5112 w 10 S f (= =)1 55 1 1325 5112 t 10 I f (s s)1 39 1 1429 5112 t 10 R f (\()1476 5112 w 10 I f (t t)1 28 1 1517 5112 t 10 R f (,)1553 5112 w 10 I f (x x)1 44 1 1586 5112 t 10 R f (,)1638 5112 w 10 I f (y y)1 44 1 1671 5112 t 10 R f (\) \(4.8b\))1 3317 1 1723 5112 t (is used, for some known)4 974 1 720 5292 t 10 I f (s s)1 39 1 1719 5292 t 10 R f (.)1758 5292 w (The)970 5448 w 10 B f (pde)1150 5448 w 10 R f (is equivalent to \(2.1\) with)4 1030 1 1331 5448 t 10 I f (f f)1 28 1 1220 5948 t 10 S f (= =)1 55 1 1313 5948 t 10 B f (u)1417 5948 w 7 I f (t t)1 20 1 1484 5968 t 10 S f (- -)1 55 1 1552 5948 t 10 I f ( .)1 0( .)1 33(g g)1 50 3 1647 5948 t (a a)1 50 1 1220 5788 t 7 R f (\( 2 \))2 91 1 1281 5748 t 10 S f ( k)1 104(= =)1 55 2 1437 5788 t 10 R f (\()1628 5788 w 10 I f (x x)1 44 1 1669 5788 t 10 R f (,)1721 5788 w 10 I f (y y)1 44 1 1754 5788 t 10 R f (\))1806 5788 w 10 B f (u)1879 5788 w 7 I f (y y)1 31 1 1946 5808 t 10 I f (a a)1 50 1 1220 5618 t 7 R f (\( 1 \))2 91 1 1281 5578 t 10 S f ( k)1 104(= =)1 55 2 1437 5618 t 10 R f (\()1628 5618 w 10 I f (x x)1 44 1 1669 5618 t 10 R f (,)1721 5618 w 10 I f (y y)1 44 1 1754 5618 t 10 R f (\))1806 5618 w 10 B f (u)1879 5618 w 7 I f (x x)1 31 1 1946 5638 t 10 R f (The bottom and top)3 786 1 720 6128 t 10 B f (bc)1531 6128 w 10 R f (s are equivalent to \(2.2\) with)5 1148 1 1631 6128 t 10 B f (b)1220 6308 w 10 S f (= =)1 55 1 1325 6308 t 10 I f (u u)1 50 1 1429 6308 t 7 I f (y y)1 31 1 1490 6328 t 10 R f (and those on the side are equivalent to)7 1528 1 720 6488 t 10 B f (b)1220 6668 w 10 S f (= =)1 55 1 1325 6668 t 10 B f (u)1429 6668 w 10 S f (- -)1 55 1 1534 6668 t 10 I f (s s)1 39 1 1638 6668 t 10 R f (\()1685 6668 w 10 I f (t t)1 28 1 1726 6668 t 10 R f (,)1762 6668 w 10 I f (x x)1 44 1 1795 6668 t 10 R f (,)1847 6668 w 10 I f (y y)1 44 1 1880 6668 t 10 R f (\))1932 6668 w 10 I f (. .)1 25 1 1981 6668 t 10 R f ( of this problem is that the normal component of)9 1971(The sporting aspect)2 788 2 720 6848 t 10 S f (k \321)1 158 1 3508 6848 t 10 B f (u)3698 6848 w 10 R f (, that is)2 300 1 3754 6848 t 10 S f (k)4083 6848 w 10 I f (u u)1 50 1 4170 6848 t 7 I f (y y)1 31 1 4231 6868 t 10 R f (, across each mate-)3 770 1 4270 6848 t ( fact forces this to)4 715( method in)2 430( Galerkin's)1 472(rial interface is expected to be continuous by any physical scientist.)10 2703 4 720 6968 t ( Example 3 of Appendix 4 for code solving this problem.)10 2284( See)1 194(be the case.)2 462 3 720 7088 t cleartomark showpage saveobj restore end %%EndPage: 6 7 %%Page: 7 8 DpostDict begin /saveobj save def mark 8 pagesetup 10 R f (- 7 -)2 166 1 2797 480 t 10 B f (Example 4 - A Non-Rectangular Domain.)5 1760 1 720 840 t 10 R f (Suppose you wanted to solve the)5 1313 1 970 996 t 10 B f (pde)2308 996 w (u)1220 1236 w 7 I f (t t)1 20 1 1287 1256 t 10 S f (= =)1 55 1 1364 1236 t (\266)1493 1306 w 10 I f (x x)1 44 1 1550 1306 t 10 S f (\266)1519 1176 w 10 S1 f (_ __)1 131 1 1478 1206 t 10 R f (\()1651 1236 w 10 I f (u u)1 50 1 1716 1236 t 7 I f (x x)1 31 1 1777 1256 t 10 S f (- -)1 55 1 1856 1236 t 10 R f (10)1976 1306 w 10 I f (u u)1 50 1 1976 1156 t 7 I f (y y)1 31 1 2037 1176 t 10 S1 f (_ __)1 130 1 1961 1206 t 10 R f (\))2133 1236 w 10 S f (+ +)1 55 1 2214 1236 t (\266)2334 1306 w 10 I f (y y)1 44 1 2391 1306 t 10 S f (\266)2360 1176 w 10 S1 f (_ __)1 131 1 2319 1206 t 10 R f (\()2492 1236 w 10 I f (u u)1 50 1 2557 1236 t 7 I f (y y)1 31 1 2618 1256 t 10 S f (- -)1 55 1 2697 1236 t 10 R f (10)2817 1306 w 10 I f (u u)1 50 1 2817 1156 t 7 I f (x x)1 31 1 2878 1176 t 10 S1 f (_ __)1 130 1 2802 1206 t 10 R f (\))2950 1236 w 10 S f (+ +)1 55 1 3031 1236 t 10 I f (g g)1 50 1 3126 1236 t 10 R f (\(4.9\))4849 1236 w (on the domain)2 572 1 720 1502 t (y)2021 1725 w (x)3914 3113 w (\(0,-1\))2103 4038 w 2047 3976 2972 3051 Dl 2046 3051 1 1 De 2972 2126 2972 3051 Dl 2971 2126 2046 3051 Dl 2046 3053 2046 3078 Dl 3812 3051 2046 3051 Dl 2046 3078 2046 3083 Dl 2046 3082 2046 3072 Dl 2046 3092 2046 3072 Dl 2046 3093 1 1 De 2046 3092 2046 3051 Dl 2046 3093 2046 1790 Dl (\(0,0\))1783 3113 w (\(1,1\))2961 2104 w (\(1,0\))2961 3029 w 2087 1831 2046 1790 Dl 2046 1791 2005 1832 Dl 3811 3052 3770 3093 Dl 3728 3093 1 1 De 3811 3050 3770 3009 Dl 2046 3076 2046 3051 Dl 2046 3078 1 1 De 2046 3051 1 1 De 2046 3976 2046 3051 Dl 10 B f (Figure 1)1 358 1 2826 4276 t 10 R f (where)720 4396 w 10 I f (g g)1 50 1 999 4396 t 10 R f (is chosen so that)3 691 1 1085 4396 t 10 I f (u u)1 50 1 1812 4396 t 10 S f (\272)1903 4396 w 10 I f ( y)1 0( y)1 76( x)1 0( x)1 76(t t)1 28 5 1999 4396 t 10 R f ( prescribe consistent Neumann data on the top and bottom, and)10 2636(. We)1 225 2 2179 4396 t (consistent Dirichlet data on the left and right hand sides.)9 2253 1 720 4516 t ( 0 , 1 ])4 190( it is easy to map that domain onto a rectangle, namely [)12 2311( Well,)1 275( you solve this?)3 643(How do)1 324 5 970 4672 t 10 S f (\264)4729 4672 w 10 R f ([ 0 , 1 ].)4 248 1 4792 4672 t (Just map)1 353 1 720 4792 t 10 I f (y y)1 44 1 1220 5092 t 10 S f ( -)1 0( -)1 112(= =)1 55 3 1313 5092 t 10 R f (1)1496 5092 w 10 S f ( h)1 109( +)1 0( +)1 104( x)1 65(+ +)1 55 5 1562 5092 t 10 I f (x x)1 44 1 1220 4952 t 10 S f ( x)1 98(= =)1 55 2 1313 4952 t 10 R f (\(4.10\))4799 5042 w ( equation may be solved on the new rectangular)8 2451(and the)1 358 2 720 5252 t 10 S f (x)3622 5252 w 10 R f (,)3679 5252 w 10 S f (h)3712 5252 w 10 R f ( 0 , 1 ])4 190(domain [)1 426 2 3865 5252 t 7 R f (2)4486 5212 w 10 R f ( is,)1 185(. That)1 326 2 4529 5252 t (\()720 5372 w 10 I f (x x)1 44 1 785 5372 t 10 R f (,)837 5372 w 10 I f (y y)1 44 1 894 5372 t 10 R f (\) \()1 106 1 970 5372 t 10 S f (x)1108 5372 w 10 R f (,)1165 5372 w 10 S f (h)1222 5372 w 10 R f (\))1314 5372 w 10 S f (\272)1396 5372 w 10 R f (\()1492 5372 w 10 I f (x x)1 44 1 1533 5372 t 10 R f (\()1609 5372 w 10 S f (x)1674 5372 w 10 R f (,)1731 5372 w 10 S f (h)1788 5372 w 10 R f (\) ,)1 74 1 1880 5372 t 10 I f (y y)1 44 1 1962 5372 t 10 R f (\()2038 5372 w 10 S f (x)2103 5372 w 10 R f (,)2160 5372 w 10 S f (h)2217 5372 w 10 R f ( 0 , 1 ])4 190( [)1 64( maps)1 243(\) \))1 106 4 2309 5372 t 10 S f (\264)2952 5372 w 10 R f ( Then,)1 286( into Figure 1.)3 585([ 0 , 1 ])4 223 3 3039 5372 t 10 CW f (TTGR)4164 5372 w 10 R f (is told to solve)3 605 1 4435 5372 t (for)720 5492 w 10 B f (w)1220 5672 w 10 R f (\()1300 5672 w 10 I f (t t)1 28 1 1365 5672 t 10 R f (,)1401 5672 w 10 S f (x)1458 5672 w 10 R f (,)1515 5672 w 10 S f (h)1572 5672 w 10 R f (\))1664 5672 w 10 S f (\272)1746 5672 w 10 B f (u)1842 5672 w 10 R f (\()1930 5672 w 10 I f (t t)1 28 1 1995 5672 t 10 R f (, \()1 90 1 2031 5672 t 10 I f (x x)1 44 1 2129 5672 t 10 R f (,)2181 5672 w 10 I f (y y)1 44 1 2214 5672 t 10 R f (\) \()1 74 1 2266 5672 t 10 S f (x)2372 5672 w 10 R f (,)2429 5672 w 10 S f (h)2486 5672 w 10 R f (,)2578 5672 w 10 I f (t t)1 28 1 2635 5672 t 10 R f (\) \))1 106 1 2695 5672 t ( 0 , 1 ])4 190(on the domain [)3 633 2 720 5852 t 7 R f (2)1548 5812 w 10 R f (, with the)2 379 1 1591 5852 t 10 B f (pde)1997 5852 w 10 R f (and)2180 5852 w 10 B f (bc)2351 5852 w 10 R f ( Example 4 of Appendix)4 990( See)1 196(s mapped into that domain as well.)6 1403 3 2451 5852 t (4 for code solving this problem.)5 1277 1 720 5972 t 10 B f (Example 5 - A Static Problem)5 1268 1 720 6212 t 10 R f (This example shows how to solve a static)7 1699 1 970 6368 t 10 B f (pde)2700 6368 w 10 R f (and also shows that using a non-uniform grid can be)9 2152 1 2888 6368 t ( the)1 147( Let)1 183(very useful and effective.)3 1013 3 720 6488 t 10 B f (pde)2088 6488 w 10 R f (be Laplace's equation)2 875 1 2269 6488 t 10 I f (u u)1 50 1 1220 6668 t 7 I f (x xx x)2 62 1 1281 6688 t 10 S f (+ +)1 55 1 1391 6668 t 10 I f (u u)1 50 1 1486 6668 t 7 I f (y yy y)2 62 1 1547 6688 t 10 S f (= =)1 55 1 1657 6668 t 10 R f (0 \(4.11\))1 3288 1 1752 6668 t ( 0 , 1 ])4 190(on the domain [)3 651 2 720 6848 t 10 S f (\264)1577 6848 w 10 R f ( any analytic function solves Laplace's equation, let's)7 2202( the Real part of)4 671( Since)1 279([ 0 , 1 ].)4 248 4 1640 6848 t (pick)720 6968 w 10 I f (u u)1 50 1 922 6968 t 10 R f (to be the Real part of)5 865 1 1002 6968 t 10 I f (z z)1 39 1 1897 6968 t 10 R f (log \()1 193 1 1968 6968 t 10 I f (z z)1 39 1 2193 6968 t 10 R f (\), where)1 331 1 2264 6968 t 10 I f (z z)1 39 1 2625 6968 t 10 S f (= =)1 55 1 2713 6968 t 10 I f (x x)1 44 1 2817 6968 t 10 S f (+ +)1 55 1 2901 6968 t 10 I f ( y)1 0( y)1 76(i i)1 28 3 2996 6968 t 10 R f ( choose Dirichlet)2 692(. We)1 218 2 3100 6968 t 10 B f (bc)4040 6968 w 10 R f ( right and)2 391(s on the left,)3 509 2 4140 6968 t (top of the domain, consistent with that choice for)8 2058 1 720 7088 t 10 I f (u u)1 50 1 2816 7088 t 10 R f ( the bottom we choose Neumann data)6 1581(. For)1 227 2 2866 7088 t 10 I f (u u)1 50 1 4712 7088 t 7 I f (y y)1 31 1 4773 7108 t 10 S f (= =)1 55 1 4861 7088 t 10 R f (0.)4965 7088 w (This gives)1 414 1 720 7208 t cleartomark showpage saveobj restore end %%EndPage: 7 8 %%Page: 8 9 DpostDict begin /saveobj save def mark 9 pagesetup 10 R f (- 8 -)2 166 1 2797 480 t 10 I f (u u)1 50 1 1220 1320 t 10 R f (\( 0 ,)2 124 1 1278 1320 t 10 I f (y y)1 44 1 1410 1320 t 10 R f (\))1462 1320 w 10 S f (= =)1 55 1 1552 1320 t 10 R f (2)1765 1390 w 10 S f ( p)1 71(- -)1 55 2 1689 1260 t 10 I f (y y)1 44 1 1847 1260 t 10 S1 f (_ ____)1 232 1 1674 1290 t 10 I f (u u)1 50 1 1220 1120 t 10 R f (\()1278 1120 w 10 I f (x x)1 44 1 1319 1120 t 10 R f (, 1 \))2 124 1 1371 1120 t 10 S f (= =)1 55 1 1552 1120 t 10 I f ( l)1 0( al)1 28( ea)1 50(R Re)1 105 4 1656 1120 t 10 R f (\()1847 1120 w 10 I f ( og g)2 50( lo)1 50( l)1 60(z z)1 39 4 1888 1120 t 10 R f (\()2095 1120 w 10 I f (z z)1 39 1 2136 1120 t 10 R f (\) \))1 74 1 2183 1120 t 10 I f (u u)1 50 1 1220 980 t 10 R f (\( 1 ,)2 124 1 1278 980 t 10 I f (y y)1 44 1 1410 980 t 10 R f (\))1462 980 w 10 S f (= =)1 55 1 1552 980 t 10 I f ( l)1 0( al)1 28( ea)1 50(R Re)1 105 4 1656 980 t 10 R f (\()1847 980 w 10 I f ( og g)2 50( lo)1 50( l)1 60(z z)1 39 4 1888 980 t 10 R f (\()2095 980 w 10 I f (z z)1 39 1 2136 980 t 10 R f (\) \))1 74 1 2183 980 t 10 I f (u u)1 50 1 1220 820 t 7 I f (y y)1 31 1 1281 840 t 10 R f (\()1328 820 w 10 I f (x x)1 44 1 1369 820 t 10 R f (, 0 \))2 124 1 1421 820 t 10 S f (= =)1 55 1 1602 820 t 10 R f (0)1706 820 w (\(4.12\))4799 1070 w ( are analytic, the solution of this static problem)8 1935(Note that even though the coefficients of \(4.11\) and \(4.12\))9 2385 2 720 1550 t (has singular first partials at)4 1082 1 720 1670 t 10 I f (z z)1 39 1 1827 1670 t 10 S f (= =)1 55 1 1890 1670 t 10 R f (0.)1961 1670 w (The)970 1826 w 10 B f (pde)1150 1826 w 10 R f (is equivalent to \(2.1\) with)4 1030 1 1331 1826 t 10 I f (f f)1 28 1 1220 2326 t 10 S f (= =)1 55 1 1313 2326 t 10 R f (0)1417 2326 w 10 I f (a a)1 50 1 1220 2166 t 7 R f (\( 2 \))2 91 1 1281 2126 t 10 S f (= =)1 55 1 1437 2166 t 10 B f (u)1541 2166 w 7 I f (y y)1 31 1 1608 2186 t 10 I f (a a)1 50 1 1220 1996 t 7 R f (\( 1 \))2 91 1 1281 1956 t 10 S f (= =)1 55 1 1437 1996 t 10 B f (u)1541 1996 w 7 I f (x x)1 31 1 1608 2016 t 10 R f (The boundary conditions are equivalent to \(2.2\) with)7 2108 1 720 2486 t 10 B f (b)1220 2951 w 10 S f (= =)1 55 1 1325 2951 t (\354)1437 2664 w (\357)1437 2764 w (\357)1437 2864 w (\355)1437 2964 w (\357)1437 3064 w (\357)1437 3164 w (\356)1437 3264 w 10 I f (u u)1 50 1 1486 3201 t 10 R f (\( 0 ,)2 124 1 1544 3201 t 10 I f (y y)1 44 1 1676 3201 t 10 R f (\))1728 3201 w 10 S f (+ +)1 55 1 1818 3201 t 10 R f (2)1987 3271 w 10 S f (p)1947 3141 w 10 I f (y y)1 44 1 2034 3141 t 10 S1 f (_ ___)1 161 1 1932 3171 t 10 I f (u u)1 50 1 1486 3001 t 10 R f (\()1544 3001 w 10 I f (x x)1 44 1 1585 3001 t 10 R f (, 1 \))2 124 1 1637 3001 t 10 S f (- -)1 55 1 1818 3001 t 10 I f ( l)1 0( al)1 28( ea)1 50(R Re)1 105 4 1922 3001 t 10 R f (\()2113 3001 w 10 I f ( og g)2 50( lo)1 50( l)1 60(z z)1 39 4 2154 3001 t 10 R f (\()2361 3001 w 10 I f (z z)1 39 1 2402 3001 t 10 R f (\) \))1 74 1 2449 3001 t 10 I f (u u)1 50 1 1486 2861 t 10 R f (\( 1 ,)2 124 1 1544 2861 t 10 I f (y y)1 44 1 1676 2861 t 10 R f (\))1728 2861 w 10 S f (- -)1 55 1 1818 2861 t 10 I f ( l)1 0( al)1 28( ea)1 50(R Re)1 105 4 1922 2861 t 10 R f (\()2113 2861 w 10 I f ( og g)2 50( lo)1 50( l)1 60(z z)1 39 4 2154 2861 t 10 R f (\()2361 2861 w 10 I f (z z)1 39 1 2402 2861 t 10 R f (\) \))1 74 1 2449 2861 t 10 I f (u u)1 50 1 1486 2701 t 7 I f (y y)1 31 1 1547 2721 t 10 R f (\()1594 2701 w 10 I f (x x)1 44 1 1635 2701 t 10 R f (, 0 \))2 124 1 1687 2701 t (The fact that there is no)5 945 1 720 3472 t 10 B f (u)1690 3472 w 7 I f (t t)1 20 1 1757 3492 t 10 R f (term in)1 286 1 1810 3472 t 10 I f (f f)1 28 1 2121 3472 t 10 R f (is ok, since)2 447 1 2174 3472 t 10 CW f (TTGR)2646 3472 w 10 R f (does not require it.)3 749 1 2911 3472 t ( See)1 203( will result in slow convergence unless a non-uniform grid is used.)11 2757(The logarithmic singularity)2 1110 3 970 3628 t (example 5 in Appendix 4 for code solving this problem.)9 2237 1 720 3748 t 10 B f (Example 6 - Estimating the Error in the Computed Solution.)9 2584 1 720 3988 t 10 R f ( the time evolution of a)5 950(Picking a spatial mesh and an accuracy requirement for)8 2235 2 970 4144 t 10 B f (pde)4184 4144 w 10 R f (guarantees abso-)1 671 1 4369 4144 t (lutely)720 4264 w 10 B f (nothing)977 4264 w 10 R f ( at least a little preliminary error esti-)7 1510( Without)1 381( of the computed solution.)4 1059(about the accuracy)2 755 4 1335 4264 t ( easily)1 260( error in a computed solution can be)7 1446( The)1 206(mation, the computed results may look nice and be garbage.)9 2408 4 720 4384 t (estimated however.)1 776 1 720 4504 t ( first is the error due to the)7 1069( The)1 206( of \(2.1\)-\(2.2\).)2 575(There are two sources of error in the computed solution)9 2220 4 970 4660 t ( speaking, the error in a solu-)6 1179( Crudely)1 374( second is due to the time discretization.)7 1618(spatial discretization and the)3 1149 4 720 4780 t (tion)720 4900 w 10 B f (u)901 4900 w 10 R f (is of the form)3 541 1 982 4900 t 10 I f (O O)1 72 1 1220 5080 t 10 R f (\()1324 5080 w 10 I f (h h)1 50 1 1389 5080 t 7 I f (k k)1 31 1 1450 5040 t 10 R f (\))1521 5080 w 10 S f (+ +)1 55 1 1611 5080 t 10 I f (O O)1 72 1 1715 5080 t 10 R f (\()1819 5080 w 10 I f (e e)1 44 1 1884 5080 t 7 R f (1)1939 5100 w 10 S f ( \357)1 0( \357)1 24(\357 \357)1 49 3 2006 5097 t 10 B f (u)2078 5080 w 10 S f ( \357)1 0( \357)1 24(\357 \357)1 49 3 2134 5097 t 7 S f (\245)2196 5100 w 10 S f (+ +)1 55 1 2304 5080 t 10 I f (e e)1 44 1 2408 5080 t 7 R f (2)2463 5100 w 10 R f (\) \(4.13\))1 2502 1 2538 5080 t (where)720 5260 w 10 I f (h h)1 50 1 992 5260 t 10 R f (is the spatial mesh spacing and)5 1255 1 1071 5260 t 10 I f (e e)1 44 1 2355 5260 t 7 R f (1)2410 5280 w 10 R f (and)2482 5260 w 10 I f (e e)1 44 1 2655 5260 t 7 R f (2)2710 5280 w 10 R f ( time evolution)2 616(are user settable numbers controlling the)5 1642 2 2782 5260 t ( make sure that)3 637(error. To)1 389 2 720 5380 t 10 B f (u)1781 5380 w 10 R f (is as accurate as we want, some testing of the size of the two components of)15 3168 1 1872 5380 t (\(4.13\) is in order.)3 696 1 720 5500 t (Assume that the solution)3 995 1 970 5656 t 10 B f (u)1991 5656 w 10 R f ( mesh)1 238(has been obtained on the)4 991 2 2073 5656 t 10 B f (x)3329 5656 w 10 S f (\264)3387 5656 w 10 B f (y)3450 5656 w 10 R f (using)3527 5656 w 10 B f (e)3771 5656 w 10 R f ( Then)1 257(for the error tolerances.)3 941 2 3842 5656 t (let)720 5776 w 10 B f (u)851 5776 w 7 I f (h h)1 35 1 918 5796 t 10 R f (be the solution of the same problem but on a mesh)10 2075 1 992 5776 t 10 B f (x)3098 5776 w 7 I f (h h)1 35 1 3159 5796 t 10 S f (\264)3210 5776 w 10 B f (y)3273 5776 w 7 I f (h h)1 35 1 3334 5796 t 10 R f (whose mesh spacing is half that of)6 1406 1 3408 5776 t 10 B f (x)4844 5776 w 10 S f (\264)4902 5776 w 10 B f (y)4965 5776 w 10 R f (.)5015 5776 w (Also, let)1 339 1 720 5896 t 10 B f (u)1084 5896 w 7 I f (e e)1 31 1 1151 5916 t 10 R f ( the same problem on the mesh)6 1249(be the solution of)3 697 2 1215 5896 t 10 B f (x)3187 5896 w 10 S f (\264)3245 5896 w 10 B f (y)3308 5896 w 10 R f (but using)1 371 1 3384 5896 t 10 B f (e)3781 5896 w 10 I f (/ /)1 28 1 3833 5896 t 10 R f ( We)1 189(10 for the error controls.)4 982 2 3869 5896 t (can then estimate the)3 840 1 720 6016 t 10 I f (O O)1 72 1 1585 6016 t 10 R f (\()1665 6016 w 10 I f (h h)1 50 1 1730 6016 t 7 I f (k k)1 31 1 1791 5976 t 10 R f (\) error term in \(4.13\) as)5 936 1 1862 6016 t 10 S f ( \357)1 0( \357)1 24(\357 \357)1 49 3 1220 6213 t 10 B f (u)1316 6196 w 10 S f (- -)1 55 1 1421 6196 t 10 B f (u)1525 6196 w 7 I f (h h)1 35 1 1592 6216 t 10 S f ( \357)1 0( \357)1 24(\357 \357)1 49 3 1659 6213 t 7 S f (\245)1721 6216 w 10 R f (,)1788 6196 w (because)720 6376 w 10 B f (u)1060 6376 w 7 I f (h h)1 35 1 1127 6396 t 10 R f (is so much more accurate than)5 1211 1 1195 6376 t 10 B f (u)2431 6376 w 10 R f (that it can be regarded as the true solution of the problem.)11 2307 1 2512 6376 t (Similarly, we estimate the)3 1044 1 970 6532 t 10 I f (O O)1 72 1 2039 6532 t 10 R f (\()2119 6532 w 10 I f (e e)1 44 1 2184 6532 t 7 R f (1)2239 6552 w 10 S f ( \357)1 0( \357)1 24(\357 \357)1 49 3 2282 6549 t 10 B f (u)2378 6532 w 10 S f ( \357)1 0( \357)1 24(\357 \357)1 49 3 2458 6549 t 7 S f (\245)2520 6552 w 10 S f (+ +)1 55 1 2628 6532 t 10 I f (e e)1 44 1 2732 6532 t 7 R f (2)2787 6552 w 10 R f (\) term in \(4.13\) as)4 718 1 2862 6532 t 10 S f ( \357)1 0( \357)1 24(\357 \357)1 49 3 1220 6729 t 10 B f (u)1316 6712 w 10 S f (- -)1 55 1 1421 6712 t 10 B f (u)1525 6712 w 7 I f (e e)1 31 1 1592 6732 t 10 S f ( \357)1 0( \357)1 24(\357 \357)1 49 3 1655 6729 t 7 S f (\245)1717 6732 w 10 R f (,)1784 6712 w (because, again,)1 609 1 720 6892 t 10 B f (u)1357 6892 w 7 I f (e e)1 31 1 1424 6912 t 10 R f (is so much more accurate than)5 1226 1 1491 6892 t 10 B f (u)2746 6892 w 10 R f (that it can be regarded as the true solution of the prob-)11 2209 1 2831 6892 t (lem.)720 7012 w (We can assemble the above two estimates for the terms in \(4.13\) to obtain)13 2956 1 970 7168 t cleartomark showpage saveobj restore end %%EndPage: 8 9 %%Page: 9 10 DpostDict begin /saveobj save def mark 10 pagesetup 10 R f (- 9 -)2 166 1 2797 480 t 10 S f ( \357)1 0( \357)1 24(\357 \357)1 49 3 1220 857 t 10 B f (u)1316 840 w 10 S f (- -)1 55 1 1421 840 t 10 B f (u)1525 840 w 7 I f ( e)1 0( ue)1 31( ru)1 35(t tr)1 47 4 1592 860 t 10 S f ( \357)1 0( \357)1 24(\357 \357)1 49 3 1737 857 t 7 S f (\245)1799 860 w 10 S f (\243)1899 840 w ( \357)1 0( \357)1 24(\357 \357)1 49 3 1987 857 t 10 B f (u)2083 840 w 10 S f (- -)1 55 1 2188 840 t 10 B f (u)2292 840 w 7 I f (h h)1 35 1 2359 860 t 10 S f ( \357)1 0( \357)1 24(\357 \357)1 49 3 2426 857 t 7 S f (\245)2488 860 w 10 S f (+ +)1 55 1 2596 840 t ( \357)1 0( \357)1 24(\357 \357)1 49 3 2692 857 t 10 B f (u)2788 840 w 10 S f (- -)1 55 1 2893 840 t 10 B f (u)2997 840 w 7 I f (e e)1 31 1 3064 860 t 10 S f ( \357)1 0( \357)1 24(\357 \357)1 49 3 3127 857 t 7 S f (\245)3189 860 w 10 R f (\(4.14\))4799 840 w (The estimate \(4.14\) is easily computable in practice since everything on the right hand side is known.)16 4050 1 720 1020 t ( second term is a little trickier to estimate, but)9 1850( The)1 207( see Appendix 4.)3 677(The first term is easily evaluated,)5 1336 4 970 1176 t ( have, using \(3.1\),)3 721( We)1 188(it has a pretty result.)4 813 3 720 1296 t 10 S f ( \357)1 0( \357)1 24(\357 \357)1 49 3 1220 1493 t 10 B f (u)1316 1476 w 10 S f (- -)1 55 1 1421 1476 t 10 B f (u)1525 1476 w 7 I f (e e)1 31 1 1592 1496 t 10 S f ( \357)1 0( \357)1 24(\357 \357)1 49 3 1655 1493 t 7 S f (\245)1717 1496 w 10 S f (\272)1817 1476 w ( \357)1 0( \357)1 24(\357 \357)1 49 3 1905 1493 t 7 I f (q qp p)2 70 1 2026 1576 t 15 S f (S)2017 1506 w 10 R f (\()2154 1476 w 10 I f (U U)1 72 1 2219 1476 t 7 I f (q qp p)2 70 1 2302 1496 t 10 S f (- -)1 55 1 2429 1476 t 10 I f (U U)1 72 1 2533 1476 t 7 I f ( qp p)2 35(e eq)1 66 2 2616 1496 t 10 R f (\))2757 1476 w 10 I f (B B)1 61 1 2830 1476 t 7 I f (p p)1 35 1 2902 1496 t 10 I f (C C)1 67 1 2977 1476 t 7 I f (q q)1 35 1 3055 1496 t 10 S f ( \357)1 0( \357)1 24(\357 \357)1 49 3 3122 1493 t 7 S f (\245)3184 1496 w 10 S f (\243)3284 1476 w 7 I f (q qp p)2 70 1 3405 1576 t 15 S f (S)3396 1506 w 10 S f (\357 \357)1 49 1 3534 1493 t 10 I f (U U)1 72 1 3606 1476 t 7 I f (q qp p)2 70 1 3689 1496 t 10 S f (- -)1 55 1 3816 1476 t 10 I f (U U)1 72 1 3920 1476 t 7 I f ( qp p)2 35(e eq)1 66 2 4003 1496 t 10 S f (\357 \357)1 49 1 4136 1493 t 10 I f (B B)1 61 1 4208 1476 t 7 I f (p p)1 35 1 4280 1496 t 10 I f (C C)1 67 1 4355 1476 t 7 I f (q q)1 35 1 4433 1496 t 10 S f (\243)4517 1476 w 7 I f (q qp p)2 70 1 1245 1836 t 15 S f (S)1236 1766 w 10 S f ( \357)1 0( \357)1 24(\357 \357)1 49 3 1374 1753 t 10 B f (U)1470 1736 w 10 S f (- -)1 55 1 1591 1736 t 10 B f (U)1695 1736 w 7 I f (e e)1 31 1 1778 1756 t 10 S f ( \357)1 0( \357)1 24(\357 \357)1 49 3 1841 1753 t 7 S f (\245)1903 1756 w 10 I f (B B)1 61 1 1994 1736 t 7 I f (p p)1 35 1 2066 1756 t 10 I f (C C)1 67 1 2141 1736 t 7 I f (q q)1 35 1 2219 1756 t 10 S f (= =)1 55 1 2311 1736 t ( \357)1 0( \357)1 24(\357 \357)1 49 3 2407 1753 t 10 B f (U)2503 1736 w 10 S f (- -)1 55 1 2624 1736 t 10 B f (U)2728 1736 w 7 I f (e e)1 31 1 2811 1756 t 10 S f ( \357)1 0( \357)1 24(\357 \357)1 49 3 2874 1753 t 7 S f (\245)2936 1756 w 7 I f (q qp p)2 70 1 3052 1836 t 15 S f (S)3043 1766 w 10 I f (B B)1 61 1 3189 1736 t 7 I f (p p)1 35 1 3261 1756 t 10 I f (C C)1 67 1 3336 1736 t 7 I f (q q)1 35 1 3414 1756 t 10 S f (= =)1 55 1 3506 1736 t ( \357)1 0( \357)1 24(\357 \357)1 49 3 3602 1753 t 10 B f (U)3698 1736 w 10 S f (- -)1 55 1 3819 1736 t 10 B f (U)3923 1736 w 7 I f (e e)1 31 1 4006 1756 t 10 S f ( \357)1 0( \357)1 24(\357 \357)1 49 3 4069 1753 t 7 S f (\245)4131 1756 w 10 I f (. .)1 25 1 4198 1736 t 10 R f (This gives the estimate)3 919 1 720 1996 t 10 S f ( \357)1 0( \357)1 24(\357 \357)1 49 3 1220 2193 t 10 B f (u)1316 2176 w 10 S f (- -)1 55 1 1421 2176 t 10 B f (u)1525 2176 w 7 I f (e e)1 31 1 1592 2196 t 10 S f ( \357)1 0( \357)1 24(\357 \357)1 49 3 1655 2193 t 7 S f (\245)1717 2196 w 10 S f (\243)1817 2176 w ( \357)1 0( \357)1 24(\357 \357)1 49 3 1905 2193 t 10 B f (U)2001 2176 w 10 S f (- -)1 55 1 2122 2176 t 10 B f (U)2226 2176 w 7 I f (e e)1 31 1 2309 2196 t 10 S f ( \357)1 0( \357)1 24(\357 \357)1 49 3 2372 2193 t 7 S f (\245)2434 2196 w 10 I f (. .)1 25 1 2501 2176 t 10 R f (\(4.15\))4799 2176 w (See example 6 in Appendix 4 for code implementing the above estimation scheme.)12 3315 1 720 2356 t 10 B f ( for the pde-bc Problem.)4 1040(5. Software)1 507 2 720 2596 t 10 R f ( called)1 266(This section is a brief user's manual for a software package)10 2391 2 970 2752 t 10 CW f (TTGR)3655 2752 w 10 R f (, \(Transient Tensor Galerkin)3 1145 1 3895 2752 t (method for)1 441 1 720 2872 t 10 B f (pde)1186 2872 w 10 R f (s on Rectangles\), implementing the algorithm outlined in section 3.)9 2691 1 1342 2872 t (Before invoking)1 652 1 970 3028 t 10 CW f (TTGR)1647 3028 w 10 R f (the user must)2 533 1 1912 3028 t 10 S f (\267)970 3184 w 10 R f (Make B-spline meshes for)3 1051 1 1041 3184 t 10 I f (x x)1 44 1 2117 3184 t 10 R f (and)2186 3184 w 10 I f (y y)1 44 1 2355 3184 t 10 R f (.)2399 3184 w 10 S f (\267)970 3340 w 10 R f (Make initial conditions for the B-spline coefficients)6 2070 1 1041 3340 t 10 B f (U)3136 3340 w 10 R f (in \(3.1\).)1 319 1 3233 3340 t 10 S f (\267)970 3496 w 10 R f (Write subroutines)1 713 1 1041 3496 t 10 S f (\267)1220 3652 w 10 CW f (AF)1326 3652 w 10 R f (- to evaluate)2 493 1 1471 3652 t 10 B f (a)1989 3652 w 10 R f (and)2064 3652 w 10 B f (f)2233 3652 w 10 R f (in \(2.1\).)1 319 1 2291 3652 t 10 S f (\267)1220 3808 w 10 CW f (BC)1326 3808 w 10 R f (- to evaluate)2 493 1 1471 3808 t 10 B f (b)1989 3808 w 10 R f (in \(2.2\).)1 319 1 2070 3808 t 10 S f (\267)1220 3964 w 10 CW f (HANDLE)1326 3964 w 10 R f (- to output \(print\) the solution results.)6 1503 1 1711 3964 t (Each of these preparatory steps are illustrated in Appendix 4 and will not be described here.)15 3664 1 720 4120 t (The outer layer of the)4 864 1 970 4276 t 10 CW f (TTGR)1859 4276 w 10 R f (package is called)2 681 1 2124 4276 t 10 CW f (TTGR)2830 4276 w 10 R f (and is invoked by)3 708 1 3095 4276 t 10 CW f (Call TTGR \(U,nu,kx,X,nx, ky,Y,ny,)3 1980 1 1500 4456 t (tstart,tstop,dt,)2160 4576 w (AF,BC,)2160 4696 w (errpar,)2160 4816 w (HANDLE\))2160 4936 w 10 R f (The input to)2 489 1 720 5116 t 10 CW f (TTGR)1234 5116 w 10 R f (is)1499 5116 w 10 CW f (U)1020 5308 w 10 R f ( initial values of the)4 926( B-spline coefficients \(3.1\) for the)5 1518(- The)1 305 3 1520 5308 t 10 B f (pde)4327 5308 w 10 R f (variables)4541 5308 w 10 B f (u)4959 5308 w 10 R f (.)5015 5308 w 10 CW f (U)1670 5428 w 10 R f (\()1730 5428 w 10 I f (i i)1 28 1 1763 5428 t 10 R f (,)1799 5428 w 10 I f (j j)1 28 1 1840 5428 t 10 R f (,)1876 5428 w 10 I f (l l)1 28 1 1909 5428 t 10 R f (\), for)1 232 1 1937 5428 t 10 I f (i i)1 28 1 2226 5428 t 10 S f (= =)1 55 1 2278 5428 t 10 R f (1 ,)1 83 1 2349 5428 t (. . .)2 125 1 2465 5403 t (,)2623 5428 w 10 CW f (nx-kx)2672 5428 w 10 R f (,)2972 5428 w 10 I f (j j)1 28 1 3054 5428 t 10 S f (= =)1 55 1 3098 5428 t 10 R f (1 ,)1 83 1 3169 5428 t (. . .)2 125 1 3285 5403 t (,)3443 5428 w 10 CW f (ny-ky)3492 5428 w 10 R f (, are the coefficients for)4 1077 1 3792 5428 t 10 I f (u u)1 50 1 4926 5428 t 7 I f (l l)1 20 1 4987 5448 t 10 R f (,)5015 5428 w 10 I f (l l)1 28 1 1670 5548 t 10 S f (= =)1 55 1 1722 5548 t 10 R f (1 ,)1 83 1 1793 5548 t (. . .)2 125 1 1909 5523 t (,)2067 5548 w 10 CW f (nu)2116 5548 w 10 R f ( below for ways to get the initial conditions,)8 1763(. See)1 219 2 2236 5548 t 10 CW f (TSL2W)4243 5548 w 10 R f (.)4543 5548 w 10 CW f (nu)1020 5704 w 10 R f ( number)1 330(- The)1 305 2 1520 5704 t 10 I f (n n)1 50 1 2180 5704 t 7 I f (u u)1 35 1 2241 5724 t 10 R f (of)2309 5704 w 10 B f (pde)2417 5704 w 10 R f (variables)2598 5704 w 10 B f (u)2983 5704 w 10 R f (.)3039 5704 w 10 CW f (kx)1020 5860 w 10 R f ( B-spline order to be used in)6 1132(- The)1 305 2 1520 5860 t 10 I f (x x)1 44 1 2982 5860 t 10 R f (.)3026 5860 w 10 CW f (kx)3171 5860 w 10 S f (\263)3316 5860 w 10 R f (2 is necessary.)2 579 1 3396 5860 t 10 CW f (X)1020 6016 w 10 R f ( B-spline mesh to be used in)6 1139(- The)1 305 2 1520 6016 t 10 I f (x x)1 44 1 2990 6016 t 10 R f (. The multiplicity of)3 809 1 3034 6016 t 10 CW f (X)3905 6016 w 10 R f (\(1\) and)1 287 1 3965 6016 t 10 CW f (X)4279 6016 w 10 R f (\()4339 6016 w 10 CW f (nx)4372 6016 w 10 R f (\) must be)2 376 1 4492 6016 t 10 CW f (kx)4895 6016 w 10 R f (,)5015 6016 w ( routines for making uniform meshes,)5 1554( Port Library)2 538( The)1 216(see Appendix 1.)2 668 4 1670 6136 t 10 CW f (UMB)4681 6136 w 10 R f (and)4896 6136 w 10 CW f (LUMB)1670 6256 w 10 R f (, guarantee the first and last mesh points have multiplicity)9 2315 1 1910 6256 t 10 CW f (kx)4250 6256 w 10 R f (.)4370 6256 w 10 CW f (nx)1020 6412 w 10 R f ( length of the mesh array)5 995(- The)1 305 2 1520 6412 t 10 CW f (X)2845 6412 w 10 R f (.)2905 6412 w 10 CW f (ky)1020 6568 w 10 R f ( B-spline order to be used in)6 1132(- The)1 305 2 1520 6568 t 10 I f (y y)1 44 1 2982 6568 t 10 R f (.)3026 6568 w 10 CW f (ky)3171 6568 w 10 S f (\263)3316 6568 w 10 R f (2 is necessary.)2 579 1 3396 6568 t 10 CW f (Y)1020 6724 w 10 R f ( B-spline mesh to be used in)6 1139(- The)1 305 2 1520 6724 t 10 I f (y y)1 44 1 2990 6724 t 10 R f (. The multiplicity of)3 809 1 3034 6724 t 10 CW f (Y)3905 6724 w 10 R f (\(1\) and)1 287 1 3965 6724 t 10 CW f (Y)4279 6724 w 10 R f (\()4339 6724 w 10 CW f (ny)4372 6724 w 10 R f (\) must be)2 376 1 4492 6724 t 10 CW f (ky)4895 6724 w 10 R f (,)5015 6724 w ( routines for making uniform meshes,)5 1554( Port Library)2 538( The)1 216(see Appendix 1.)2 668 4 1670 6844 t 10 CW f (UMB)4681 6844 w 10 R f (and)4896 6844 w 10 CW f (LUMB)1670 6964 w 10 R f (, guarantee the first and last mesh points have multiplicity)9 2315 1 1910 6964 t 10 CW f (ky)4250 6964 w 10 R f (.)4370 6964 w cleartomark showpage saveobj restore end %%EndPage: 9 10 %%Page: 11 11 DpostDict begin /saveobj save def mark 11 pagesetup 10 R f (- 11 -)2 216 1 2772 480 t 10 CW f (ny)1020 840 w 10 R f ( length of the mesh array)5 995(- The)1 305 2 1520 840 t 10 CW f (Y)2845 840 w 10 R f (.)2905 840 w 10 CW f (tstart)1020 996 w 10 R f ( integration at time)3 758(- Start)1 339 2 1520 996 t 10 CW f (tstart)2642 996 w 10 R f (.)3002 996 w 10 CW f (tstop)1020 1152 w 10 R f ( integration at time)3 779(- Stop)1 334 2 1520 1152 t 10 CW f (tstop)2665 1152 w 10 R f (.)2965 1152 w 10 CW f (tstop)3117 1152 w 10 R f (should be a variable,)3 850 1 3450 1152 t 10 I f ( t)1 0(n no ot)2 128 2 4333 1152 t 10 R f (a constant, in)2 546 1 4494 1152 t (the program calling)2 782 1 1670 1272 t 10 CW f (TTGR)2477 1272 w 10 R f (; see output description below.)4 1224 1 2717 1272 t 10 CW f (dt)1020 1428 w 10 R f ( performance of)2 636( The)1 205( initial choice for the time-step.)5 1254(- The)1 305 4 1520 1428 t 10 CW f (TTGR)3945 1428 w 10 R f (is substantially inde-)2 830 1 4210 1428 t (pendent of the initial value of)5 1209 1 1670 1548 t 10 CW f (dt)2910 1548 w 10 R f ( is sufficient that)3 687(chosen. It)1 419 2 3061 1548 t 10 CW f (dt)4198 1548 w 10 R f (be within several)2 692 1 4348 1548 t ( value of)2 389( The)1 225(orders of magnitude of being "correct.")5 1662 3 1670 1668 t 10 CW f (dt)3991 1668 w 10 R f (will automatically be)2 884 1 4156 1668 t (adjusted by)1 464 1 1670 1788 t 10 CW f (TTGR)2200 1788 w 10 R f ( least possi-)2 482(to obtain the solution to the desired accuracy at the)9 2087 2 2471 1788 t ( Thus,)1 300(ble cost.)1 358 2 1670 1908 t 10 CW f (dt)2378 1908 w 10 R f (should be a variable,)3 901 1 2548 1908 t 10 I f ( t)1 0(n no ot)2 128 2 3499 1908 t 10 R f ( calling)1 323(a constant, in the user's)4 1040 2 3677 1908 t (program.)1670 2028 w 10 CW f (AF)1020 2184 w 10 R f ( subroutine for specifying the)4 1260(- A)1 222 2 1520 2184 t 10 B f (a)3048 2184 w 10 R f (and)3144 2184 w 10 B f (f)3334 2184 w 10 R f (terms in the)2 516 1 3413 2184 t 10 B f (pde)3976 2184 w 10 R f (\(2.1\).)4179 2184 w 10 CW f (AF)4537 2184 w 10 R f (must be)1 336 1 4704 2184 t ( subprogram)1 521( user-supplied)1 583( This)1 248(declared External in the user's calling program.)6 2018 4 1670 2304 t (will be described later.)3 909 1 1670 2424 t 10 CW f (BC)1020 2580 w 10 R f ( specifying the boundary conditions)4 1524( subroutine for)2 632(- A)1 222 3 1520 2580 t 10 B f (b)3946 2580 w 10 R f (in \(2.2\).)1 342 1 4050 2580 t 10 CW f (BC)4535 2580 w 10 R f (must be)1 337 1 4703 2580 t ( subprogram)1 521( user-supplied)1 583( This)1 248(declared External in the user's calling program.)6 2018 4 1670 2700 t ( there are no)3 495( If)1 116(will be described later.)3 909 3 1670 2820 t 10 B f (bc)3216 2820 w 10 R f (s then the dummy subroutine)4 1165 1 3316 2820 t 10 CW f (TTGRP)4542 2820 w 10 R f (may)4868 2820 w (be used in place of)4 748 1 1670 2940 t 10 CW f (BC)2443 2940 w 10 R f (.)2563 2940 w 10 CW f (errpar)1020 3096 w 10 R f ( in the)2 272( Real vector of length 2 for determining the error desired \(to be allowed\))13 3026(- A)1 222 3 1520 3096 t ( two components govern, roughly, the relative)6 1842( The)1 206(solution of the equations in time.)5 1322 3 1670 3216 t ( the)1 148( For)1 190(and absolute error in the computed solution.)6 1768 3 1670 3336 t 10 I f (i i)1 28 1 3802 3336 t 7 I f ( h)1 0(t th)1 55 2 3841 3296 t 10 R f (component of the)2 701 1 3930 3336 t 10 B f (pde)4657 3336 w 10 R f (solu-)4840 3336 w (tion)1670 3456 w 10 B f (u)1851 3456 w 10 R f (, the error at each time-step in the time integration will be at most)13 2619 1 1907 3456 t 10 CW f (errpar)1920 3612 w 10 R f (\( 1 \))2 132 1 2288 3612 t 10 I f (* *)1 50 1 2436 3612 t 10 S f ( \357)1 0( \357)1 24(\357 \357)1 49 3 2486 3629 t 10 I f (u u)1 50 1 2558 3612 t 7 I f (i i)1 20 1 2619 3632 t 10 S f ( \357)1 0( \357)1 24(\357 \357)1 49 3 2647 3629 t 7 S f (\245)2709 3632 w 10 S f (+ +)1 55 1 2808 3612 t 10 CW f (errpar)2903 3612 w 10 R f (\( 2 \))2 132 1 3271 3612 t (Thus,)1670 3768 w 10 CW f (errpar)1921 3768 w 10 R f ( solution accurate to an absolute error of)7 1624(\(1\)=0 gives the)2 607 2 2281 3768 t 10 CW f (errpar)4539 3768 w 10 R f (\(2\),)4899 3768 w (and)1670 3888 w 10 CW f (errpar)1888 3888 w 10 R f ( a relative error of)4 771(\(2\)=0 gives the solution accurate to)5 1482 2 2248 3888 t 10 CW f (errpar)4539 3888 w 10 R f (\(1\).)4899 3888 w (The choice of)2 560 1 1670 4008 t 10 CW f (errpar)2261 4008 w 10 R f (\(1\) and)1 292 1 2621 4008 t 10 CW f (errpar)2945 4008 w 10 R f ( sound)1 271( A)1 129(\(2\) is highly problem dependent.)4 1335 3 3305 4008 t ( the scale of the problem is such that)8 1499(technique is the following: If)4 1184 2 1670 4128 t 10 I f (S S)1 50 1 4383 4128 t 10 R f (is the smallest)2 577 1 4463 4128 t (value for which a prescribed relative error tolerance is desired, then the choice)12 3126 1 1670 4248 t 10 CW f (errpar)2280 4428 w 10 R f (\(1\))2640 4428 w 10 S f (= =)1 55 1 2781 4428 t 10 R f (10)2885 4428 w 7 S f (- -)1 39 1 2996 4388 t 7 R f (2)3046 4388 w 10 R f (;)3139 4428 w 10 CW f (errpar)3392 4428 w 10 R f (\(2\))3752 4428 w 10 S f (= =)1 55 1 3893 4428 t 10 R f (10)3997 4428 w 7 S f (- -)1 39 1 4108 4388 t 7 R f (2)4158 4388 w 10 S f (\264)4209 4428 w 10 R f (S)4289 4428 w ( values down to around S in size.)7 1405(will essentially give 1% relative accuracy in all)7 1965 2 1670 4608 t ( will have absolute error smaller than 10)7 1702(Values below S)2 655 2 1670 4728 t 7 S f (- -)1 39 1 4038 4688 t 7 R f (2)4088 4688 w 10 S f (\264)4139 4728 w 10 R f ( should be)2 437(S. Users)1 371 2 4232 4728 t (very careful to avoid setting)4 1132 1 1670 4848 t 10 CW f (errpar)2830 4848 w 10 R f (\(2\))3190 4848 w 10 S f (= =)1 55 1 3334 4848 t 10 R f ( the solution is zero at any point)7 1308(0 when)1 294 2 3438 4848 t (in either space or time, or the integration will die for the obvious reasons.)13 2932 1 1670 4968 t 10 CW f (HANDLE)1020 5124 w 10 R f ( be called by)3 510( user-supplied subroutine that will)4 1366(- A)1 222 3 1520 5124 t 10 CW f (TTGR)3644 5124 w 10 R f (at the end of each time-step.)5 1130 1 3910 5124 t 10 CW f (HANDLE)1670 5244 w 10 R f ( subprogram)1 503( This)1 229( calling program.)2 687(must be declared External in the user's)6 1564 4 2057 5244 t ( no output is desired, then the dummy subroutine)8 1974( If)1 118(will be described later.)3 915 3 1670 5364 t 10 CW f (TTGRH)4740 5364 w 10 R f (may be used in place of)5 945 1 1670 5484 t 10 CW f (HANDLE)2640 5484 w 10 R f (.)3000 5484 w (The output from)2 655 1 720 5640 t 10 CW f (TTGR)1400 5640 w 10 R f (is)1665 5640 w 10 CW f (U)1020 5832 w 10 R f ( B-spline coefficients for the)4 1142(- The)1 305 2 1370 5832 t 10 B f (pde)2842 5832 w 10 R f (solution)3023 5832 w 10 B f (u)3371 5832 w 10 R f (at time)1 275 1 3452 5832 t 10 CW f (tstop)3752 5832 w 10 R f (.)4052 5832 w 10 CW f (tstop)1020 5988 w 10 R f ( time at which integration stopped.)5 1433(- The)1 305 2 1370 5988 t 10 CW f (tstop)3237 5988 w 10 R f ( the user-supplied)2 730(may be altered by)3 739 2 3571 5988 t (subroutine)1520 6108 w 10 CW f (HANDLE)1972 6108 w 10 R f ( exists on return \(see Appendix 5\),)6 1402( an error state)3 560(. If)1 146 3 2332 6108 t 10 CW f (tstop)4504 6108 w 10 R f (is set)1 207 1 4833 6108 t ( Thus,)1 295(to the last instant in time when the solution was known accurately.)11 2879 2 1520 6228 t 10 CW f (tstop)4740 6228 w 10 R f (should be a variable,)3 826 1 1520 6348 t 10 I f ( t)1 0(n no ot)2 128 2 2371 6348 t 10 R f (a constant, in the user's call to)6 1212 1 2524 6348 t 10 CW f (TTGR)3761 6348 w 10 R f (.)4001 6348 w 10 CW f (dt)1020 6504 w 10 R f ( final value of the "optimal" time-step.)6 1539(- The)1 305 2 1370 6504 t 10 B f (Static Problems)1 674 1 720 6744 t 10 R f (For static problems, where)3 1065 1 970 6900 t 10 I f (t t)1 28 1 2060 6900 t 10 R f (,)2088 6900 w 10 I f (u u)1 50 1 2138 6900 t 7 I f (t t)1 20 1 2199 6920 t 10 R f (,)2227 6900 w 10 I f (u u)1 50 1 2277 6900 t 7 I f ( t)1 0(x xt)1 51 2 2338 6920 t 10 R f (and)2422 6900 w 10 I f (u u)1 50 1 2591 6900 t 7 I f ( t)1 0(y yt)1 51 2 2652 6920 t 10 R f (do not appear in the)4 793 1 2736 6900 t 10 B f (pde)3554 6900 w 10 R f (,)3710 6900 w 10 CW f (tstart)3795 6900 w 10 R f (,)4155 6900 w 10 CW f (tstop)4205 6900 w 10 R f (and)4530 6900 w 10 CW f (dt)4699 6900 w 10 R f (must)4845 6900 w ( example,)1 388( For)1 189(be chosen consistently but they may be otherwise arbitrary.)8 2371 3 720 7020 t cleartomark showpage saveobj restore end %%EndPage: 11 11 %%Page: 12 12 DpostDict begin /saveobj save def mark 12 pagesetup 10 R f (- 12 -)2 216 1 2772 480 t 10 CW f (tstart = 0)2 600 1 1080 900 t ( 1)1 120(tstop =)1 480 2 1080 1020 t ( tstop)1 360(dt =)1 480 2 1080 1140 t 10 R f (is a fine choice.)3 626 1 720 1320 t (Choosing the initial)2 790 1 970 1476 t 10 CW f (dt)1785 1476 w 10 R f (to go less than all the way to the final time will waste run-time by solving the)16 3109 1 1931 1476 t ( it again and again until the final time is reached.)10 1970(static problem once on the first time-step and then solving)9 2350 2 720 1596 t 10 CW f (TTGR)720 1716 w 10 R f ( wasteful to solve a problem more)6 1377(raises the time-step rapidly when solving a static problem, but it is)11 2676 2 987 1716 t (than once.)1 410 1 720 1836 t (A "restart" in)2 549 1 970 1992 t 10 CW f (HANDLE)1553 1992 w 10 R f (, see below, for a static problem is a disaster)9 1856 1 1913 1992 t 10 S f (-)3804 1992 w 10 R f (the user should)2 625 1 3894 1992 t 10 B f (STOP)4554 1992 w 10 R f (right)4851 1992 w (there.)720 2112 w 10 CW f (TTGR)1067 2112 w 10 R f (thinks that by lowering)3 934 1 1335 2112 t 10 CW f (dt)2297 2112 w 10 R f ( make the problem easier, a correct assumption if the prob-)10 2373(it can)1 222 2 2445 2112 t ( difficulty is that)3 709( The)1 220( static problem changes nothing.)4 1356(lem is transient, but lowering the time-step for a)8 2035 4 720 2232 t (Newton's method cannot converge from the initial conditions, or the initial guess in this case.)14 3741 1 720 2352 t 10 B f (Scratch Space Used.)2 863 1 720 2592 t 10 R f ( neglecting lower order)3 946(The amount of scratch space used on the dynamic stack of the Port Library [14] is,)15 3374 2 720 2748 t (terms, when the default setting are used,)6 1608 1 720 2868 t 10 I f (n n)1 50 1 970 3024 t 7 I f (u u)1 35 1 1031 3044 t 10 I f (n n)1 50 1 1106 3024 t 7 I f (x x)1 31 1 1167 3044 t 10 I f (n n)1 50 1 1238 3024 t 7 I f (y y)1 31 1 1299 3044 t 10 R f (\( 3)1 115 1 1370 3024 t 10 I f (H H)1 72 1 1517 3024 t 10 S f (- -)1 55 1 1629 3024 t 10 R f (1 \))1 91 1 1724 3024 t (Real words \(storage units\), where)4 1420 1 720 3180 t 10 I f (H H)1 72 1 2184 3180 t 10 S f (\272)2297 3180 w 10 I f (n n)1 50 1 2393 3180 t 7 I f (u u)1 35 1 2454 3200 t 10 R f (\()2529 3180 w 10 I f (k k)1 44 1 2594 3180 t 7 I f (x x)1 31 1 2649 3200 t 10 S f (+ +)1 55 1 2728 3180 t 10 R f (\()2823 3180 w 10 I f (n n)1 50 1 2888 3180 t 7 I f (x x)1 31 1 2949 3200 t 10 S f (- -)1 55 1 3028 3180 t 10 I f (k k)1 44 1 3123 3180 t 7 I f (x x)1 31 1 3178 3200 t 10 R f (\) \()1 106 1 3249 3180 t 10 I f (k k)1 44 1 3363 3180 t 7 I f (y y)1 31 1 3418 3200 t 10 S f (- -)1 55 1 3497 3180 t 10 R f ( half-band-width of the)3 983( is the)2 277( \))1 73(1 \))1 115 4 3592 3180 t (Jacobian,)720 3300 w 10 I f (n n)1 50 1 1124 3300 t 7 I f (u u)1 35 1 1185 3320 t 10 R f (is the number of)3 667 1 1258 3300 t 10 B f (pde)1955 3300 w 10 R f (s,)2111 3300 w 10 I f (n n)1 50 1 2204 3300 t 7 I f (x x)1 31 1 2265 3320 t 10 R f (is the number of points in the spatial mesh for)9 1871 1 2333 3300 t 10 I f (x x)1 44 1 4233 3300 t 10 R f (,)4277 3300 w 10 I f (k k)1 44 1 4331 3300 t 7 I f (x x)1 31 1 4386 3320 t 10 R f (is the B-spline)2 586 1 4454 3300 t (order for the mesh)3 737 1 720 3420 t 10 I f (x x)1 44 1 1483 3420 t 10 R f (,)1527 3420 w 10 I f (n n)1 50 1 1578 3420 t 7 I f (y y)1 31 1 1639 3440 t 10 R f (is the number of points in the spatial mesh for)9 1844 1 1704 3420 t 10 I f (y y)1 44 1 3574 3420 t 10 R f (and)3644 3420 w 10 I f (k k)1 44 1 3814 3420 t 7 I f (y y)1 31 1 3869 3440 t 10 R f (is the B-spline order for the)5 1106 1 3934 3420 t (mesh in)1 325 1 720 3540 t 10 I f (y y)1 44 1 1081 3540 t 10 R f ( large values of)3 642( sufficiently)1 491(. For)1 225 3 1125 3540 t 10 I f (n n)1 50 1 2518 3540 t 7 I f (x x)1 31 1 2579 3560 t 10 R f (and)2653 3540 w 10 I f (n n)1 50 1 2832 3540 t 7 I f (y y)1 31 1 2893 3560 t 10 R f ( \()1 65(, the storage is roughly 3)5 1038 2 2932 3540 t 10 I f (k k)1 44 1 4067 3540 t 7 I f (y y)1 31 1 4122 3560 t 10 S f (- -)1 55 1 4201 3540 t 10 R f (1 \))1 115 1 4296 3540 t 10 I f (n n)1 50 1 4451 3540 t 7 I f (u u)1 35 1 4506 3559 t 7 R f (2)4506 3500 w 10 I f (n n)1 50 1 4581 3540 t 7 I f (x x)1 31 1 4636 3559 t 7 R f (2)4636 3500 w 10 I f (n n)1 50 1 4711 3540 t 7 I f (y y)1 31 1 4772 3560 t 10 R f (. See)1 229 1 4811 3540 t (section 6 for ways of causing)5 1167 1 720 3660 t 10 CW f (TTGR)1912 3660 w 10 R f (to use much less space.)4 929 1 2177 3660 t (All scratch space for)3 840 1 970 3816 t 10 CW f (TTGR)1841 3816 w 10 R f ( Solving)1 368(is taken from the stack.)4 953 2 2112 3816 t 10 B f (pde)3464 3816 w 10 R f ( non-trivial process, requiring)3 1208(s is a)2 212 2 3620 3816 t ( virtually all problems solved by)5 1307( For)1 191(substantial work space.)2 933 3 720 3936 t 10 CW f (TTGR)3178 3936 w 10 R f ( and initial-)2 463(the user will have to declare)5 1132 2 3445 3936 t ( process will be illus-)4 895( This)1 238( than the default size of 1000 Real words.)8 1741(ize the PORT stack to a size larger)7 1446 4 720 4056 t (trated in the first example of Appendix 4.)7 1653 1 720 4176 t 10 B f (Run-time.)720 4416 w 10 R f (The run-time of)2 642 1 970 4572 t 10 CW f (TTGR)1642 4572 w 10 R f (is proportional to)2 699 1 1912 4572 t 10 I f (n n)1 50 1 2641 4572 t 7 I f (u u)1 35 1 2702 4592 t 10 I f (n n)1 50 1 2777 4572 t 7 I f (x x)1 31 1 2838 4592 t 10 I f (n n)1 50 1 2909 4572 t 7 I f (y y)1 31 1 2970 4592 t 10 R f (\()3041 4572 w 10 I f (H H)1 72 1 3106 4572 t 10 S f (- -)1 55 1 3218 4572 t 10 R f (1 \))1 115 1 3313 4572 t 7 R f (2)3433 4532 w 10 R f ( section 6)2 395( See)1 200(for the default settings.)3 939 3 3506 4572 t (for ways to make)3 690 1 720 4692 t 10 CW f (TTGR)1435 4692 w 10 R f (run much faster.)2 651 1 1700 4692 t (The storage and run-time of)4 1134 1 970 4848 t 10 CW f (TTGR)2135 4848 w 10 R f (are far from optimal with the default settings, being on the order)11 2634 1 2406 4848 t (of)720 4968 w 10 I f (n n)1 50 1 833 4968 t 7 R f (3)894 4928 w 10 R f (and)967 4968 w 10 I f (n n)1 50 1 1141 4968 t 7 R f (4)1202 4928 w 10 R f (, respectively, for an)3 829 1 1245 4968 t 10 I f (n n)1 50 1 2103 4968 t 10 R f (by)2182 4968 w 10 I f (n n)1 50 1 2311 4968 t 10 R f ( space and time would be)5 1032(grid. Optimal)1 568 2 2390 4968 t 10 I f (O O)1 72 1 4019 4968 t 10 R f (\()4099 4968 w 10 I f (n n)1 50 1 4164 4968 t 7 R f (2)4225 4928 w 10 R f ( reasons for)2 473(\). The)1 267 2 4300 4968 t ( section also shows how to)5 1074( That)1 234( in section 6.)3 514(the default use of a banded, pivoting matrix solver are detailed)10 2498 4 720 5088 t ( on a smaller class of problems \(parabolic\), and use much less)11 2605(make the package run much faster, albeit)6 1715 2 720 5208 t (space.)720 5328 w 10 B f (Double Precision Version.)2 1108 1 720 5568 t 10 R f (The Double Precision version of)4 1314 1 720 5724 t 10 CW f (TTGR)2063 5724 w 10 R f (is called)1 334 1 2332 5724 t 10 CW f (DTTGR)2695 5724 w 10 R f ( sequence for)2 541( calling)1 301(. The)1 234 3 2995 5724 t 10 CW f (DTTGR)4101 5724 w 10 R f (is precisely the)2 609 1 4431 5724 t (same as that for)3 704 1 720 5844 t 10 CW f (TTGR)1474 5844 w 10 R f (, with)1 253 1 1714 5844 t 10 I f ( ll l)2 28(a al)1 78 2 2016 5844 t 10 R f (floating-point arguments Double Precision,)3 1804 1 2171 5844 t 10 I f ( t)1 0( pt)1 28(e ex xc ce ep)4 226 3 4024 5844 t 10 CW f (errpar)4362 5844 w 10 R f (, which)1 318 1 4722 5844 t ( amount of scratch space used by)6 1349( The)1 210(remains Real.)1 553 3 720 5964 t 10 CW f (DTTGR)2862 5964 w 10 R f (on the dynamic stack of the Port Library [14])8 1848 1 3192 5964 t (is, neglecting lower order terms,)4 1292 1 720 6084 t 10 I f (n n)1 50 1 970 6240 t 7 I f (u u)1 35 1 1031 6260 t 10 I f (n n)1 50 1 1106 6240 t 7 I f (x x)1 31 1 1167 6260 t 10 I f (n n)1 50 1 1238 6240 t 7 I f (y y)1 31 1 1299 6260 t 10 R f (\( 3)1 115 1 1370 6240 t 10 I f (H H)1 72 1 1517 6240 t 10 S f (- -)1 55 1 1629 6240 t 10 R f (1 \))1 91 1 1724 6240 t (Double Precision words \(storage units\).)4 1584 1 720 6396 t 10 CW f (AF)720 6636 w 10 B f (and)865 6636 w 10 CW f (BC)1052 6636 w 10 B f (Descriptions.)1197 6636 w 10 R f (The user-supplied subroutines)2 1218 1 970 6792 t 10 CW f (AF)2220 6792 w 10 R f (and)2372 6792 w 10 CW f (BC)2548 6792 w 10 R f (, which define the)3 736 1 2668 6792 t 10 B f (pde)3436 6792 w 10 R f (-)3625 6792 w 10 B f (bc)3658 6792 w 10 R f (problem to be solved, are now)5 1249 1 3791 6792 t (described. When)1 695 1 720 6912 t 10 CW f (TTGR)1440 6912 w 10 R f (needs to compute)2 699 1 1705 6912 t 10 B f (a)2429 6912 w 10 R f (and)2504 6912 w 10 B f (f)2673 6912 w 10 R f (, it will)2 287 1 2706 6912 t cleartomark showpage saveobj restore end %%EndPage: 12 12 %%Page: 13 13 DpostDict begin /saveobj save def mark 13 pagesetup 10 R f (- 13 -)2 216 1 2772 480 t 10 CW f (Call AF\(t,Xe,Ye,nxe,nye,Nu, U,Ut,Ux,Uy,Uyt,Uxt,)2 2820 1 1080 900 t (A,AU,AUt,AUx,AUy,AUxt,AUyt,)1560 1020 w (F,FU,FUt,FUx,FUy,FUxt,FUyt\))1560 1140 w 10 R f (Before)720 1320 w 10 CW f (TTGR)1016 1320 w 10 R f (calls)1281 1320 w 10 CW f (AF)1489 1320 w 10 R f (, it sets to)3 384 1 1609 1320 t 10 B f (0)2018 1320 w 10 R f (the 14 arrays)2 515 1 2093 1320 t 10 CW f (A)2668 1320 w 10 R f (through)2753 1320 w 10 CW f (FUyt)3089 1320 w 10 R f (and provides the)2 660 1 3354 1320 t 10 I f ( t)1 0( np pu ut)3 128(i in)1 78 3 4039 1320 t 10 R f (values)4270 1320 w 10 CW f (t)1020 1512 w 10 R f ( current value of time.)4 884(- The)1 305 2 1320 1512 t 10 CW f (Xe)1020 1668 w 10 R f ( list of points)3 532(- A)1 222 2 1320 1668 t 10 I f (x x)1 44 1 2101 1668 t 10 R f (where)2172 1668 w 10 B f (a)2442 1668 w 10 R f (and)2519 1668 w 10 B f (f)2690 1668 w 10 R f ( This)1 230(are to be evaluated.)3 781 2 2750 1668 t 10 CW f (Xe)3788 1668 w 10 R f (is)3936 1668 w 10 I f ( t)1 0(n no ot)2 128 2 4031 1668 t 10 R f (the B-spline mesh X.)3 853 1 4187 1668 t (The points)1 426 1 1470 1788 t 10 CW f (Xe)1922 1788 w 10 R f (at which)1 342 1 2068 1788 t 10 B f (a)2436 1788 w 10 R f (and)2512 1788 w 10 B f (f)2682 1788 w 10 R f (are desired are determined by the quadrature rule used by)9 2299 1 2741 1788 t 10 CW f (TTGR)1470 1908 w 10 R f (to implement Galerkin's method.)3 1327 1 1735 1908 t 10 CW f (nxe)1020 2064 w 10 R f ( length of Xe.)3 549(- The)1 305 2 1320 2064 t 10 CW f (Ye)1020 2220 w 10 R f ( list of points)3 532(- A)1 222 2 1320 2220 t 10 I f (y y)1 44 1 2101 2220 t 10 R f (where)2172 2220 w 10 B f (a)2442 2220 w 10 R f (and)2519 2220 w 10 B f (f)2690 2220 w 10 R f ( This)1 230(are to be evaluated.)3 781 2 2750 2220 t 10 CW f (Ye)3788 2220 w 10 R f (is)3936 2220 w 10 I f ( t)1 0(n no ot)2 128 2 4031 2220 t 10 R f (the B-spline mesh Y.)3 853 1 4187 2220 t (The points)1 426 1 1470 2340 t 10 CW f (Ye)1922 2340 w 10 R f (at which)1 342 1 2068 2340 t 10 B f (a)2436 2340 w 10 R f (and)2512 2340 w 10 B f (f)2682 2340 w 10 R f (are desired are determined by the quadrature rule used by)9 2299 1 2741 2340 t 10 CW f (TTGR)1470 2460 w 10 R f (to implement Galerkin's method.)3 1327 1 1735 2460 t 10 CW f (nye)1020 2616 w 10 R f ( length of Ye.)3 549(- The)1 305 2 1320 2616 t 10 CW f (nu)1020 2772 w 10 R f ( number)1 330(- The)1 305 2 1320 2772 t 10 I f (n n)1 50 1 1980 2772 t 7 I f (u u)1 35 1 2041 2792 t 10 R f (of)2109 2772 w 10 B f (pde)2217 2772 w 10 R f (variables)2398 2772 w 10 B f (u)2783 2772 w 10 R f (.)2839 2772 w 10 CW f (U)1020 2928 w 10 R f (- The)1 305 1 1320 2928 t 10 I f ( s)1 0( es)1 39( ue)1 44( lu)1 50( al)1 28(v va)1 94 6 1675 2928 t 10 R f (of)1980 2928 w 10 B f (u)2113 2928 w 10 R f (at the)1 244 1 2219 2928 t 10 CW f (Xe)2513 2928 w 10 R f (\()2633 2928 w 10 I f (p p)1 50 1 2666 2928 t 10 R f (\) and)1 227 1 2716 2928 t 10 CW f (Ye)2993 2928 w 10 R f (\()3113 2928 w 10 I f (q q)1 50 1 3146 2928 t 10 R f (\),)3196 2928 w 10 B f (not)3304 2928 w 10 R f (the B-spline coefficients)2 1026 1 3493 2928 t 10 B f (U)4569 2928 w 10 R f (of \(3.1\).)1 349 1 4691 2928 t 10 CW f (U)1470 3048 w 10 R f (\()1530 3048 w 10 I f (p p)1 50 1 1563 3048 t 10 R f (,)1621 3048 w 10 I f (q q)1 50 1 1654 3048 t 10 R f (,)1712 3048 w 10 I f (j j)1 28 1 1753 3048 t 10 R f (\) =)1 114 1 1781 3048 t 10 I f (u u)1 50 1 1920 3048 t 7 I f (j j)1 20 1 1981 3068 t 10 R f (\(t,)2009 3048 w 10 CW f (Xe)2095 3048 w 10 R f (\()2215 3048 w 10 I f (p p)1 50 1 2248 3048 t 10 R f (\),)2298 3048 w 10 CW f (Ye)2356 3048 w 10 R f (\()2476 3048 w 10 I f (q q)1 50 1 2509 3048 t 10 R f (\)\),)2559 3048 w 10 I f (p p)1 50 1 2675 3048 t 10 S f (= =)1 55 1 2749 3048 t 10 R f (1 ,)1 83 1 2820 3048 t (. . .)2 125 1 2936 3023 t (,)3094 3048 w 10 CW f (nxe)3143 3048 w 10 R f (,)3323 3048 w 10 I f (q q)1 50 1 3373 3048 t 10 S f (= =)1 55 1 3447 3048 t 10 R f (1 ,)1 83 1 3518 3048 t (. . .)2 125 1 3634 3023 t (,)3792 3048 w 10 CW f (nye)3841 3048 w 10 R f (and)4046 3048 w 10 I f (j j)1 28 1 4215 3048 t 10 S f (= =)1 55 1 4259 3048 t 10 R f (1 ,)1 83 1 4330 3048 t (. . .)2 125 1 4446 3023 t (,)4604 3048 w 10 CW f (nu)4653 3048 w 10 R f (.)4773 3048 w 10 CW f (Ut)1020 3204 w 10 R f ( values of)2 388(- The)1 305 2 1320 3204 t 10 B f (u)2038 3204 w 7 I f (t t)1 20 1 2105 3224 t 10 R f (at the)1 219 1 2158 3204 t 10 CW f (Xe)2402 3204 w 10 R f (\()2522 3204 w 10 I f (p p)1 50 1 2555 3204 t 10 R f (\) and)1 202 1 2605 3204 t 10 CW f (Ye)2832 3204 w 10 R f (\()2952 3204 w 10 I f (q q)1 50 1 2985 3204 t 10 R f (\), stored as above.)3 723 1 3035 3204 t 10 CW f (Ux)1020 3360 w 10 R f ( values of)2 388(- The)1 305 2 1320 3360 t 10 B f (u)2038 3360 w 7 I f (x x)1 31 1 2105 3380 t 10 R f (, stored as above.)3 690 1 2144 3360 t 10 CW f (Uy)1020 3516 w 10 R f ( values of)2 388(- The)1 305 2 1320 3516 t 10 B f (u)2038 3516 w 7 I f (y y)1 31 1 2105 3536 t 10 R f (, stored as above.)3 690 1 2144 3516 t 10 CW f (Uxt)1020 3672 w 10 R f ( values of)2 388(- The)1 305 2 1320 3672 t 10 B f (u)2038 3672 w 7 I f ( t)1 0(x xt)1 51 2 2105 3692 t 10 R f (, stored as above.)3 690 1 2164 3672 t 10 CW f (Uyt)1020 3828 w 10 R f ( values of)2 388(- The)1 305 2 1320 3828 t 10 B f (u)2038 3828 w 7 I f ( t)1 0(y yt)1 51 2 2105 3848 t 10 R f (, stored as above.)3 690 1 2164 3828 t 10 CW f (AF)720 3984 w 10 R f (must return as)2 566 1 865 3984 t 10 I f ( t)1 0( pu ut)2 78( tp)1 50(o ou ut)2 128 4 1456 3984 t 10 CW f (A)1020 4176 w 10 R f ( value of)2 441(- The)1 305 2 1320 4176 t 10 B f (a)2137 4176 w 10 R f (at the)1 265 1 2258 4176 t 10 CW f (Xe)2594 4176 w 10 R f (\()2714 4176 w 10 I f (p p)1 50 1 2747 4176 t 10 R f (\) and)1 248 1 2797 4176 t 10 CW f (Xe)3151 4176 w 10 R f (\()3271 4176 w 10 I f (q q)1 50 1 3304 4176 t 10 R f (\).)3354 4176 w 10 CW f (A)3578 4176 w 10 R f (\()3638 4176 w 10 I f (p p)1 50 1 3671 4176 t 10 R f (,)3729 4176 w 10 I f (q q)1 50 1 3762 4176 t 10 R f (,)3820 4176 w 10 I f (j j)1 28 1 3861 4176 t 10 R f (\) =)1 161 1 3889 4176 t 10 I f (a a)1 50 1 4122 4176 t 7 I f (j j)1 20 1 4183 4196 t 10 R f (\(t,)4211 4176 w 10 CW f (Xe)4297 4176 w 10 R f (\()4417 4176 w 10 I f (p p)1 50 1 4450 4176 t 10 R f (\),)4500 4176 w 10 CW f (Ye)4558 4176 w 10 R f (\()4678 4176 w 10 I f (q q)1 50 1 4711 4176 t 10 R f (\)\), for)1 279 1 4761 4176 t 10 I f (p p)1 50 1 1470 4296 t 10 S f (= =)1 55 1 1544 4296 t 10 R f (1 ,)1 83 1 1615 4296 t (. . .)2 125 1 1731 4271 t (,)1889 4296 w 10 CW f (nxe)1938 4296 w 10 R f (,)2118 4296 w 10 I f (q q)1 50 1 2168 4296 t 10 S f (= =)1 55 1 2242 4296 t 10 R f (1 ,)1 83 1 2313 4296 t 10 I f (. .)1 25 1 2404 4296 t 10 R f (.. ,)1 83 1 2429 4296 t 10 CW f (nye)2536 4296 w 10 R f (and)2741 4296 w 10 I f (j j)1 28 1 2910 4296 t 10 S f (= =)1 55 1 2954 4296 t 10 R f (1 ,)1 83 1 3025 4296 t (. . .)2 125 1 3141 4271 t (,)3299 4296 w 10 CW f (nu)3348 4296 w 10 R f (.)3468 4296 w 10 CW f (AU)1020 4452 w 10 R f ( partial derivatives of)3 881(- The)1 305 2 1320 4452 t 10 B f (a)2541 4452 w 10 R f (with respect to)2 610 1 2627 4452 t 10 B f (u)3273 4452 w 10 R f (at the)1 230 1 3365 4452 t 10 CW f (Xe)3631 4452 w 10 R f (\()3751 4452 w 10 I f (p p)1 50 1 3784 4452 t 10 R f (\) and)1 213 1 3834 4452 t 10 CW f (Ye)4083 4452 w 10 R f (\()4203 4452 w 10 I f (q q)1 50 1 4236 4452 t 10 R f (\).)4286 4452 w 10 CW f (AU)4475 4452 w 10 R f (\()4595 4452 w 10 I f (p p)1 50 1 4628 4452 t 10 R f (,)4686 4452 w 10 I f (q q)1 50 1 4719 4452 t 10 R f (,)4777 4452 w 10 I f (i i)1 28 1 4810 4452 t 10 R f (,)4846 4452 w 10 I f (j j)1 28 1 4887 4452 t 10 R f (\) =)1 125 1 4915 4452 t 10 S f (\266)1470 4572 w 10 I f (a a)1 50 1 1527 4572 t 7 I f (i i)1 20 1 1588 4592 t 10 I f (/ /)1 28 1 1624 4572 t 10 S f (\266)1660 4572 w 10 I f (u u)1 50 1 1717 4572 t 7 I f (j j)1 20 1 1778 4592 t 10 R f (\(t,)1831 4572 w 10 CW f (Xe)1942 4572 w 10 R f (\()2062 4572 w 10 I f (p p)1 50 1 2095 4572 t 10 R f (\),)2145 4572 w 10 CW f (Ye)2203 4572 w 10 R f (\()2323 4572 w 10 I f (q q)1 50 1 2356 4572 t 10 R f (\)\), for)1 232 1 2406 4572 t 10 I f (p p)1 50 1 2663 4572 t 10 S f (= =)1 55 1 2737 4572 t 10 R f (1 ,)1 83 1 2808 4572 t (. . .)2 125 1 2924 4547 t (,)3082 4572 w 10 CW f (nxe)3131 4572 w 10 R f (,)3311 4572 w 10 I f (q q)1 50 1 3361 4572 t 10 S f (= =)1 55 1 3435 4572 t 10 R f (1 ,)1 83 1 3506 4572 t 10 I f (. .)1 25 1 3597 4572 t 10 R f (.. ,)1 83 1 3622 4572 t 10 CW f (nye)3729 4572 w 10 R f (and)3934 4572 w 10 I f (i i)1 28 1 4103 4572 t 10 R f (,)4139 4572 w 10 I f (j j)1 28 1 4180 4572 t 10 S f (= =)1 55 1 4224 4572 t 10 R f (1 ,)1 83 1 4295 4572 t (. . .)2 125 1 4411 4547 t (,)4569 4572 w 10 CW f (nu)4618 4572 w 10 R f (.)4738 4572 w 10 CW f (AUt)1020 4728 w 10 R f ( partial derivatives of)3 851(- The)1 305 2 1320 4728 t 10 B f (a)2501 4728 w 10 R f (with respect to)2 588 1 2576 4728 t 10 B f (u)3189 4728 w 7 I f (t t)1 20 1 3256 4748 t 10 R f (, as above.)2 421 1 3309 4728 t 10 CW f (AUx)1020 4884 w 10 R f ( partial derivatives of)3 851(- The)1 305 2 1320 4884 t 10 B f (a)2501 4884 w 10 R f (with respect to)2 588 1 2576 4884 t 10 B f (u)3189 4884 w 7 I f (x x)1 31 1 3256 4904 t 10 R f (, as above.)2 421 1 3295 4884 t 10 CW f (AUy)1020 5040 w 10 R f ( partial derivatives of)3 851(- The)1 305 2 1320 5040 t 10 B f (a)2501 5040 w 10 R f (with respect to)2 588 1 2576 5040 t 10 B f (u)3189 5040 w 7 I f (y y)1 31 1 3256 5060 t 10 R f (, as above.)2 421 1 3295 5040 t 10 CW f (AUxt)1020 5196 w 10 R f ( partial derivatives of)3 851(- The)1 305 2 1320 5196 t 10 B f (a)2501 5196 w 10 R f (with respect to)2 588 1 2576 5196 t 10 B f (u)3189 5196 w 7 I f ( t)1 0(x xt)1 51 2 3256 5216 t 10 R f (, as above.)2 421 1 3315 5196 t 10 CW f (AUyt)1020 5352 w 10 R f ( partial derivatives of)3 851(- The)1 305 2 1320 5352 t 10 B f (a)2501 5352 w 10 R f (with respect to)2 588 1 2576 5352 t 10 B f (u)3189 5352 w 7 I f ( t)1 0(y yt)1 51 2 3256 5372 t 10 R f (, as above.)2 421 1 3315 5352 t 10 CW f (F)1020 5508 w 10 R f ( value of)2 447(- The)1 305 2 1320 5508 t 10 B f (f)2146 5508 w 10 R f (at the)1 268 1 2253 5508 t 10 CW f (Xe)2595 5508 w 10 R f (\()2715 5508 w 10 I f (p p)1 50 1 2748 5508 t 10 R f (\) and)1 252 1 2798 5508 t 10 CW f (Ye)3160 5508 w 10 R f (\()3280 5508 w 10 I f (q q)1 50 1 3313 5508 t 10 R f (\).)3363 5508 w 10 CW f (F)3591 5508 w 10 R f (\()3651 5508 w 10 I f (p p)1 50 1 3684 5508 t 10 R f (,)3742 5508 w 10 I f (q q)1 50 1 3775 5508 t 10 R f (,)3833 5508 w 10 I f (j j)1 28 1 3874 5508 t 10 R f (\) =)1 164 1 3902 5508 t 10 I f (f f)1 28 1 4141 5508 t 7 I f (j j)1 20 1 4180 5528 t 10 R f (\(t,)4208 5508 w 10 CW f (Xe)4294 5508 w 10 R f (\()4414 5508 w 10 I f (p p)1 50 1 4447 5508 t 10 R f (\),)4497 5508 w 10 CW f (Ye)4555 5508 w 10 R f (\()4675 5508 w 10 I f (q q)1 50 1 4708 5508 t 10 R f (\)\), for)1 282 1 4758 5508 t 10 I f (p p)1 50 1 1470 5628 t 10 S f (= =)1 55 1 1544 5628 t 10 R f (1 ,)1 83 1 1615 5628 t (. . .)2 125 1 1731 5603 t (,)1889 5628 w 10 CW f (nxe)1938 5628 w 10 R f (,)2118 5628 w 10 I f (q q)1 50 1 2168 5628 t 10 S f (= =)1 55 1 2242 5628 t 10 R f (1 ,)1 83 1 2313 5628 t 10 I f (. .)1 25 1 2404 5628 t 10 R f (.. ,)1 83 1 2429 5628 t 10 CW f (nye)2536 5628 w 10 R f (and)2741 5628 w 10 I f (j j)1 28 1 2910 5628 t 10 S f (= =)1 55 1 2954 5628 t 10 R f (1 ,)1 83 1 3025 5628 t (. . .)2 125 1 3141 5603 t (,)3299 5628 w 10 CW f (nu)3348 5628 w 10 R f (.)3468 5628 w 10 CW f (FU)1020 5784 w 10 R f ( of)1 120( partial derivatives)2 765(- The)1 305 3 1320 5784 t 10 B f (f)2547 5784 w 10 R f (with respect to)2 612 1 2617 5784 t 10 B f (u)3266 5784 w 10 R f (at the)1 231 1 3359 5784 t 10 CW f (Xe)3627 5784 w 10 R f (\()3747 5784 w 10 I f (p p)1 50 1 3780 5784 t 10 R f (\) and)1 214 1 3830 5784 t 10 CW f (Ye)4081 5784 w 10 R f (\()4201 5784 w 10 I f (q q)1 50 1 4234 5784 t 10 R f (\).)4284 5784 w 10 CW f (FU)4474 5784 w 10 R f (\()4594 5784 w 10 I f (p p)1 50 1 4627 5784 t 10 R f (,)4685 5784 w 10 I f (q q)1 50 1 4718 5784 t 10 R f (,)4776 5784 w 10 I f (i i)1 28 1 4809 5784 t 10 R f (,)4845 5784 w 10 I f (j j)1 28 1 4886 5784 t 10 R f (\) =)1 126 1 4914 5784 t 10 S f (\266)1470 5904 w 10 I f (f f)1 28 1 1535 5904 t 7 I f (i i)1 20 1 1574 5924 t 10 I f (/ /)1 28 1 1634 5904 t 10 S f (\266)1670 5904 w 10 I f (u u)1 50 1 1727 5904 t 7 I f (j j)1 20 1 1788 5924 t 10 R f (\(t,)1841 5904 w 10 CW f (Xe)1952 5904 w 10 R f (\()2072 5904 w 10 I f (p p)1 50 1 2105 5904 t 10 R f (\),)2155 5904 w 10 CW f (Ye)2213 5904 w 10 R f (\()2333 5904 w 10 I f (q q)1 50 1 2366 5904 t 10 R f (\)\), for)1 232 1 2416 5904 t 10 I f (p p)1 50 1 2673 5904 t 10 S f (= =)1 55 1 2747 5904 t 10 R f (1 ,)1 83 1 2818 5904 t (. . .)2 125 1 2934 5879 t (,)3092 5904 w 10 CW f (nxe)3141 5904 w 10 R f (,)3321 5904 w 10 I f (q q)1 50 1 3371 5904 t 10 S f (= =)1 55 1 3445 5904 t 10 R f (1 ,)1 83 1 3516 5904 t 10 I f (. .)1 25 1 3607 5904 t 10 R f (.. ,)1 83 1 3632 5904 t 10 CW f (nye)3739 5904 w 10 R f (and)3944 5904 w 10 I f (i i)1 28 1 4113 5904 t 10 R f (,)4149 5904 w 10 I f (j j)1 28 1 4190 5904 t 10 S f (= =)1 55 1 4234 5904 t 10 R f (1 ,)1 83 1 4305 5904 t (. . .)2 125 1 4421 5879 t (,)4579 5904 w 10 CW f (nu)4628 5904 w 10 R f (.)4748 5904 w 10 CW f (FUt)1020 6060 w 10 R f ( partial derivatives of)3 851(- The)1 305 2 1320 6060 t 10 B f (f)2501 6060 w 10 R f (with respect to)2 588 1 2559 6060 t 10 B f (u)3172 6060 w 7 I f (t t)1 20 1 3239 6080 t 10 R f (, as above.)2 421 1 3292 6060 t 10 CW f (FUx)1020 6216 w 10 R f ( partial derivatives of)3 851(- The)1 305 2 1320 6216 t 10 B f (f)2501 6216 w 10 R f (with respect to)2 588 1 2559 6216 t 10 B f (u)3172 6216 w 7 I f (x x)1 31 1 3239 6236 t 10 R f (, as above.)2 421 1 3278 6216 t 10 CW f (FUy)1020 6372 w 10 R f ( partial derivatives of)3 851(- The)1 305 2 1320 6372 t 10 B f (f)2501 6372 w 10 R f (with respect to)2 588 1 2559 6372 t 10 B f (u)3172 6372 w 7 I f (y y)1 31 1 3239 6392 t 10 R f (, as above.)2 421 1 3278 6372 t 10 CW f (FUxt)1020 6528 w 10 R f ( partial derivatives of)3 851(- The)1 305 2 1320 6528 t 10 B f (f)2501 6528 w 10 R f (with respect to)2 588 1 2559 6528 t 10 B f (u)3172 6528 w 7 I f ( t)1 0(x xt)1 51 2 3239 6548 t 10 R f (, as above.)2 421 1 3298 6528 t 10 CW f (FUyt)1020 6684 w 10 R f ( partial derivatives of)3 851(- The)1 305 2 1320 6684 t 10 B f (f)2501 6684 w 10 R f (with respect to)2 588 1 2559 6684 t 10 B f (u)3172 6684 w 7 I f ( t)1 0(y yt)1 51 2 3239 6704 t 10 R f (, as above.)2 421 1 3298 6684 t (When)720 6840 w 10 CW f (TTGR)983 6840 w 10 R f (needs the boundary conditions it will)5 1480 1 1248 6840 t cleartomark showpage saveobj restore end %%EndPage: 13 13 %%Page: 14 14 DpostDict begin /saveobj save def mark 14 pagesetup 10 R f (- 14 -)2 216 1 2772 480 t 10 CW f (Call BC\(t,Xe,nxe,Ye,nye,Lx,Rx,Ly,Ry,)1 2160 1 1080 900 t (U,Ut,Ux,Uy,Uxt,Uyt,nu,)1560 1020 w (B,BU,BUt,BUx,BUy,BUxt,BUyt\))1560 1140 w 10 R f (Before)720 1356 w 10 CW f (TTGR)1016 1356 w 10 R f (calls)1281 1356 w 10 CW f (BC)1489 1356 w 10 R f (, it sets to)3 384 1 1609 1356 t 10 B f (0)2018 1356 w 10 R f (the 7 arrays)2 465 1 2093 1356 t 10 CW f (B)2618 1356 w 10 R f (through)2703 1356 w 10 CW f (BUyt)3039 1356 w 10 R f (, and provides the)3 710 1 3279 1356 t 10 I f ( t)1 0( np pu ut)3 128(i in)1 78 3 4014 1356 t 10 CW f (t)1020 1548 w 10 R f ( current value of time.)4 884(- The)1 305 2 1370 1548 t 10 CW f (Xe)1020 1704 w 10 R f ( list of points)3 535(- A)1 222 2 1370 1704 t 10 I f (x x)1 44 1 2155 1704 t 10 R f (where)2227 1704 w 10 B f (b)2498 1704 w 10 R f ( This)1 232(is to be evaluated.)3 730 2 2582 1704 t 10 CW f (Xe)3573 1704 w 10 R f (is)3722 1704 w 10 I f ( t)1 0(n no ot)2 128 2 3818 1704 t 10 R f ( The)1 209(the B-spline mesh X.)3 856 2 3975 1704 t (points)1520 1824 w 10 CW f (Xe)1794 1824 w 10 R f (at which)1 345 1 1943 1824 t 10 B f (b)2317 1824 w 10 R f ( are determined by the quadrature rule used by)8 1880(is desired)1 384 2 2402 1824 t 10 CW f (TTGR)4694 1824 w 10 R f (to)4962 1824 w (implement Galerkin's method.)2 1224 1 1520 1944 t 10 CW f (nxe)1020 2100 w 10 R f ( length of Xe.)3 549(- The)1 305 2 1370 2100 t 10 CW f (Ye)1020 2256 w 10 R f ( list of points)3 535(- A)1 222 2 1370 2256 t 10 I f (y y)1 44 1 2155 2256 t 10 R f (where)2227 2256 w 10 B f (b)2498 2256 w 10 R f ( This)1 232(is to be evaluated.)3 730 2 2582 2256 t 10 CW f (Ye)3573 2256 w 10 R f (is)3722 2256 w 10 I f ( t)1 0(n no ot)2 128 2 3818 2256 t 10 R f ( The)1 209(the B-spline mesh Y.)3 856 2 3975 2256 t (points)1520 2376 w 10 CW f (Ye)1794 2376 w 10 R f (at which)1 345 1 1943 2376 t 10 B f (b)2317 2376 w 10 R f ( are determined by the quadrature rule used by)8 1880(is desired)1 384 2 2402 2376 t 10 CW f (TTGR)4694 2376 w 10 R f (to)4962 2376 w (implement Galerkin's method.)2 1224 1 1520 2496 t 10 CW f (nye)1020 2652 w 10 R f ( length of Ye.)3 549(- The)1 305 2 1370 2652 t 10 CW f (Lx)1020 2808 w 10 R f ( left-hand end-point of the)4 1048(- The)1 305 2 1370 2808 t 10 I f (x x)1 44 1 2748 2808 t 10 R f (spatial domain.)1 611 1 2817 2808 t 10 CW f (Rx)1020 2964 w 10 R f ( right-hand end-point of the)4 1104(- The)1 305 2 1370 2964 t 10 I f (x x)1 44 1 2804 2964 t 10 R f (spatial domain.)1 611 1 2873 2964 t 10 CW f (Ly)1020 3120 w 10 R f ( left-hand end-point of the)4 1048(- The)1 305 2 1370 3120 t 10 I f (y y)1 44 1 2748 3120 t 10 R f (spatial domain.)1 611 1 2817 3120 t 10 CW f (Ry)1020 3276 w 10 R f ( right-hand end-point of the)4 1104(- The)1 305 2 1370 3276 t 10 I f (y y)1 44 1 2804 3276 t 10 R f (spatial domain.)1 611 1 2873 3276 t 10 CW f (U)1020 3432 w 10 R f (-)1370 3432 w 10 CW f (U)1520 3432 w 10 R f (\()1580 3432 w 10 I f (p p)1 50 1 1613 3432 t 10 R f (,)1671 3432 w 10 I f (q q)1 50 1 1704 3432 t 10 R f (,)1762 3432 w 10 I f (i i)1 28 1 1795 3432 t 10 R f (\) =)1 114 1 1823 3432 t 10 I f (u u)1 50 1 1962 3432 t 7 I f (i i)1 20 1 2023 3452 t 10 R f (\(t,)2051 3432 w 10 CW f (Xe)2137 3432 w 10 R f (\()2257 3432 w 10 I f (p p)1 50 1 2290 3432 t 10 R f (\),)2340 3432 w 10 CW f (Ye)2398 3432 w 10 R f (\()2518 3432 w 10 I f (q q)1 50 1 2551 3432 t 10 R f (\)\) for)1 207 1 2601 3432 t 10 I f (i i)1 28 1 2833 3432 t 10 S f (= =)1 55 1 2885 3432 t 10 R f (1 ,)1 83 1 2956 3432 t (. . .)2 125 1 3072 3407 t (,)3230 3432 w 10 CW f (nu)3279 3432 w 10 R f (,)3399 3432 w 10 I f (p p)1 50 1 3449 3432 t 10 S f (= =)1 55 1 3523 3432 t 10 R f (1 ,)1 83 1 3594 3432 t 10 I f (. .)1 25 1 3685 3432 t 10 R f (.. ,)1 83 1 3710 3432 t 10 CW f (nxe)3817 3432 w 10 R f (and)4022 3432 w 10 I f (q q)1 50 1 4191 3432 t 10 S f (= =)1 55 1 4265 3432 t 10 R f (1 ,)1 83 1 4336 3432 t (. . .)2 125 1 4452 3407 t (,)4610 3432 w 10 CW f (nye)4659 3432 w 10 R f (.)4839 3432 w 10 CW f (Ut)1020 3588 w 10 R f (-)1370 3588 w 10 CW f (Ut)1520 3588 w 10 S f (= =)1 55 1 1665 3588 t 10 B f (u)1745 3588 w 7 I f (t t)1 20 1 1812 3608 t 10 R f (, stored as above.)3 690 1 1840 3588 t 10 CW f (Ux)1020 3744 w 10 R f (-)1370 3744 w 10 CW f (Ux)1520 3744 w 10 S f (= =)1 55 1 1665 3744 t 10 B f (u)1745 3744 w 7 I f (x x)1 31 1 1812 3764 t 10 R f (, as above.)2 421 1 1851 3744 t 10 CW f (Uy)1020 3900 w 10 R f (-)1370 3900 w 10 CW f (Uy)1520 3900 w 10 S f (= =)1 55 1 1665 3900 t 10 B f (u)1745 3900 w 7 I f (y y)1 31 1 1812 3920 t 10 R f (, as above.)2 421 1 1851 3900 t 10 CW f (Uxt)1020 4056 w 10 R f (-)1370 4056 w 10 CW f (Uxt)1520 4056 w 10 S f (= =)1 55 1 1725 4056 t 10 B f (u)1805 4056 w 7 I f ( t)1 0(x xt)1 51 2 1872 4076 t 10 R f (, as above.)2 421 1 1931 4056 t 10 CW f (Uyt)1020 4212 w 10 R f (-)1370 4212 w 10 CW f (Uyt)1520 4212 w 10 S f (= =)1 55 1 1725 4212 t 10 B f (u)1805 4212 w 7 I f ( t)1 0(y yt)1 51 2 1872 4232 t 10 R f (, as above.)2 421 1 1931 4212 t 10 CW f (nu)1020 4368 w 10 R f ( number)1 330(- The)1 305 2 1370 4368 t 10 I f (n n)1 50 1 2030 4368 t 7 I f (u u)1 35 1 2091 4388 t 10 R f (of)2159 4368 w 10 B f (pde)2267 4368 w 10 R f (variables)2448 4368 w 10 B f (u)2833 4368 w 10 R f (.)2889 4368 w 10 CW f (BC)720 4524 w 10 R f (must return as)2 566 1 865 4524 t 10 I f ( t)1 0( pu ut)2 78( tp)1 50(o ou ut)2 128 4 1456 4524 t 10 CW f (B)1020 4716 w 10 R f (-)1370 4716 w 10 CW f (B)1520 4716 w 10 R f (\()1580 4716 w 10 I f (p p)1 50 1 1613 4716 t 10 R f (,)1671 4716 w 10 I f (q q)1 50 1 1704 4716 t 10 R f (,)1762 4716 w 10 I f (i i)1 28 1 1795 4716 t 10 R f (\) =)1 114 1 1823 4716 t 10 B f (b)1962 4716 w 10 R f (,)2018 4716 w 10 I f (i i)1 28 1 2068 4716 t 10 S f (= =)1 55 1 2120 4716 t 10 R f (1 ,)1 83 1 2191 4716 t (. . .)2 125 1 2307 4691 t (,)2465 4716 w 10 CW f (nu)2514 4716 w 10 R f (,)2634 4716 w 10 I f (p p)1 50 1 2684 4716 t 10 S f (= =)1 55 1 2758 4716 t 10 R f (1 ,)1 83 1 2829 4716 t 10 I f (. .)1 25 1 2920 4716 t 10 R f (.. ,)1 83 1 2945 4716 t 10 CW f (nxe)3052 4716 w 10 R f (and)3257 4716 w 10 I f (q q)1 50 1 3426 4716 t 10 S f (= =)1 55 1 3500 4716 t 10 R f (1 ,)1 83 1 3571 4716 t (. . .)2 125 1 3687 4691 t (,)3845 4716 w 10 CW f (nye)3894 4716 w 10 R f ( \(2.2\).)1 241(. see)1 202 2 4074 4716 t 10 CW f (BU)1020 4872 w 10 R f (-)1370 4872 w 10 CW f (BU)1520 4872 w 10 R f (\()1640 4872 w 10 I f (p p)1 50 1 1673 4872 t 10 R f (,)1731 4872 w 10 I f (q q)1 50 1 1764 4872 t 10 R f (,)1822 4872 w 10 I f (i i)1 28 1 1855 4872 t 10 R f (,)1891 4872 w 10 I f (j j)1 28 1 1932 4872 t 10 R f (\) =)1 189 1 1960 4872 t 10 S f (\266)2250 4872 w 10 I f (b b)1 50 1 2307 4872 t 7 I f (i i)1 20 1 2368 4892 t 10 I f (/ /)1 28 1 2428 4872 t 10 S f (\266)2464 4872 w 10 I f (u u)1 50 1 2521 4872 t 7 I f (j j)1 20 1 2582 4892 t 10 R f (\()2610 4872 w 10 CW f (Xe)2643 4872 w 10 R f (\()2763 4872 w 10 I f (p p)1 50 1 2796 4872 t 10 R f (\),)2846 4872 w 10 CW f (Ye)2904 4872 w 10 R f (\()3024 4872 w 10 I f (q q)1 50 1 3057 4872 t 10 R f (\),)3107 4872 w 10 I f (i i)1 28 1 3266 4872 t 10 R f (,)3302 4872 w 10 I f (j j)1 28 1 3343 4872 t 10 S f (= =)1 55 1 3387 4872 t 10 R f (1 ,)1 83 1 3458 4872 t (. . .)2 125 1 3574 4847 t (,)3732 4872 w 10 CW f (nu)3781 4872 w 10 R f (and)4002 4872 w 10 I f (p p)1 50 1 4247 4872 t 10 S f (= =)1 55 1 4321 4872 t 10 R f (1 ,)1 83 1 4392 4872 t 10 I f (. .)1 25 1 4483 4872 t 10 R f (.. ,)1 83 1 4508 4872 t 10 CW f (nxe)4615 4872 w 10 R f (and)4896 4872 w 10 I f (q q)1 50 1 1520 4992 t 10 S f (= =)1 55 1 1594 4992 t 10 R f (1 ,)1 83 1 1665 4992 t (. . .)2 125 1 1781 4967 t (,)1939 4992 w 10 CW f (nye)1988 4992 w 10 R f (.)2168 4992 w 10 CW f (BUt)1020 5148 w 10 R f (-)1370 5148 w 10 CW f (BUt)1520 5148 w 10 R f (\()1700 5148 w 10 I f (p p)1 50 1 1733 5148 t 10 R f (,)1791 5148 w 10 I f (q q)1 50 1 1824 5148 t 10 R f (,)1882 5148 w 10 I f (i i)1 28 1 1915 5148 t 10 R f (,)1951 5148 w 10 I f (j j)1 28 1 1992 5148 t 10 R f (\) =)1 114 1 2020 5148 t 10 S f (\266)2159 5148 w 10 I f (b b)1 50 1 2216 5148 t 7 I f (i i)1 20 1 2277 5168 t 10 I f (/ /)1 28 1 2337 5148 t 10 S f (\266)2373 5148 w 10 I f (u u)1 50 1 2430 5148 t 7 I f ( t)1 0( t)1 25(j j)1 20 3 2491 5168 t 10 R f (, as above.)2 421 1 2544 5148 t 10 CW f (BUx)1020 5304 w 10 R f (-)1370 5304 w 10 CW f (BUx)1520 5304 w 10 R f (\()1700 5304 w 10 I f (p p)1 50 1 1733 5304 t 10 R f (,)1791 5304 w 10 I f (q q)1 50 1 1824 5304 t 10 R f (,)1882 5304 w 10 I f (i i)1 28 1 1915 5304 t 10 R f (,)1951 5304 w 10 I f (j j)1 28 1 1992 5304 t 10 R f (\) =)1 114 1 2020 5304 t 10 S f (\266)2159 5304 w 10 I f (b b)1 50 1 2216 5304 t 7 I f (i i)1 20 1 2277 5324 t 10 I f (/ /)1 28 1 2337 5304 t 10 S f (\266)2373 5304 w 10 I f (u u)1 50 1 2430 5304 t 7 I f ( x)1 0( x)1 36(j j)1 20 3 2491 5324 t 10 R f (, as above.)2 421 1 2555 5304 t 10 CW f (BUy)1020 5460 w 10 R f (-)1370 5460 w 10 CW f (BUy)1520 5460 w 10 R f (\()1700 5460 w 10 I f (p p)1 50 1 1733 5460 t 10 R f (,)1791 5460 w 10 I f (q q)1 50 1 1824 5460 t 10 R f (,)1882 5460 w 10 I f (i i)1 28 1 1915 5460 t 10 R f (,)1951 5460 w 10 I f (j j)1 28 1 1992 5460 t 10 R f (\) =)1 114 1 2020 5460 t 10 S f (\266)2159 5460 w 10 I f (b b)1 50 1 2216 5460 t 7 I f (i i)1 20 1 2277 5480 t 10 I f (/ /)1 28 1 2337 5460 t 10 S f (\266)2373 5460 w 10 I f (u u)1 50 1 2430 5460 t 7 I f ( y)1 0( y)1 36(j j)1 20 3 2491 5480 t 10 R f (, as above.)2 421 1 2555 5460 t 10 CW f (BUxt)1020 5616 w 10 R f (-)1370 5616 w 10 CW f (BUxt)1520 5616 w 10 R f (\()1760 5616 w 10 I f (p p)1 50 1 1793 5616 t 10 R f (,)1851 5616 w 10 I f (q q)1 50 1 1884 5616 t 10 R f (,)1942 5616 w 10 I f (i i)1 28 1 1975 5616 t 10 R f (,)2011 5616 w 10 I f (j j)1 28 1 2052 5616 t 10 R f (\) =)1 114 1 2080 5616 t 10 S f (\266)2219 5616 w 10 I f (b b)1 50 1 2276 5616 t 7 I f (i i)1 20 1 2337 5636 t 10 I f (/ /)1 28 1 2397 5616 t 10 S f (\266)2433 5616 w 10 I f (u u)1 50 1 2490 5616 t 7 I f ( x)1 0( tx)1 31( t)1 25(j j)1 20 4 2551 5636 t 10 R f (, as above.)2 421 1 2635 5616 t 10 CW f (BUyt)1020 5772 w 10 R f (-)1370 5772 w 10 CW f (BUyt)1520 5772 w 10 R f (\()1760 5772 w 10 I f (p p)1 50 1 1793 5772 t 10 R f (,)1851 5772 w 10 I f (q q)1 50 1 1884 5772 t 10 R f (,)1942 5772 w 10 I f (i i)1 28 1 1975 5772 t 10 R f (,)2011 5772 w 10 I f (j j)1 28 1 2052 5772 t 10 R f (\) =)1 114 1 2080 5772 t 10 S f (\266)2219 5772 w 10 I f (b b)1 50 1 2276 5772 t 7 I f (i i)1 20 1 2337 5792 t 10 I f (/ /)1 28 1 2397 5772 t 10 S f (\266)2433 5772 w 10 I f (u u)1 50 1 2490 5772 t 7 I f ( y)1 0( ty)1 31( t)1 25(j j)1 20 4 2551 5792 t 10 R f (, as above.)2 421 1 2635 5772 t 10 CW f (HANDLE)720 6012 w 10 B f (Description.)1105 6012 w 10 R f (The user-supplied output and control subroutine)5 1958 1 720 6168 t 10 CW f (HANDLE)2710 6168 w 10 R f ( numerical solution at an)4 1016( The)1 212(is now described.)2 710 3 3102 6168 t ( lengthy and complex calculations involving trying several)7 2359(instant in time is obtained only after some rather)8 1961 2 720 6288 t ( When)1 294( detail.)1 278(small sub-steps in time; see Appendix 2 for more)8 2005 3 720 6408 t 10 CW f (TTGR)3328 6408 w 10 R f (has finally come up with a solution)6 1441 1 3599 6408 t ( the end of each time-step,)5 1063( At)1 152( as requested by the user, it just has to tell the user the good news.)15 2663(as accurate)1 442 4 720 6528 t 10 CW f (TTGR)720 6648 w 10 R f (will)985 6648 w 10 CW f (Call HANDLE\(t0,U0,t,U,lU,dt,tstop\))1 2040 1 1080 6828 t 10 R f ( the out-)2 343( If)1 121( whatever is desired with the solution.)6 1548(so that the user may look at, print out, plot, fondle, or do)12 2308 4 720 7008 t ( only the solution at time)5 1008(put at the end of each time-step is not desired, and)10 2035 2 720 7128 t 10 CW f (tstop)3790 7128 w 10 R f (is needed, the "Return-)3 923 1 4117 7128 t (End")720 7248 w 10 CW f (HANDLE)947 7248 w 10 R f (subroutine)1332 7248 w 10 CW f (TTGRH)1779 7248 w 10 R f (may be used.)2 524 1 2104 7248 t cleartomark showpage saveobj restore end %%EndPage: 14 14 %%Page: 15 15 DpostDict begin /saveobj save def mark 15 pagesetup 10 R f (- 15 -)2 216 1 2772 480 t 10 CW f (TTGR)970 840 w 10 R f (also invokes)1 501 1 1239 840 t 10 CW f (HANDLE)1769 840 w 10 R f (whenever it tries to take a time step and fails to obtain the user desired)14 2882 1 2158 840 t ( may be caused by the time step)7 1287(accuracy. This)1 608 2 720 960 t 10 CW f (dt)2642 960 w 10 R f ( it may mean that there is something)7 1466( Or)1 157(being too large.)2 628 3 2789 960 t ("funny" going on near time)4 1100 1 720 1080 t 10 CW f (t)1847 1080 w 10 R f ( case, the user may want to know that)8 1524( either)1 254(. In)1 160 3 1907 1080 t 10 CW f (TTGR)3873 1080 w 10 R f (failed at time)2 533 1 4141 1080 t 10 CW f (t)4702 1080 w 10 R f (. Such)1 278 1 4762 1080 t (things are called "restarts" and are typically expensive and worth knowing about.)11 3237 1 720 1200 t (The input provided by)3 891 1 970 1356 t 10 CW f (TTGR)1886 1356 w 10 R f (to)2151 1356 w 10 CW f (HANDLE)2254 1356 w 10 R f (is)2639 1356 w 10 CW f (t0)1020 1548 w 10 R f ( at the beginning of the time-step just completed.)8 1957(- Time)1 361 2 1370 1548 t 10 CW f (U0)1020 1704 w 10 R f (-)1370 1704 w 10 B f (pde)1520 1704 w 10 R f (solution)1708 1704 w 10 B f (u)2063 1704 w 10 R f ( given by B-spline coefficients)4 1258(at time t0 is)3 491 2 2151 1704 t 10 CW f (U0)3933 1704 w 10 R f ( array is)2 337(. This)1 261 2 4053 1704 t 10 CW f (Real)4684 1704 w 10 R f (of)4957 1704 w (length)1520 1824 w 10 CW f (lU)1832 1824 w 10 S f (= =)1 55 1 2014 1824 t 10 I f (n n)1 50 1 2118 1824 t 7 I f (x x)1 31 1 2179 1844 t 10 I f (n n)1 50 1 2250 1824 t 7 I f (y y)1 31 1 2311 1844 t 10 I f (n n)1 50 1 2382 1824 t 7 I f (u u)1 35 1 2443 1844 t 10 R f ( are stored as if)4 753( coefficients)1 527(. The)1 267 3 2486 1824 t 10 CW f (U0)4094 1824 w 10 R f (were dimensioned)1 765 1 4275 1824 t 10 CW f (\(nx,ny,Nu\))1520 1944 w 10 R f (, where the)2 440 1 2120 1944 t 10 I f (x x)1 44 1 2585 1944 t 10 R f (grid has)1 319 1 2654 1944 t 10 CW f (nx)2998 1944 w 10 R f (points, similarly for)2 792 1 3143 1944 t 10 I f (y y)1 44 1 3960 1944 t 10 R f (and there are)2 514 1 4029 1944 t 10 CW f (Nu)4568 1944 w 10 B f (pde)4713 1944 w 10 R f (s.)4869 1944 w 10 CW f (t)1020 2100 w 10 R f ( "current" value of time.)4 966( The)1 205( at the end of the time-step just completed.)8 1701(- Time)1 361 4 1370 2100 t 10 CW f (U)1020 2256 w 10 R f (-)1370 2256 w 10 B f (pde)1520 2256 w 10 R f (solution)1704 2256 w 10 B f (u)2055 2256 w 10 R f ( given by B-spline coefficients)4 1242(at time t is)3 429 2 2139 2256 t 10 CW f (U)3839 2256 w 10 R f (. If)1 145 1 3899 2256 t 10 CW f (t0)4073 2256 w 10 R f (=)4222 2256 w 10 CW f (t)4307 2256 w 10 R f (, then a restart is)4 673 1 4367 2256 t (in progress and the values in)5 1140 1 1520 2376 t 10 CW f (U)2685 2376 w 10 R f (are meaningless.)1 665 1 2770 2376 t 10 CW f (lU)1020 2532 w 10 R f ( length of the array)4 759(- The)1 305 2 1370 2532 t 10 CW f (U)2459 2532 w 10 R f (.)2519 2532 w 10 CW f (dt)1020 2688 w 10 R f ( current "optimal" value of)4 1069(- The)1 305 2 1370 2688 t 10 CW f (dt)2769 2688 w 10 R f (.)2889 2688 w 10 CW f (tstop)1020 2844 w 10 R f ( current final value for time.)5 1125(- The)1 305 2 1370 2844 t (The use of)2 425 1 720 3000 t 10 CW f (lU)1172 3000 w 10 R f ( use of)2 272(above is a botch required by the)6 1287 2 1319 3000 t 10 CW f (IODE)2906 3000 w 10 R f (to solve the spatially discretized problem - the)7 1866 1 3174 3000 t ( Appendix 4, the needed values of)6 1385( the example code of)4 851( In)1 138(output routine calling sequence is fixed.)5 1623 4 720 3120 t 10 CW f (nx)4746 3120 w 10 R f (,)4866 3120 w 10 CW f (ny)4920 3120 w 10 R f (and)720 3240 w 10 CW f (Nu)938 3240 w 10 R f (are obtained, magically, from)3 1220 1 1097 3240 t 10 CW f (Common)2356 3240 w 10 R f (regions internal to)2 755 1 2755 3240 t 10 CW f (TTGR)3549 3240 w 10 R f ( be)1 134( botch will probably)3 850(. This)1 267 3 3789 3240 t (fixed in the next edition of)5 1063 1 720 3360 t 10 CW f (TTGR)1808 3360 w 10 R f (.)2048 3360 w (The output from)2 655 1 720 3516 t 10 CW f (HANDLE)1400 3516 w 10 R f (is)1785 3516 w 10 CW f (t)1020 3708 w 10 R f ( too many people want to do this, but it is allowed.)11 2025( Not)1 200( be altered by the user.)5 903(- May)1 333 4 1370 3708 t 10 CW f (U)1020 3864 w 10 R f ( solution to be)3 582( example, the user may want to force the)8 1645( For)1 192( be altered by the user.)5 918(- May)1 333 5 1370 3864 t (non-negative or monotone.)2 1079 1 1520 3984 t 10 CW f (dt)1020 4140 w 10 R f ( user may want to choose)5 1057( The)1 214( user.)1 225( be altered by the)4 719(- May)1 333 5 1370 4140 t 10 CW f (dt)3952 4140 w 10 R f (so that some particular)3 934 1 4106 4140 t (value of time is achieved on the next time-step.)8 1889 1 1520 4260 t 10 CW f (tstop)1020 4416 w 10 R f ( example, the user may only want to integrate until)9 2031( For)1 189( be altered by the user.)5 903(- May)1 333 4 1370 4416 t 10 B f (u)4852 4416 w 7 I f (t t)1 20 1 4919 4436 t 10 R f (is)4973 4416 w ("small enough" and then stop.)4 1201 1 1520 4536 t 10 B f (Evaluating the Solution.)2 1033 1 720 4776 t 10 R f (To evaluate the solution created by)5 1400 1 970 4932 t 10 CW f (TTGR)2395 4932 w 10 R f (, simply)1 323 1 2635 4932 t 10 CW f ( k, t, it, nt, a, e, ie, ne, m, f\) .)11 2160(Call TSD1\(p,)1 780 2 1200 5112 t 10 R f ( Grosse and)2 491(This is general purpose, multi-dimensional tensor spline evaluation software written by E. H.)12 3829 2 720 5292 t (distributed with)1 636 1 720 5412 t 10 CW f (TTGR)1386 5412 w 10 R f ( 1 in Appendix 4)4 688( Example)1 409(to make evaluation easier for both users and the authors.)9 2287 3 1656 5412 t (shows)720 5532 w 10 CW f (TSD1)996 5532 w 10 R f ( should think of)3 637( argument descriptions below are from that software, and users)9 2520( The)1 206(at work.)1 328 4 1262 5532 t 10 CW f (p)4980 5532 w 10 R f ( input to)2 334( The)1 205(as 2.)1 183 3 720 5652 t 10 CW f (TSD1)1467 5652 w 10 R f (is)1732 5652 w 10 CW f (p)770 5808 w 10 R f ( number of coordinates, that is, 2.)6 1340(- The)1 305 2 1070 5808 t 10 CW f (k)770 5964 w 10 R f ( that)1 175( Note)1 244( order of the tensor product spline.)6 1378(- The)1 305 4 1070 5964 t 10 CW f (k)3197 5964 w 10 R f (,)3257 5964 w 10 CW f (it)3307 5964 w 10 R f (,)3427 5964 w 10 CW f (nt)3477 5964 w 10 R f (,)3597 5964 w 10 CW f (ie)3647 5964 w 10 R f (,)3767 5964 w 10 CW f (ne)3818 5964 w 10 R f (and)3964 5964 w 10 CW f (m)4134 5964 w 10 R f (are vectors of length)3 820 1 4220 5964 t 10 CW f (p)1220 6084 w 10 R f ( of)1 108( Think)1 289(with an independent value for each coordinate.)6 1870 3 1305 6084 t 10 CW f (k\(1\)=kx)3597 6084 w 10 R f (and)4042 6084 w 10 CW f (k\(2\)=ky)4211 6084 w 10 R f (.)4631 6084 w 10 CW f (t)770 6240 w 10 R f ( containing the meshes for each spatial coordinate.)7 2074( array)1 236(- An)1 272 3 1070 6240 t 10 CW f (t\(it\(1\)\),...,t\(it\(1\)-)3780 6240 w (1+nt\(1\)\))1220 6360 w 10 R f (contains the mesh for the first coordinate \()7 1848 1 1748 6360 t 10 I f (x x)1 44 1 3596 6360 t 10 R f (\),)3640 6360 w 10 CW f (t\(it\(2\)\),...,t\(it\(2\)-)3780 6360 w (1+nt\(2\)\))1220 6480 w 10 R f (for the second \()3 623 1 1750 6480 t 10 I f (y y)1 44 1 2373 6480 t 10 R f (\), and so on.)3 491 1 2417 6480 t 10 CW f (it)770 6636 w 10 R f ( pointers to the meshes in each coordinate, as used above.)10 2301(- The)1 305 2 1070 6636 t 10 CW f (nt)770 6792 w 10 R f ( of mesh points in each dimension.)6 1391(- Number)1 477 2 1070 6792 t 10 CW f (a)770 6948 w 10 R f ( coefficients, stored as if dimensioned)5 1514(- B-spline)1 489 2 1070 6948 t 10 CW f (\( nt\(1\)-k\(1\),...,nt\(p\)-k\(p\) \))2 1740 1 3133 6948 t 10 R f (.)4873 6948 w cleartomark showpage saveobj restore end %%EndPage: 15 15 %%Page: 16 16 DpostDict begin /saveobj save def mark 16 pagesetup 10 R f (- 16 -)2 216 1 2772 480 t 10 CW f (e)770 840 w 10 R f ( is stored like)3 536( It)1 111( grid on which evaluation is to be done.)8 1579(- The)1 305 4 1070 840 t 10 CW f (t)3626 840 w 10 R f (,)3686 840 w 10 CW f (it)3736 840 w 10 R f (and)3881 840 w 10 CW f (nt)4050 840 w 10 R f (.)4170 840 w 10 CW f (ie)770 996 w 10 R f ( indices in each dimension, as with)6 1396(- Evaluation)1 583 2 1070 996 t 10 CW f (it)3074 996 w 10 R f (above.)3219 996 w 10 CW f (ne)770 1152 w 10 R f ( number of evaluation points in each dimension.)7 1926(- The)1 305 2 1070 1152 t 10 CW f (m)770 1308 w 10 R f ( of the partial derivatives. See)5 1192(- Order)1 382 2 1070 1308 t 10 CW f (f)2669 1308 w 10 R f (below.)2754 1308 w (The output of)2 544 1 720 1464 t 10 CW f (TSD1)1289 1464 w 10 R f (is)1554 1464 w 10 CW f (f)770 1620 w 10 R f ( values, stored as if dimensioned)5 1319(- Derivative)1 571 2 1070 1620 t 10 CW f ( \))1 124(\( ne\(1\),...,ne\(p\))1 1023 2 3023 1620 t 10 R f ( derivative is of)3 636(. The)1 234 2 4170 1620 t (order)1220 1740 w 10 CW f (m\(1\))1464 1740 w 10 R f (with respect to the first coordinate,)5 1437 1 1738 1740 t 10 CW f (m\(2\))3209 1740 w 10 R f ( on.)1 158(with respect to the second, and so)6 1399 2 3483 1740 t (That is,)1 305 1 1220 1910 t 10 CW f (f\(i,j\))1555 1910 w 10 S f (= =)1 55 1 1946 1910 t (\266)2075 1990 w 10 I f (x x)1 44 1 2132 1990 t 7 I f (m m)1 50 1 2187 1950 t 7 R f (\( 1 \))2 91 1 2242 1950 t 10 S f (\266)2104 1850 w 7 I f (m m)1 50 1 2158 1810 t 7 R f (\( 1 \))2 91 1 2213 1810 t 10 S1 f (_ _____)1 296 1 2060 1880 t 10 S f (\266)2440 1990 w 10 I f (y y)1 44 1 2497 1990 t 7 I f (m m)1 50 1 2552 1950 t 7 R f (\( 2 \))2 91 1 2607 1950 t 10 S f (\266)2469 1850 w 7 I f (m m)1 50 1 2523 1810 t 7 R f (\( 2 \))2 91 1 2578 1810 t 10 S1 f (_ _____)1 296 1 2425 1880 t 10 I f (s s)1 39 1 2747 1910 t 10 R f (\()2794 1910 w 10 I f (x x)1 44 1 2835 1910 t 7 I f (i i)1 20 1 2890 1930 t 10 R f (,)2926 1910 w 10 I f (y y)1 44 1 2983 1910 t 7 I f (j j)1 20 1 3038 1930 t 10 R f (\), where)1 332 1 3074 1910 t 10 I f (s s)1 39 1 3437 1910 t 10 R f (is the spline whose coefficients are in)6 1533 1 3507 1910 t 10 CW f (a)1220 2090 w 10 R f (, for)1 172 1 1280 2090 t 10 CW f (i)1483 2090 w 10 S f (= =)1 55 1 1584 2090 t 10 R f (1 ,)1 83 1 1688 2090 t 10 I f (. .)1 25 1 1779 2090 t 10 R f (.. ,)1 83 1 1804 2090 t 10 CW f (ne\(1\))1887 2090 w 10 R f (and)2218 2090 w 10 CW f (j)2393 2090 w 10 S f (= =)1 55 1 2494 2090 t 10 R f (1 ,)1 83 1 2598 2090 t 10 I f (. .)1 25 1 2689 2090 t 10 R f (.. ,)1 83 1 2714 2090 t 10 CW f (ne\(2\))2797 2090 w 10 R f ( means that)2 467(. This)1 259 2 3097 2090 t 10 CW f (m = \(0,0\))2 552 1 3854 2090 t 10 R f (gives the func-)2 603 1 4437 2090 t (tion value, for example.)3 951 1 1220 2210 t (The double precision version of)4 1270 1 970 2366 t 10 CW f (TSD1)2265 2366 w 10 R f (is)2530 2366 w 10 CW f (DTSD1)2622 2366 w 10 R f (with all Real arguments typed double precision.)6 1912 1 2947 2366 t 10 B f (Obtaining Initial Conditions.)2 1240 1 720 2606 t 10 R f (If your initial conditions are constant, setting the initial values for)10 2783 1 970 2762 t 10 B f (U)3794 2762 w 10 R f ( easy: simply set)3 715(in \(3.1\) is)2 418 2 3907 2762 t 10 I f (U U)1 72 1 720 2882 t 7 I f (q qp p)2 70 1 803 2902 t 10 S f (\272)922 2882 w 10 R f ( your initial data is not constant, then you can)9 1821(constant. If)1 474 2 1035 2882 t 10 CW f (Call TSL2W\(p, k, t, it, nt, e, ie, w, iw, ne, f, a\))12 3060 1 1200 3062 t 10 R f ( Grosse [19] and)3 686(This is general purpose, multi-dimensional tensor spline fitting software written by E. H.)12 3634 2 720 3242 t (distributed with)1 645 1 720 3362 t 10 CW f (TTGR)1404 3362 w 10 R f ( argument descriptions)2 936( The)1 218( and the authors.)3 699(to make fitting easier for both users)6 1504 4 1683 3362 t (below are from that software, and users should think of)9 2207 1 720 3482 t 10 CW f (p)2952 3482 w 10 R f ( input to)2 334( The)1 205(as 2.)1 183 3 3037 3482 t 10 CW f (TSL2W)3784 3482 w 10 R f (is)4109 3482 w 10 CW f (p)770 3638 w 10 R f ( number of coordinates, that is, 2.)6 1340(- The)1 305 2 1070 3638 t 10 CW f (k)770 3794 w 10 R f ( that)1 185( Note)1 254( the tensor product spline.)4 1075( order of)2 361(- The)1 305 5 1070 3794 t 10 CW f (k)3285 3794 w 10 R f (,)3345 3794 w 10 CW f (it)3405 3794 w 10 R f (,)3525 3794 w 10 CW f (nt)3585 3794 w 10 R f (,)3705 3794 w 10 CW f (ie)3765 3794 w 10 R f (and)3920 3794 w 10 CW f (ne)4099 3794 w 10 R f (are vectors with an)3 786 1 4254 3794 t (independent value for each coordinate.)4 1548 1 1220 3914 t 10 CW f (t)770 4070 w 10 R f ( chapter on approx-)3 784( the introduction to the PORT)5 1197( See)1 195( mesh on which the spline is defined.)7 1489(- The)1 305 5 1070 4070 t ( the mesh;)2 437(imation for advice on laying out)5 1355 2 1220 4190 t 10 CW f (t\(it\(1\)\),...,t\(it\(1\)-1+nt\(1\)\))3085 4190 w 10 R f (con-)4863 4190 w ( coordinate,)1 472(tains the mesh for the first)5 1046 2 1220 4310 t 10 CW f (t\(it\(2\)\),...,t\(it\(2\)-1+nt\(2\)\))2799 4310 w 10 R f (for the sec-)2 450 1 4590 4310 t (ond, and so on.)3 608 1 1220 4430 t 10 CW f (it)770 4586 w 10 R f ( point indices.)2 564(- Mesh)1 372 2 1070 4586 t 10 CW f (nt)770 4742 w 10 R f ( of mesh points in each dimension.)6 1391(- Number)1 477 2 1070 4742 t 10 CW f (e)770 4898 w 10 R f ( is stored like)3 536( It)1 111( grid on which data is defined.)6 1212(- The)1 305 4 1070 4898 t 10 CW f (t)3259 4898 w 10 R f (,)3319 4898 w 10 CW f (it)3369 4898 w 10 R f (and)3514 4898 w 10 CW f (nt)3683 4898 w 10 R f (.)3803 4898 w 10 CW f (ie)770 5054 w 10 R f ( indices in each dimension.)4 1085(- Data)1 338 2 1070 5054 t 10 CW f (w)770 5210 w 10 R f ( stored like)2 456(- Weights,)1 508 2 1070 5210 t 10 CW f (e)2065 5210 w 10 R f ( for the)2 302( weight)1 303(. The)1 236 3 2125 5210 t 10 CW f (\(i1,i2,...\))2998 5210 w 10 R f (data point is)2 503 1 3690 5210 t 10 CW f (w\(iw\(1\)-1+i1\))4260 5210 w (* w\(iw\(2\)-1+i2\) * ...)3 1260 1 1220 5330 t (iw)770 5486 w 10 R f ( for the weights in each dimension.)6 1401(- Indices)1 438 2 1070 5486 t 10 CW f (ne)770 5642 w 10 R f ( number of data points in each dimension.)7 1676(- The)1 305 2 1070 5642 t 10 CW f (f)770 5798 w 10 R f ( values, stored as if dimensioned)5 1304(- Data)1 338 2 1070 5798 t 10 CW f (\(ne\(1\),...,ne\(p\)\))2737 5798 w 10 R f (.)3757 5798 w (The output of)2 544 1 720 5954 t 10 CW f (TSL2W)1289 5954 w 10 R f (is)1614 5954 w 10 CW f (a)770 6110 w 10 R f ( coefficients, stored as if dimensioned)5 1514(- B-spline)1 489 2 1070 6110 t 10 CW f (\(nt\(1\)-k\(1\), ... , nt\(p\)-k\(p\)\))3 1800 1 3098 6110 t 10 R f (.)4898 6110 w (The double precision version of)4 1270 1 970 6266 t 10 CW f (TSL2W)2265 6266 w 10 R f (is)2590 6266 w 10 CW f (DTSL2W)2682 6266 w 10 R f (with all Real arguments typed double precision.)6 1912 1 3067 6266 t 10 B f (Other Ways to Use and Speed-Up)5 1436 1 720 6506 t 10 CW f (TTGR)2181 6506 w 10 B f (.)2421 6506 w 10 R f (There are many "knobs" in)4 1074 1 970 6662 t 10 CW f (TTGR)2069 6662 w 10 R f (; section 6 describes their use.)5 1198 1 2309 6662 t cleartomark showpage saveobj restore end %%EndPage: 16 16 %%Page: 17 17 DpostDict begin /saveobj save def mark 17 pagesetup 10 R f (- 17 -)2 216 1 2772 480 t 10 B f (Trouble in)1 454 1 720 840 t 10 CW f (AF)1199 840 w 10 B f (or)1344 840 w 10 CW f (BC)1463 840 w 10 B f (.)1583 840 w 10 R f ( or another, the user cannot evaluate the appropriate functions when)10 2787(If, for one reason)3 707 2 970 996 t 10 CW f (AF)4497 996 w 10 R f (or)4650 996 w 10 CW f (BC)4766 996 w 10 R f (are)4919 996 w (called, this fact can be communicated to)6 1605 1 720 1116 t 10 CW f (TTGR)2350 1116 w 10 R f (through the named Common region)4 1427 1 2615 1116 t 10 CW f (Common / TTGRF / Failed ; Logical Failed .)8 2520 1 1080 1296 t 10 R f (Before)720 1512 w 10 CW f (TTGR)1020 1512 w 10 R f (calls)1289 1512 w 10 CW f (AF)1501 1512 w 10 R f (or)1650 1512 w 10 CW f (BC)1762 1512 w 10 R f (, it sets)2 289 1 1882 1512 t 10 CW f (Failed)2200 1512 w 10 R f (=)2589 1512 w 10 CW f (.False.)2674 1512 w 10 R f ( if the user doesn't use or even know of)9 1613(. Thus,)1 304 2 3123 1512 t (the existence of)2 634 1 720 1632 t 10 CW f (TTGRF)1382 1632 w 10 R f (,)1682 1632 w 10 CW f (TTGR)1770 1632 w 10 R f (assumes that everything has been correctly computed on return from those)10 3002 1 2038 1632 t ( however, the user has a problem, uses)7 1580(subroutines. If,)1 633 2 720 1752 t 10 CW f (TTGRF)2965 1752 w 10 R f (, and sets)2 383 1 3265 1752 t 10 CW f (Failed)3680 1752 w 10 R f (=)4072 1752 w 10 CW f (.True.)4160 1752 w 10 R f (,)4520 1752 w 10 CW f (TTGR)4612 1752 w 10 R f (will)4884 1752 w ( in an attempt to obtain a more accurate, and hence more reasonable,)12 2929(automatically lower the time-step)3 1391 2 720 1872 t (numerical solution so that)3 1036 1 720 1992 t 10 CW f (AF)1781 1992 w 10 R f (and)1926 1992 w 10 CW f (BC)2095 1992 w 10 R f (can do their job.)3 649 1 2240 1992 t (The Double Precision version of)4 1298 1 970 2148 t 10 CW f (TTGRF)2293 2148 w 10 R f (is)2618 2148 w 10 CW f (DTTGRF)2710 2148 w 10 R f (.)3070 2148 w 10 B f (Mapping Software)1 797 1 720 2388 t 10 R f ( How-)1 285( rectangle, with conformal mapping preferable.)5 1921(There are many ways to map ink-blots onto a)8 1864 3 970 2544 t ( and is not strictly speaking necessary when solving)8 2182(ever, conformal mapping is a tricky business)6 1879 2 720 2664 t 10 B f (pde)4820 2664 w 10 R f (s.)4976 2664 w ( suppose you want to map a domain)7 1505( example,)1 397( For)1 198(Much simpler non-conformal maps can often be used.)7 2220 4 720 2784 t (like)720 2904 w (R\(t\) L\(t\))1 -1157 1 3428 5120 t (f\(t,x\))2892 4785 w (g\(t,x\))2471 3754 w (x)3922 5223 w (y)1833 3122 w 1861 5181 1861 3147 Dl 1835 5181 1 1 De 3895 5181 1861 5181 Dl 3895 5180 3870 5155 Dl 3895 5181 3870 5181 Dl 3895 5181 3870 5181 Dl 3870 5207 1 1 De 3870 5207 1 1 De 3895 5182 3870 5207 Dl 1809 3173 1 1 De 1860 3148 1835 3173 Dl 1886 3172 1861 3147 Dl 2222 3999 2222 4433 Dl 3508 4535 3508 3662 Dl 2222 5130 2222 5207 Dl 3508 5206 3508 5129 Dl 2222 3998 2222 3998 2222 3998 Ds 2222 3998 2222 3998 2608 3741 Ds 2222 3998 2608 3741 3017 3972 Ds 2608 3741 3017 3972 3248 3612 Ds 3017 3972 3248 3612 3505 3663 Ds 3248 3612 3505 3663 3505 3663 Ds 2222 4433 2222 4433 2222 4433 Ds 2222 4433 2222 4433 2505 4458 Ds 2222 4433 2505 4458 2711 4689 Ds 2505 4458 2711 4689 3275 4689 Ds 2711 4689 3275 4689 3403 4561 Ds 3275 4689 3403 4561 3505 4536 Ds 3403 4561 3505 4536 3505 4536 Ds 10 B f (Figure 2)1 358 1 2826 5445 t 10 R f (onto the unit rectangle, where)4 1189 1 720 5565 t 10 I f (f f)1 28 1 1934 5565 t 10 R f (and)1987 5565 w 10 I f (g g)1 50 1 2156 5565 t 10 R f (are nice, smooth functions of)4 1162 1 2231 5565 t 10 I f (x x)1 44 1 3418 5565 t 10 R f ( mapping)1 375(. The)1 230 2 3462 5565 t 10 I f (y y)1 44 1 1220 5865 t 10 S f (= =)1 55 1 1313 5865 t 10 I f (f f)1 28 1 1425 5865 t 10 R f (\()1501 5865 w 10 I f (t t)1 28 1 1566 5865 t 10 R f (,)1602 5865 w 10 I f (x x)1 44 1 1635 5865 t 10 R f (\))1711 5865 w 10 S f ( h)1 109(+ +)1 55 2 1801 5865 t 10 R f (\()1997 5865 w 10 I f (g g)1 50 1 2062 5865 t 10 R f (\()2120 5865 w 10 I f (t t)1 28 1 2161 5865 t 10 R f (,)2197 5865 w 10 I f (x x)1 44 1 2230 5865 t 10 R f (\))2282 5865 w 10 S f (- -)1 55 1 2363 5865 t 10 I f (f f)1 28 1 2466 5865 t 10 R f (\()2510 5865 w 10 I f (t t)1 28 1 2551 5865 t 10 R f (,)2587 5865 w 10 I f (x x)1 44 1 2620 5865 t 10 R f (\) \))1 106 1 2672 5865 t 10 I f (x x)1 44 1 1220 5725 t 10 S f (= =)1 55 1 1313 5725 t 10 I f (L L)1 56 1 1417 5725 t 10 R f (\()1481 5725 w 10 I f (t t)1 28 1 1522 5725 t 10 R f (\))1558 5725 w 10 S f ( x)1 65(+ +)1 55 2 1607 5725 t 10 R f (\()1759 5725 w 10 I f (R R)1 61 1 1824 5725 t 10 R f (\()1893 5725 w 10 I f (t t)1 28 1 1934 5725 t 10 R f (\))1970 5725 w 10 S f (- -)1 55 1 2051 5725 t 10 I f (L L)1 56 1 2146 5725 t 10 R f (\()2210 5725 w 10 I f (t t)1 28 1 2251 5725 t 10 R f (\) \))1 106 1 2287 5725 t (\(4.1\))4849 5815 w (takes the unit square in)4 937 1 720 6025 t 10 S f (x)1686 6025 w 10 R f (,)1767 6025 w 10 S f (h)1800 6025 w 10 R f ( implies)1 325( This)1 233( the mapping is smooth.)4 979( Moreover,)1 472(into the domain of Figure 2.)5 1142 5 1889 6025 t (that the mesh in)3 636 1 720 6145 t 10 S f (x)1381 6145 w 10 R f (,)1462 6145 w 10 S f (h)1495 6145 w 10 R f (need only be fine where the domain or the solution is kinky.)11 2403 1 1580 6145 t ( have been canned for use by)6 1217(Mappings like the one above are so common that they)9 2242 2 970 6301 t 10 CW f (TTGR)4464 6301 w 10 R f ( you)1 185(. If)1 151 2 4704 6301 t (wish to map some)3 725 1 720 6421 t 10 S f (x)1470 6421 w 10 R f (,)1551 6421 w 10 S f (h)1584 6421 w 10 R f (pairs into)1 375 1 1669 6421 t 10 I f (x x)1 44 1 2069 6421 t 10 R f (,)2121 6421 w 10 I f (y y)1 44 1 2154 6421 t 10 R f (pairs, then the)2 563 1 2223 6421 t 10 CW f (Call BTMAP\(t,c,e,nx,ny, LR,BT, x,y,D\))3 2220 1 1200 6601 t 10 R f ( input arguments are)3 818( The)1 205(will do the job.)3 606 3 720 6781 t 10 CW f (t)770 6937 w 10 R f ( time)1 203(- The)1 305 2 1120 6937 t 10 I f (t t)1 28 1 1653 6937 t 10 R f (.)1681 6937 w cleartomark showpage saveobj restore end %%EndPage: 17 17 %%Page: 18 18 DpostDict begin /saveobj save def mark 18 pagesetup 10 R f (- 18 -)2 216 1 2772 480 t 10 CW f (c)770 840 w 10 R f (- The)1 305 1 1120 840 t 10 S f (x)1450 840 w 10 R f (values to be used in the mapping.)6 1335 1 1524 840 t 10 CW f (e)770 996 w 10 R f (- The)1 305 1 1120 996 t 10 S f (h)1450 996 w 10 R f (values to be used in the mapping.)6 1335 1 1535 996 t 10 CW f (nx)770 1152 w 10 R f ( length of the)3 530(- The)1 305 2 1120 1152 t 10 S f (x)1980 1152 w 10 R f (array.)2054 1152 w 10 CW f (ny)770 1308 w 10 R f ( length of the)3 530(- The)1 305 2 1120 1308 t 10 S f (h)1980 1308 w 10 R f (array.)2065 1308 w 10 CW f (LR)770 1464 w 10 R f ( name of a subroutine, declared)5 1317(- The)1 305 2 1120 1464 t 10 CW f (External)2780 1464 w 10 R f (in the program calling)3 927 1 3299 1464 t 10 CW f (BTMAP)4265 1464 w 10 R f (, for giving)2 475 1 4565 1464 t (information about)1 719 1 1270 1584 t 10 I f (L L)1 56 1 2014 1584 t 10 R f (\()2078 1584 w 10 I f (t t)1 28 1 2119 1584 t 10 R f (\) and)1 202 1 2155 1584 t 10 I f (R R)1 61 1 2382 1584 t 10 R f (\()2451 1584 w 10 I f (t t)1 28 1 2492 1584 t 10 R f (\).)2528 1584 w (When)1270 1740 w 10 CW f (BTMAP)1533 1740 w 10 R f (wants the values of)3 768 1 1858 1740 t 10 I f (L L)1 56 1 2651 1740 t 10 R f (\()2715 1740 w 10 I f (t t)1 28 1 2756 1740 t 10 R f (\) and)1 202 1 2792 1740 t 10 I f (R R)1 61 1 3019 1740 t 10 R f (\()3088 1740 w 10 I f (t t)1 28 1 3129 1740 t 10 R f (\) it will)2 295 1 3165 1740 t 10 CW f (Call LR\(t,L,R,Lt,Rt\))1 1200 1 1630 1920 t 10 R f (The input to)2 489 1 1270 2100 t 10 CW f (LR)1784 2100 w 10 R f (is)1929 2100 w 10 CW f (t)1320 2256 w 10 R f (- Time)1 361 1 1670 2256 t 10 I f (t t)1 28 1 2056 2256 t 10 R f (.)2084 2256 w (The output from)2 655 1 1270 2412 t 10 CW f (LR)1950 2412 w 10 R f (is)2095 2412 w 10 CW f (L)1320 2568 w 10 R f ( value of)2 349(- The)1 305 2 1670 2568 t 10 I f (L L)1 56 1 2349 2568 t 10 R f (\()2413 2568 w 10 I f (t t)1 28 1 2454 2568 t 10 R f (\).)2490 2568 w 10 CW f (R)1320 2724 w 10 R f ( value of)2 349(- The)1 305 2 1670 2724 t 10 I f (R R)1 61 1 2349 2724 t 10 R f (\()2418 2724 w 10 I f (t t)1 28 1 2459 2724 t 10 R f (\).)2495 2724 w 10 CW f (Lt)1320 2880 w 10 R f ( value of)2 349(- The)1 305 2 1670 2880 t 10 I f (L L)1 56 1 2349 2880 t 10 S f (\242 \242)1 25 1 2413 2880 t 10 R f (\()2446 2880 w 10 I f (t t)1 28 1 2487 2880 t 10 R f (\).)2523 2880 w 10 CW f (Rt)1320 3036 w 10 R f ( value of)2 349(- The)1 305 2 1670 3036 t 10 I f (R R)1 61 1 2349 3036 t 10 S f (\242 \242)1 25 1 2418 3036 t 10 R f (\()2451 3036 w 10 I f (t t)1 28 1 2492 3036 t 10 R f (\).)2528 3036 w 10 CW f (BT)770 3192 w 10 R f ( name of a subroutine, declared)5 1317(- The)1 305 2 1120 3192 t 10 CW f (External)2780 3192 w 10 R f (in the program calling)3 927 1 3299 3192 t 10 CW f (BTMAP)4265 3192 w 10 R f (, for giving)2 475 1 4565 3192 t (information about)1 719 1 1270 3312 t 10 I f (f f)1 28 1 2014 3312 t 10 R f (\()2058 3312 w 10 I f (t t)1 28 1 2099 3312 t 10 R f (,)2135 3312 w 10 I f (x x)1 44 1 2168 3312 t 10 R f (\) and)1 202 1 2220 3312 t 10 I f (g g)1 50 1 2447 3312 t 10 R f (\()2505 3312 w 10 I f (t t)1 28 1 2546 3312 t 10 R f (,)2582 3312 w 10 I f (x x)1 44 1 2615 3312 t 10 R f (\).)2667 3312 w (When)1270 3468 w 10 CW f (BTMAP)1533 3468 w 10 R f (wants to evaluate)2 693 1 1858 3468 t 10 I f (f f)1 28 1 2576 3468 t 10 R f (and)2629 3468 w 10 I f (g g)1 50 1 2798 3468 t 10 R f (it will)1 237 1 2873 3468 t 10 CW f (Call BT\(t,x,f,g,fx,gx,ft,gt\))1 1680 1 1630 3648 t 10 R f (The input to)2 489 1 1270 3828 t 10 CW f (BT)1784 3828 w 10 R f (is)1929 3828 w 10 CW f (t)1320 3984 w 10 R f ( time)1 203(- The)1 305 2 1670 3984 t 10 I f (t t)1 28 1 2203 3984 t 10 R f (.)2231 3984 w 10 CW f (x)1320 4140 w 10 R f ( point)1 231(- The)1 305 2 1670 4140 t 10 I f (x x)1 44 1 2231 4140 t 10 R f (for which)1 385 1 2300 4140 t 10 I f (f f)1 28 1 2710 4140 t 10 R f (\()2754 4140 w 10 I f (t t)1 28 1 2795 4140 t 10 R f (,)2831 4140 w 10 I f (x x)1 44 1 2864 4140 t 10 R f (\) and)1 202 1 2916 4140 t 10 I f (g g)1 50 1 3143 4140 t 10 R f (\()3201 4140 w 10 I f (t t)1 28 1 3242 4140 t 10 R f (,)3278 4140 w 10 I f (x x)1 44 1 3311 4140 t 10 R f (\) are desired.)2 517 1 3363 4140 t (The output from)2 655 1 1270 4296 t 10 CW f (BT)1950 4296 w 10 R f (is)2095 4296 w 10 CW f (f)1320 4452 w 10 R f ( value of)2 349(- The)1 305 2 1670 4452 t 10 I f (f f)1 28 1 2349 4452 t 10 R f (\()2393 4452 w 10 I f (t t)1 28 1 2434 4452 t 10 R f (,)2470 4452 w 10 I f (x x)1 44 1 2503 4452 t 10 R f (\).)2555 4452 w 10 CW f (g)1320 4608 w 10 R f ( value of)2 349(- The)1 305 2 1670 4608 t 10 I f (g g)1 50 1 2349 4608 t 10 R f (\()2407 4608 w 10 I f (t t)1 28 1 2448 4608 t 10 R f (,)2484 4608 w 10 I f (x x)1 44 1 2517 4608 t 10 R f (\).)2569 4608 w 10 CW f (fx)1320 4804 w 10 R f ( value of)2 349(- The)1 305 2 1670 4804 t 10 S f (\266)2374 4874 w 10 I f (x x)1 44 1 2431 4874 t 10 S f (\266)2378 4744 w 10 I f (f f)1 28 1 2443 4744 t 10 S1 f (_ __)1 131 1 2359 4774 t 10 R f (\()2516 4804 w 10 I f (t t)1 28 1 2557 4804 t 10 R f (,)2593 4804 w 10 I f (x x)1 44 1 2626 4804 t 10 R f (\).)2678 4804 w 10 CW f (gx)1320 5050 w 10 R f ( value of)2 349(- The)1 305 2 1670 5050 t 10 S f (\266)2377 5120 w 10 I f (x x)1 44 1 2434 5120 t 10 S f (\266)2374 4990 w 10 I f (g g)1 50 1 2431 4990 t 10 S1 f (_ __)1 137 1 2359 5020 t 10 R f (\()2514 5050 w 10 I f (t t)1 28 1 2555 5050 t 10 R f (,)2591 5050 w 10 I f (x x)1 44 1 2624 5050 t 10 R f (\).)2676 5050 w 10 CW f (ft)1320 5296 w 10 R f ( value of)2 349(- The)1 305 2 1670 5296 t 10 S f (\266)2378 5366 w 10 I f (t t)1 28 1 2435 5366 t 10 S f (\266)2374 5236 w 10 I f (f f)1 28 1 2439 5236 t 10 S1 f (_ __)1 123 1 2359 5266 t 10 R f (\()2508 5296 w 10 I f (t t)1 28 1 2549 5296 t 10 R f (,)2585 5296 w 10 I f (x x)1 44 1 2618 5296 t 10 R f (\).)2670 5296 w 10 CW f (gt)1320 5542 w 10 R f ( value of)2 349(- The)1 305 2 1670 5542 t 10 S f (\266)2385 5612 w 10 I f (t t)1 28 1 2442 5612 t 10 S f (\266)2374 5482 w 10 I f (g g)1 50 1 2431 5482 t 10 S1 f (_ __)1 137 1 2359 5512 t 10 R f (\()2514 5542 w 10 I f (t t)1 28 1 2555 5542 t 10 R f (,)2591 5542 w 10 I f (x x)1 44 1 2624 5542 t 10 R f (\).)2676 5542 w (The output from)2 655 1 720 5748 t 10 CW f (BTMAP)1400 5748 w 10 R f (is)1725 5748 w 10 CW f (x)770 5904 w 10 R f (-)1120 5904 w 10 CW f (x)1270 5904 w 10 R f (\()1330 5904 w 10 I f (p p)1 50 1 1363 5904 t 10 R f (,)1421 5904 w 10 I f (q q)1 50 1 1454 5904 t 10 R f (\))1504 5904 w 10 S f (= =)1 55 1 1562 5904 t 10 I f (x x)1 44 1 1666 5904 t 10 R f (\()1718 5904 w 10 S f (x)1759 5904 w 10 R f (\()1816 5904 w 10 I f (p p)1 50 1 1857 5904 t 10 R f (\) ,)1 74 1 1915 5904 t 10 S f (h)2021 5904 w 10 R f (\()2113 5904 w 10 I f (q q)1 50 1 2154 5904 t 10 R f ( for)1 141(\) \),)1 131 2 2212 5904 t 10 I f (p p)1 50 1 2509 5904 t 10 S f (= =)1 55 1 2583 5904 t 10 R f (1 ,)1 83 1 2654 5904 t 10 I f (. .)1 25 1 2745 5904 t 10 R f (.. ,)1 83 1 2770 5904 t 10 CW f (nx)2877 5904 w 10 R f (and)3022 5904 w 10 I f (q q)1 50 1 3191 5904 t 10 S f (= =)1 55 1 3265 5904 t 10 R f (1 ,)1 83 1 3336 5904 t 10 I f (. .)1 25 1 3427 5904 t 10 R f (.. ,)1 83 1 3452 5904 t 10 CW f (ny)3584 5904 w 10 R f (.)3704 5904 w 10 CW f (y)770 6060 w 10 R f (-)1120 6060 w 10 CW f (y)1270 6060 w 10 R f (\()1330 6060 w 10 I f (p p)1 50 1 1363 6060 t 10 R f (,)1421 6060 w 10 I f (q q)1 50 1 1454 6060 t 10 R f (\))1504 6060 w 10 S f (= =)1 55 1 1562 6060 t 10 I f (y y)1 44 1 1666 6060 t 10 R f (\()1718 6060 w 10 S f (x)1759 6060 w 10 R f (\()1816 6060 w 10 I f (p p)1 50 1 1857 6060 t 10 R f (\) ,)1 74 1 1915 6060 t 10 S f (h)2021 6060 w 10 R f (\()2113 6060 w 10 I f (q q)1 50 1 2154 6060 t 10 R f ( for)1 141(\) \),)1 131 2 2212 6060 t 10 I f (p p)1 50 1 2509 6060 t 10 S f (= =)1 55 1 2583 6060 t 10 R f (1 ,)1 83 1 2654 6060 t 10 I f (. .)1 25 1 2745 6060 t 10 R f (.. ,)1 83 1 2770 6060 t 10 CW f (nx)2877 6060 w 10 R f (and)3022 6060 w 10 I f (q q)1 50 1 3191 6060 t 10 S f (= =)1 55 1 3265 6060 t 10 R f (1 ,)1 83 1 3336 6060 t 10 I f (. .)1 25 1 3427 6060 t 10 R f (.. ,)1 83 1 3452 6060 t 10 CW f (ny)3584 6060 w 10 R f (.)3704 6060 w 10 CW f (D)770 6216 w 10 R f ( partial derivatives of)3 872( array giving the)3 675(- An)1 272 3 1120 6216 t 10 I f (x x)1 44 1 2971 6216 t 10 R f (and)3047 6216 w 10 I f (y y)1 44 1 3223 6216 t 10 R f (wrt.)3299 6216 w 10 S f (x)3489 6216 w 10 R f (,)3538 6216 w 10 S f (h)3595 6216 w 10 R f (and)3687 6216 w 10 I f (t t)1 28 1 3863 6216 t 10 R f ( information is neces-)3 889(. This)1 260 2 3891 6216 t (sary in the)2 416 1 1270 6336 t 10 B f (pde)1711 6336 w 10 R f (mapping software to be described shortly.)5 1675 1 1892 6336 t cleartomark showpage saveobj restore end %%EndPage: 18 18 %%Page: 19 19 DpostDict begin /saveobj save def mark 19 pagesetup 10 R f (- 19 -)2 216 1 2772 480 t 10 CW f (D)1270 940 w 10 R f (\()1330 940 w 10 I f (p p)1 50 1 1363 940 t 10 R f (,)1421 940 w 10 I f (q q)1 50 1 1454 940 t 10 R f (, 1 , 1\))3 207 1 1512 940 t 10 S f (= =)1 55 1 1744 940 t (\266 x)1 106 1 1873 1010 t (\266)1876 880 w 10 I f (x x)1 44 1 1933 880 t 10 S1 f (_ __)1 136 1 1859 910 t 10 R f (\()2037 940 w 10 I f (t t)1 28 1 2078 940 t 10 R f (,)2114 940 w 10 S f (x)2147 940 w 10 R f (\()2204 940 w 10 I f (p p)1 50 1 2245 940 t 10 R f (\) ,)1 74 1 2303 940 t 10 S f (h)2385 940 w 10 R f (\()2477 940 w 10 I f (q q)1 50 1 2518 940 t 10 R f (\) \))1 74 1 2576 940 t 10 CW f (D)1270 1150 w 10 R f (\()1330 1150 w 10 I f (p p)1 50 1 1363 1150 t 10 R f (,)1421 1150 w 10 I f (q q)1 50 1 1454 1150 t 10 R f (, 1 , 2\))3 207 1 1512 1150 t 10 S f (= =)1 55 1 1744 1150 t (\266 h)1 117 1 1873 1220 t (\266)1881 1090 w 10 I f (x x)1 44 1 1938 1090 t 10 S1 f (_ __)1 147 1 1858 1120 t 10 R f (\()2047 1150 w 10 I f (t t)1 28 1 2088 1150 t 10 R f (,)2124 1150 w 10 S f (x)2157 1150 w 10 R f (\()2214 1150 w 10 I f (p p)1 50 1 2255 1150 t 10 R f (\) ,)1 74 1 2313 1150 t 10 S f (h)2395 1150 w 10 R f (\()2487 1150 w 10 I f (q q)1 50 1 2528 1150 t 10 R f (\) \))1 74 1 2586 1150 t 10 CW f (D)1270 1360 w 10 R f (\()1330 1360 w 10 I f (p p)1 50 1 1363 1360 t 10 R f (,)1421 1360 w 10 I f (q q)1 50 1 1454 1360 t 10 R f (, 1 , 3\))3 207 1 1512 1360 t 10 S f (= =)1 55 1 1744 1360 t (\266)1881 1430 w 10 I f (t t)1 28 1 1938 1430 t 10 S f (\266)1873 1300 w 10 I f (x x)1 44 1 1930 1300 t 10 S1 f (_ __)1 131 1 1858 1330 t 10 R f (\()2031 1360 w 10 I f (t t)1 28 1 2072 1360 t 10 R f (,)2108 1360 w 10 S f (x)2141 1360 w 10 R f (\()2198 1360 w 10 I f (p p)1 50 1 2239 1360 t 10 R f (\) ,)1 74 1 2297 1360 t 10 S f (h)2379 1360 w 10 R f (\()2471 1360 w 10 I f (q q)1 50 1 2512 1360 t 10 R f (\) \))1 74 1 2570 1360 t 10 CW f (D)1270 1570 w 10 R f (\()1330 1570 w 10 I f (p p)1 50 1 1363 1570 t 10 R f (,)1421 1570 w 10 I f (q q)1 50 1 1454 1570 t 10 R f (, 2 , 1\))3 207 1 1512 1570 t 10 S f (= =)1 55 1 1744 1570 t (\266 x)1 106 1 1873 1640 t (\266)1876 1510 w 10 I f (y y)1 44 1 1933 1510 t 10 S1 f (_ __)1 136 1 1859 1540 t 10 R f (\()2037 1570 w 10 I f (t t)1 28 1 2078 1570 t 10 R f (,)2114 1570 w 10 S f (x)2147 1570 w 10 R f (\()2204 1570 w 10 I f (p p)1 50 1 2245 1570 t 10 R f (\) ,)1 74 1 2303 1570 t 10 S f (h)2385 1570 w 10 R f (\()2477 1570 w 10 I f (q q)1 50 1 2518 1570 t 10 R f (\) \))1 74 1 2576 1570 t 10 CW f (D)1270 1780 w 10 R f (\()1330 1780 w 10 I f (p p)1 50 1 1363 1780 t 10 R f (,)1421 1780 w 10 I f (q q)1 50 1 1454 1780 t 10 R f (, 2 , 2\))3 207 1 1512 1780 t 10 S f (= =)1 55 1 1744 1780 t (\266 h)1 117 1 1873 1850 t (\266)1881 1720 w 10 I f (y y)1 44 1 1938 1720 t 10 S1 f (_ __)1 147 1 1858 1750 t 10 R f (\()2047 1780 w 10 I f (t t)1 28 1 2088 1780 t 10 R f (,)2124 1780 w 10 S f (x)2157 1780 w 10 R f (\()2214 1780 w 10 I f (p p)1 50 1 2255 1780 t 10 R f (\) ,)1 74 1 2313 1780 t 10 S f (h)2395 1780 w 10 R f (\()2487 1780 w 10 I f (q q)1 50 1 2528 1780 t 10 R f (\) \))1 74 1 2586 1780 t 10 CW f (D)1270 1990 w 10 R f (\()1330 1990 w 10 I f (p p)1 50 1 1363 1990 t 10 R f (,)1421 1990 w 10 I f (q q)1 50 1 1454 1990 t 10 R f (, 2 , 3\))3 207 1 1512 1990 t 10 S f (= =)1 55 1 1744 1990 t (\266)1881 2060 w 10 I f (t t)1 28 1 1938 2060 t 10 S f (\266)1873 1930 w 10 I f (y y)1 44 1 1930 1930 t 10 S1 f (_ __)1 131 1 1858 1960 t 10 R f (\()2031 1990 w 10 I f (t t)1 28 1 2072 1990 t 10 R f (,)2108 1990 w 10 S f (x)2141 1990 w 10 R f (\()2198 1990 w 10 I f (p p)1 50 1 2239 1990 t 10 R f (\) ,)1 74 1 2297 1990 t 10 S f (h)2379 1990 w 10 R f (\()2471 1990 w 10 I f (q q)1 50 1 2512 1990 t 10 R f (\) \))1 74 1 2570 1990 t ( In)1 135( rectangles.)1 456(The above example is given to show the ease with which domains can be mapped onto)15 3479 3 970 2256 t (fact, here is the)3 609 1 720 2376 t 10 CW f (Ratfor)1354 2376 w 10 R f (source for)1 401 1 1739 2376 t 10 CW f (BTMAP)2165 2376 w 10 R f (, which maps the domain of Figure 2 onto the unit square)11 2287 1 2465 2376 t 10 CW f (Subroutine BTMAP\(t,c,e,nx,ny, LR,BT, x,y,D\))3 2580 1 1200 2556 t (# To map \(c,e\): \(0,1\)x\(0,1\) to Left-right x Bottom-top.)8 3300 1 1080 2796 t (Double Precision t,c\(nx\),e\(ny\),x\(nx,ny\),y\(nx,ny\),D\(nx,ny,2,3\))2 3660 1 1200 3036 t (Integer nx,ny)1 780 1 1200 3156 t (External LR,BT)1 840 1 1200 3276 t (Double Precision L,R,Lt,Rt,f,g,fx,gx,ft,gt,l)2 2640 1 1200 3516 t (Integer p,q)1 660 1 1200 3636 t (Do p = 1, nx)4 720 1 1200 3876 t ({)1320 3996 w ( Get \(L,R\)\(t\) and \(L',R'\)\(t\).)4 1740( #)1 300(Call LR\(t,L,R,Lt,Rt\))1 1200 3 1320 4116 t (l = R-L)2 420 1 1320 4236 t (Do q = 1, ny)4 720 1 1320 4476 t ({)1440 4596 w (x\(p,q\) = L+c\(p\)*l)2 1020 1 1440 4716 t (D\(p,q,1,1\) = l)2 840 1 1440 4836 t (D\(p,q,1,2\) = 0)2 840 1 1440 4956 t (D\(p,q,1,3\) = Lt+c\(p\)*\(Rt-Lt\))2 1680 1 1440 5076 t ( Get \(f,g,fx,gx,ft,gt\)\(t,x\).)2 1680( #)1 300(Call BT\(t,x\(p,q\),f,g,fx,gx,ft,gt\))1 1980 3 1440 5316 t (y\(p,q\) = f+e\(q\)*\(g-f\))2 1260 1 1440 5556 t (D\(p,q,2,1\) = \(fx+e\(q\)*\(gx-fx\)\)*l)2 1920 1 1440 5676 t (D\(p,q,2,2\) = g-f)2 960 1 1440 5796 t (D\(p,q,2,3\) = ft + e\(q\)*\(gt-ft\) + \( fx + e\(q\)* \( gx-fx \) \)*D\(p,q,1,3\))13 4080 1 1440 5916 t (})1440 6036 w (})1320 6156 w (Return)1200 6396 w (End)1200 6636 w 10 R f ( such maps are trivial and new ones can easily be created for other types of)15 3124(The programs for performing)3 1196 2 720 6816 t (domains.)720 6936 w cleartomark showpage saveobj restore end %%EndPage: 19 19 %%Page: 20 20 DpostDict begin /saveobj save def mark 20 pagesetup 10 R f (- 20 -)2 216 1 2772 480 t 10 B f (pde Mapping Software.)2 1003 1 720 840 t 10 R f ( specific example of a general mapping \()7 1654(The mapping \(4.1\) is a)4 919 2 970 996 t 10 I f (x x)1 44 1 3551 996 t 10 R f (,)3603 996 w 10 I f (y y)1 44 1 3636 996 t 10 R f (\) \()1 106 1 3688 996 t 10 S f (x)3826 996 w 10 R f (,)3907 996 w 10 S f (h)3940 996 w 10 R f (,)4032 996 w 10 I f (t t)1 28 1 4089 996 t 10 R f (\) which can be used to)5 915 1 4125 996 t (transform a)1 458 1 720 1116 t 10 B f (pde)1204 1116 w 10 R f (for solution by)2 591 1 1386 1116 t 10 CW f (TTGR)2003 1116 w 10 R f (.)2243 1116 w 10 CW f (TTGR)2389 1116 w 10 R f ( mapping transformation procedures for)4 1591(has general purpose)2 794 2 2655 1116 t ( results of a map \()5 863(taking the)1 425 2 720 1236 t 10 I f (x x)1 44 1 2016 1236 t 10 R f (,)2068 1236 w 10 I f (y y)1 44 1 2101 1236 t 10 R f (\) \()1 106 1 2153 1236 t 10 S f (x)2291 1236 w 10 R f (,)2372 1236 w 10 S f (h)2429 1236 w 10 R f (,)2521 1236 w 10 I f (t t)1 28 1 2578 1236 t 10 R f (\) and converting the)3 888 1 2638 1236 t 10 B f (pde)3580 1236 w 10 R f (s and)1 237 1 3736 1236 t 10 B f (bc)4027 1236 w 10 R f (s from)1 287 1 4127 1236 t 10 B f (u)4468 1236 w 10 R f (\()4556 1236 w 10 I f (t t)1 28 1 4621 1236 t 10 R f (,)4657 1236 w 10 I f (x x)1 44 1 4714 1236 t 10 R f (,)4766 1236 w 10 I f (y y)1 44 1 4823 1236 t 10 R f (\) to)1 165 1 4875 1236 t 10 B f (w)720 1356 w 10 R f (\()800 1356 w 10 I f (t t)1 28 1 865 1356 t 10 R f (,)901 1356 w 10 S f (x)958 1356 w 10 R f (,)1039 1356 w 10 S f (h)1096 1356 w 10 R f (\))1188 1356 w 10 S f (\272)1270 1356 w 10 B f (u)1366 1356 w 10 R f (\()1454 1356 w 10 I f (t t)1 28 1 1519 1356 t 10 R f (, \()1 90 1 1555 1356 t 10 I f (x x)1 44 1 1653 1356 t 10 R f (,)1705 1356 w 10 I f (y y)1 44 1 1738 1356 t 10 R f (\) \()1 74 1 1790 1356 t 10 S f (x)1872 1356 w 10 R f (,)1953 1356 w 10 S f (h)2010 1356 w 10 R f (,)2102 1356 w 10 I f (t t)1 28 1 2159 1356 t 10 R f ( coordinates, where \()3 848(\) \))1 74 2 2195 1356 t 10 I f (x x)1 44 1 3149 1356 t 10 R f (,)3201 1356 w 10 I f (y y)1 44 1 3258 1356 t 10 R f (\) \()1 106 1 3334 1356 t 10 S f (x)3472 1356 w 10 R f (,)3529 1356 w 10 S f (h)3586 1356 w 10 R f ( onto the user's)3 622(\) maps a rectangle)3 740 2 3678 1356 t (physical domain.)1 683 1 720 1476 t (The user tells)2 544 1 970 1632 t 10 CW f (TTGR)1542 1632 w 10 R f (to solve for)2 461 1 1810 1632 t 10 B f (w)2300 1632 w 10 R f (, but the)2 333 1 2372 1632 t 10 B f (pde)2734 1632 w 10 R f (and)2919 1632 w 10 B f (bc)3092 1632 w 10 R f (definition can be made in the physical)6 1544 1 3221 1632 t 10 B f (u)4794 1632 w 10 R f (sys-)4879 1632 w (tem using the software to be described shortly.)7 1864 1 720 1752 t (The)970 1908 w 10 B f (pde)1150 1908 w 10 R f ( have)1 213( We)1 188(mapping proceeds as follows.)3 1187 3 1331 1908 t 10 B f (w)2077 2088 w 10 R f (\()2157 2088 w 10 I f (t t)1 28 1 2222 2088 t 10 R f (,)2258 2088 w 10 S f (x)2315 2088 w 10 R f (,)2396 2088 w 10 S f (h)2453 2088 w 10 R f (\))2545 2088 w 10 S f (\272)2627 2088 w 10 B f (u)2723 2088 w 10 R f (\()2787 2088 w 10 I f (t t)1 28 1 2852 2088 t 10 R f (, \()1 90 1 2888 2088 t 10 I f (x x)1 44 1 2986 2088 t 10 R f (,)3038 2088 w 10 I f (y y)1 44 1 3071 2088 t 10 R f (\) \()1 74 1 3123 2088 t 10 S f (x)3229 2088 w 10 R f (,)3310 2088 w 10 S f (h)3367 2088 w 10 R f (,)3459 2088 w 10 I f (t t)1 28 1 3516 2088 t 10 R f (\) \))1 106 1 3576 2088 t (and thus)1 345 1 720 2308 t 10 B f (w)1099 2308 w 7 I f (t t)1 20 1 1182 2328 t 10 S f (\272)1251 2308 w 10 B f (u)1347 2308 w 7 I f (x x)1 31 1 1414 2328 t 10 I f ( t)1 0(d dt)1 78 2 1518 2378 t ( x)1 0(d dx)1 94 2 1510 2248 t 10 S1 f (_ __)1 124 1 1495 2278 t 10 S f (+ +)1 55 1 1678 2308 t 10 B f (u)1782 2308 w 7 I f (y y)1 31 1 1849 2328 t 10 I f ( t)1 0(d dt)1 78 2 1953 2378 t ( y)1 0(d dy)1 94 2 1945 2248 t 10 S1 f (_ __)1 124 1 1930 2278 t 10 S f (+ +)1 55 1 2113 2308 t 10 B f (u)2217 2308 w 7 I f (t t)1 20 1 2284 2328 t 10 R f (,)2312 2308 w 10 B f (w)2371 2308 w 7 S f (x)2454 2328 w 10 S f (\272)2537 2308 w 10 B f (u)2633 2308 w 7 I f (x x)1 31 1 2700 2328 t 10 I f (x x)1 44 1 2771 2308 t 7 S f (x)2826 2328 w 10 S f (+ +)1 55 1 2917 2308 t 10 B f (u)3021 2308 w 7 I f (y y)1 31 1 3088 2328 t 10 I f (y y)1 44 1 3159 2308 t 7 S f (x)3214 2328 w 10 R f (,)3288 2308 w 10 B f (w)3347 2308 w 7 S f (h)3430 2328 w 10 S f (\272)3521 2308 w 10 B f (u)3617 2308 w 7 I f (x x)1 31 1 3684 2328 t 10 I f (x x)1 44 1 3755 2308 t 7 S f (h)3810 2328 w 10 S f (+ +)1 55 1 3909 2308 t 10 B f (u)4013 2308 w 7 I f (y y)1 31 1 4080 2328 t 10 I f (y y)1 44 1 4151 2308 t 7 S f (h)4206 2328 w 10 I f (. .)1 25 1 4264 2308 t 10 R f ( that)1 185(This implies)1 507 2 4348 2308 t 10 B f (pde)720 2478 w 10 R f (s which began in the form \(2.1\) now have the form)10 2032 1 876 2478 t 10 S f (\266 x)1 106 1 1245 2768 t (\266)1274 2638 w 10 S1 f (_ __)1 136 1 1231 2668 t 10 R f (\()1409 2698 w 10 S f (- -)1 55 1 1482 2698 t 10 I f (x x)1 44 1 1553 2698 t 7 S f (h)1608 2718 w 10 B f (a)1690 2698 w 7 R f (\( 2 \))2 91 1 1745 2658 t 10 S f (+ +)1 55 1 1901 2698 t 10 I f (y y)1 44 1 2005 2698 t 7 S f (h)2060 2718 w 10 B f (a)2142 2698 w 7 R f (\( 1 \))2 91 1 2197 2658 t 10 R f (\))2336 2698 w 10 S f (+ +)1 55 1 2426 2698 t (\266 h)1 117 1 2555 2768 t (\266)2589 2638 w 10 S1 f (_ __)1 147 1 2540 2668 t 10 R f (\()2729 2698 w 10 S f (- -)1 55 1 2802 2698 t 10 I f (y y)1 44 1 2873 2698 t 7 S f (x)2928 2718 w 10 B f (a)3002 2698 w 7 R f (\( 2 \))2 91 1 3057 2658 t 10 S f (+ +)1 55 1 3213 2698 t 10 I f (x x)1 44 1 3317 2698 t 7 S f (x)3372 2718 w 10 B f (a)3446 2698 w 7 R f (\( 1 \))2 91 1 3501 2658 t 10 R f (\))3640 2698 w 10 S f (= =)1 55 1 3730 2698 t 10 R f (\()3834 2698 w 10 I f (x x)1 44 1 3899 2698 t 7 S f (x)3954 2718 w 10 I f (y y)1 44 1 4028 2698 t 7 S f (h)4083 2718 w 10 S f (- -)1 55 1 4173 2698 t 10 I f (x x)1 44 1 4268 2698 t 7 S f (h)4323 2718 w 10 I f (y y)1 44 1 4405 2698 t 7 S f (x)4460 2718 w 10 R f (\))4534 2698 w (.)4607 2668 w 10 B f (f)4640 2698 w 10 I f (. .)1 25 1 4681 2698 t 10 R f (assuming that the mapping \()4 1133 1 720 2928 t 10 I f (x x)1 44 1 1861 2928 t 10 R f (,)1913 2928 w 10 I f (y y)1 44 1 1946 2928 t 10 R f (\) \()1 74 1 1998 2928 t 10 S f (x)2080 2928 w 10 R f (,)2161 2928 w 10 S f (h)2194 2928 w 10 R f (\) has continuous second partials.)4 1301 1 2286 2928 t ( of the user's)3 548(At the beginning)2 690 2 970 3084 t 10 CW f (AF)2243 3084 w 10 R f (procedure it is necessary to convert from the internal coordinate)9 2642 1 2398 3084 t (system of)1 394 1 720 3204 t 10 CW f (TTGR)1146 3204 w 10 R f (, namely)1 351 1 1386 3204 t 10 B f (w)1769 3204 w 10 R f (and)1873 3204 w 10 S f (x)2049 3204 w 10 R f (,)2130 3204 w 10 S f (h)2163 3204 w 10 R f (, to the user's physical coordinate system,)6 1712 1 2223 3204 t 10 B f (u)3967 3204 w 10 R f (and)4055 3204 w 10 I f (x x)1 44 1 4231 3204 t 10 R f (,)4283 3204 w 10 I f (y y)1 44 1 4316 3204 t 10 R f ( accomplish)1 487(. To)1 193 2 4360 3204 t (this the user must calculate the value of the map \()10 2080 1 720 3324 t 10 I f (x x)1 44 1 2808 3324 t 10 R f (,)2860 3324 w 10 I f (y y)1 44 1 2893 3324 t 10 R f (\) \()1 106 1 2945 3324 t 10 S f (x)3083 3324 w 10 R f (,)3164 3324 w 10 S f (h)3197 3324 w 10 R f (,)3289 3324 w 10 I f (t t)1 28 1 3346 3324 t 10 R f (\) at the)2 297 1 3382 3324 t 10 S f (x)3714 3324 w 10 R f (,)3795 3324 w 10 S f (h)3828 3324 w 10 R f (input to)1 320 1 3923 3324 t 10 CW f (AF)4279 3324 w 10 R f (and its partials)2 605 1 4435 3324 t (with respect to time,)3 822 1 720 3444 t 10 S f (x)1569 3444 w 10 R f (,)1618 3444 w 10 S f (h)1669 3444 w 10 R f ( is, the "x" in the calling sequence to)8 1469( That)1 234(, etc.)1 192 3 1729 3444 t 10 CW f (AF)3650 3444 w 10 R f (should be called "xi", similarly)4 1244 1 3796 3444 t (for "y", and the values of \()6 1060 1 720 3564 t 10 I f (x x)1 44 1 1788 3564 t 10 R f (,)1840 3564 w 10 I f (y y)1 44 1 1873 3564 t 10 R f (\) \()1 74 1 1925 3564 t 10 S f (x)2007 3564 w 10 R f (,)2088 3564 w 10 S f (h)2145 3564 w 10 R f (,)2237 3564 w 10 I f (t t)1 28 1 2294 3564 t 10 R f (\) should be computed by the user inside)7 1590 1 2330 3564 t 10 CW f (AF)3945 3564 w 10 R f (. Then)1 280 1 4065 3564 t 10 CW f (Call TTGRU\(nx,ny,D, Ux,Uy,Ut,Nu\))2 1920 1 1080 3744 t 10 R f (will do the mapping from internal \()6 1434 1 720 3924 t 10 B f (w)2183 3924 w 10 R f (\) to User \()3 419 1 2284 3924 t 10 B f (u)2732 3924 w 10 R f ( of Appendix 4 illustrates the)5 1188( 4)1 79( Example)1 409(\) coordinates.)1 547 4 2817 3924 t (use of)1 241 1 720 4044 t 10 CW f (TTGRU)986 4044 w 10 R f ( arguments to)2 544(. The)1 230 2 1286 4044 t 10 CW f (TTGRU)2085 4044 w 10 R f ( input to)2 334( The)1 205(are now described.)2 750 3 2410 4044 t 10 CW f (TTGRU)3724 4044 w 10 R f (is)4049 4044 w 10 CW f (nx)770 4200 w 10 R f ( number of)2 438(- The)1 305 2 1120 4200 t 10 S f (x)1888 4200 w 10 R f (points.)1962 4200 w 10 CW f (ny)770 4356 w 10 R f ( number of)2 438(- The)1 305 2 1120 4356 t 10 S f (h)1888 4356 w 10 R f (points.)1973 4356 w 10 CW f (D)770 4512 w 10 R f ( output array as described above in)6 1391(- An)1 272 2 1120 4512 t 10 CW f (BTMAP)2808 4512 w 10 R f (.)3108 4512 w 10 CW f (Ux)770 4668 w 10 R f ( input argument of)3 741(- The)1 305 2 1120 4668 t 10 CW f (AF)2191 4668 w 10 R f (of the same name.)3 726 1 2336 4668 t 10 CW f (Uy)770 4824 w 10 R f ( input argument of)3 741(- The)1 305 2 1120 4824 t 10 CW f (AF)2191 4824 w 10 R f (of the same name.)3 726 1 2336 4824 t 10 CW f (Ut)770 4980 w 10 R f ( input argument of)3 741(- The)1 305 2 1120 4980 t 10 CW f (AF)2191 4980 w 10 R f (of the same name.)3 726 1 2336 4980 t 10 CW f (Nu)770 5136 w 10 R f ( input argument of)3 741(- The)1 305 2 1120 5136 t 10 CW f (AF)2191 5136 w 10 R f (of the same name.)3 726 1 2336 5136 t (The output of)2 544 1 970 5292 t 10 CW f (TTGRU)1539 5292 w 10 R f (is)1864 5292 w 10 CW f (Ux)770 5448 w 10 R f ( derivatives)1 463(- The)1 305 2 1120 5448 t 10 B f (u)1913 5448 w 7 I f (x x)1 31 1 1980 5468 t 10 R f (.)2019 5448 w 10 CW f (Uy)770 5604 w 10 R f ( derivatives)1 463(- The)1 305 2 1120 5604 t 10 B f (u)1913 5604 w 7 I f (y y)1 31 1 1980 5624 t 10 R f (.)2019 5604 w 10 CW f (Ut)770 5760 w 10 R f ( derivatives)1 463(- The)1 305 2 1120 5760 t 10 B f (u)1913 5760 w 7 I f (t t)1 20 1 1980 5780 t 10 R f (.)2008 5760 w (With the map of)3 655 1 970 5916 t 10 B f (w)1651 5916 w 10 R f (into user coordinates)2 834 1 1749 5916 t 10 B f (u)2609 5916 w 10 R f (done using)1 437 1 2691 5916 t 10 CW f (TTGRU)3154 5916 w 10 R f (above, the user then continues with)5 1414 1 3480 5916 t 10 CW f (AF)4920 5916 w 10 R f ( terms of the original)4 850( is, the user can then think only in)8 1384( That)1 237(as if there were no mapping being used at all.)9 1849 4 720 6036 t (variables,)720 6156 w 10 B f (u)1130 6156 w 10 R f (and)1211 6156 w 10 I f (x x)1 44 1 1380 6156 t 10 R f (,)1432 6156 w 10 I f (y y)1 44 1 1465 6156 t 10 R f (.)1509 6156 w (However, before leaving)2 994 1 970 6312 t 10 CW f (AF)1992 6312 w 10 R f (the user must map the physical coordinates)6 1738 1 2140 6312 t 10 B f (u)3907 6312 w 10 R f (back into the internal sys-)4 1048 1 3992 6312 t (tem)720 6432 w 10 CW f (TTGR)895 6432 w 10 R f (is using,)1 334 1 1160 6432 t 10 B f (w)1519 6432 w 10 R f ( is done by)3 436(. This)1 253 2 1591 6432 t 10 CW f (Call TTGRG\(nx,ny,D, Nu, A,AU,AUx,AUy, F,FU,FUx,FUy\))4 3060 1 1080 6612 t 10 R f (just before leaving)2 743 1 720 6792 t 10 CW f (AF)1488 6792 w 10 R f (.)1608 6792 w (The input arguments to)3 930 1 970 6948 t 10 CW f (TTGRG)1925 6948 w 10 R f (are as in)2 332 1 2250 6948 t 10 CW f (TTGRU)2607 6948 w 10 R f (above.)2932 6948 w (The output of)2 544 1 720 7104 t 10 CW f (TTGRG)1289 7104 w 10 R f (is)1614 7104 w cleartomark showpage saveobj restore end %%EndPage: 20 20 %%Page: 21 21 DpostDict begin /saveobj save def mark 21 pagesetup 10 R f (- 21 -)2 216 1 2772 480 t 10 CW f (A)770 840 w 10 R f (-)1120 840 w 10 B f (a)1270 840 w 10 R f (for the)1 263 1 1345 840 t 10 I f (w w)1 67 1 1633 840 t 10 R f (and)1725 840 w 10 S f (x)1894 840 w 10 R f (,)1975 840 w 10 S f (h)2008 840 w 10 R f (coordinate system.)1 749 1 2093 840 t 10 CW f (AU)770 996 w 10 R f ( partials of)2 427(- The)1 305 2 1120 996 t 10 B f (a)1877 996 w 10 R f (with respect to)2 588 1 1952 996 t 10 B f (w)2565 996 w 10 R f (.)2637 996 w 10 CW f (AUx)770 1152 w 10 R f ( partials of)2 427(- The)1 305 2 1120 1152 t 10 B f (a)1877 1152 w 10 R f (with respect to)2 588 1 1952 1152 t 10 B f (w)2565 1152 w 7 S f (x)2648 1172 w 10 R f (.)2690 1152 w 10 CW f (AUy)770 1308 w 10 R f ( partials of)2 427(- The)1 305 2 1120 1308 t 10 B f (a)1877 1308 w 10 R f (with respect to)2 588 1 1952 1308 t 10 B f (w)2565 1308 w 7 S f (h)2648 1328 w 10 R f (.)2698 1308 w 10 CW f (F)770 1464 w 10 R f (-)1120 1464 w 10 B f (f)1270 1464 w 10 R f (for the)1 263 1 1328 1464 t 10 B f (w)1616 1464 w 10 R f (and)1713 1464 w 10 S f (x)1882 1464 w 10 R f (,)1963 1464 w 10 S f (h)1996 1464 w 10 R f (system.)2081 1464 w 10 CW f (FU)770 1620 w 10 R f ( partials of)2 427(- The)1 305 2 1120 1620 t 10 B f (f)1877 1620 w 10 R f (with respect to)2 588 1 1935 1620 t 10 B f (w)2548 1620 w 10 R f (.)2620 1620 w 10 CW f (FUx)770 1776 w 10 R f ( partials of)2 427(- The)1 305 2 1120 1776 t 10 B f (f)1877 1776 w 10 R f (with respect to)2 588 1 1935 1776 t 10 B f (w)2548 1776 w 7 S f (x)2631 1796 w 10 R f (.)2673 1776 w 10 CW f (FUy)770 1932 w 10 R f ( partials of)2 427(- The)1 305 2 1120 1932 t 10 B f (f)1877 1932 w 10 R f (with respect to)2 588 1 1935 1932 t 10 B f (w)2548 1932 w 7 S f (h)2631 1952 w 10 R f (.)2681 1932 w (The Double Precision versions of)4 1337 1 970 2088 t 10 CW f (TTGRG)2333 2088 w 10 R f (and)2659 2088 w 10 CW f (TTGRU)2829 2088 w 10 R f (are called)1 385 1 3155 2088 t 10 CW f (DTTGRG)3566 2088 w 10 R f (and)3952 2088 w 10 CW f (DTTGRU)4122 2088 w 10 R f (, respectively,)1 558 1 4482 2088 t (with all Real arguments typed Double Precision.)6 1940 1 720 2208 t (Example 4 of Appendix 4 shows)5 1307 1 970 2364 t 10 CW f (TTGRU)2302 2364 w 10 R f (and)2627 2364 w 10 CW f (TTGRG)2796 2364 w 10 R f (at work.)1 327 1 3121 2364 t 10 B f (Mapping Boundary Conditions)2 1337 1 720 2604 t 10 R f (The)970 2760 w 10 B f (bc)1154 2760 w 10 R f (mapping paradigm generally follows that for the)6 1960 1 1283 2760 t 10 B f (pde)3272 2760 w 10 R f ( At)1 155(mapping software described above.)3 1428 2 3457 2760 t ( user's)1 270(the beginning of the)3 826 2 720 2880 t 10 CW f (BC)1848 2880 w 10 R f (procedure it is necessary to convert from the internal coordinate system of)11 3040 1 2000 2880 t 10 CW f (TTGR)720 3000 w 10 R f (, namely)1 345 1 960 3000 t 10 B f (w)1331 3000 w 10 R f (and)1429 3000 w 10 S f (x)1599 3000 w 10 R f (,)1680 3000 w 10 S f (h)1713 3000 w 10 R f (, to the user's physical coordinate system,)6 1676 1 1773 3000 t 10 B f (u)3475 3000 w 10 R f (and)3557 3000 w 10 I f (x x)1 44 1 3727 3000 t 10 R f (,)3779 3000 w 10 I f (y y)1 44 1 3812 3000 t 10 R f ( accomplish this the user)4 996(. To)1 188 2 3856 3000 t (must calculate the value of the map \()7 1500 1 720 3120 t 10 I f (x x)1 44 1 2228 3120 t 10 R f (,)2280 3120 w 10 I f (y y)1 44 1 2313 3120 t 10 R f (\) \()1 106 1 2365 3120 t 10 S f (x)2503 3120 w 10 R f (,)2584 3120 w 10 S f (h)2617 3120 w 10 R f (,)2709 3120 w 10 I f (t t)1 28 1 2766 3120 t 10 R f (\) at the)2 285 1 2826 3120 t 10 S f (x)3139 3120 w 10 R f (,)3220 3120 w 10 S f (h)3253 3120 w 10 R f (input to)1 312 1 3341 3120 t 10 CW f (BC)3681 3120 w 10 R f (and its partials with respect to)5 1211 1 3829 3120 t (time,)720 3240 w 10 S f (x)956 3240 w 10 R f (,)1005 3240 w 10 S f (h)1063 3240 w 10 R f ( Then)1 263(, etc.)1 199 2 1123 3240 t 10 CW f (TTGRU)1618 3240 w 10 R f ( the)1 156(should be invoked,)2 774 2 1951 3240 t 10 B f (bc)2915 3240 w 10 R f (s entered in terms of)4 851 1 3015 3240 t 10 B f (u)3900 3240 w 10 R f (and)3990 3240 w 10 I f (x x)1 44 1 4168 3240 t 10 R f (,)4220 3240 w 10 I f (y y)1 44 1 4253 3240 t 10 R f (, and then)2 409 1 4297 3240 t 10 CW f (TTGRB)4740 3240 w 10 R f (invoked before returning from)3 1211 1 720 3360 t 10 CW f (BC)1956 3360 w 10 R f (.)2076 3360 w ( start of)2 315(At the)1 251 2 970 3516 t 10 CW f (BC)1566 3516 w 10 R f (the user calls a mapping procedure to do the map from internal)11 2564 1 1716 3516 t 10 CW f (TTGR)4310 3516 w 10 R f (coordinates)4580 3516 w 10 S f (x)720 3636 w 10 R f (,)801 3636 w 10 S f (h)834 3636 w 10 R f (, the)1 172 1 894 3636 t 10 CW f (BC)1091 3636 w 10 R f (input arguments "xi,eta", to)3 1101 1 1236 3636 t 10 CW f (x,y)2362 3636 w 10 R f (coordinates. The)1 690 1 2567 3636 t 10 CW f (Call TTGRU\(nx,ny,D, Ux,Uy,Ut,Nu\))2 1920 1 1080 3816 t 10 R f (will do this mapping from internal \()6 1505 1 720 3996 t 10 B f (w)2262 3996 w 10 R f (\) to user \()3 421 1 2371 3996 t 10 B f (u)2829 3996 w 10 R f ( input to)2 360( The)1 218(\) coordinates.)1 556 3 2922 3996 t 10 CW f (TTGRU)4094 3996 w 10 R f (is as described)2 608 1 4432 3996 t (above.)720 4116 w (With the map of)3 655 1 970 4272 t 10 B f (w)1651 4272 w 10 R f (into user coordinates)2 834 1 1749 4272 t 10 B f (u)2609 4272 w 10 R f (done using)1 437 1 2691 4272 t 10 CW f (TTGRU)3154 4272 w 10 R f (above, the user continues with)4 1216 1 3480 4272 t 10 CW f (BC)4722 4272 w 10 R f (as if)1 171 1 4869 4272 t ( can then think only in terms of the original variables,)10 2145( is, the user)3 458( That)1 234(there were no mapping being used.)5 1402 4 720 4392 t 10 B f (u)4984 4392 w 10 R f (and)720 4512 w 10 I f (x x)1 44 1 889 4512 t 10 R f (,)941 4512 w 10 I f (y y)1 44 1 974 4512 t 10 R f (.)1018 4512 w (However, before leaving)2 994 1 970 4668 t 10 CW f (BC)1992 4668 w 10 R f (the user must map the physical coordinates)6 1738 1 2140 4668 t 10 B f (u)3907 4668 w 10 R f (back into the internal sys-)4 1048 1 3992 4668 t (tem)720 4788 w 10 CW f (TTGR)895 4788 w 10 R f (is using,)1 334 1 1160 4788 t 10 B f (w)1519 4788 w 10 R f ( is done by)3 436(. This)1 253 2 1591 4788 t 10 CW f (Call TTGRB\(nx,ny,D, BUx,BUy,BUt\))2 1920 1 1080 4968 t 10 R f (just before leaving)2 743 1 720 5148 t 10 CW f (BC)1488 5148 w 10 R f (.)1608 5148 w (The input arguments to)3 930 1 970 5344 t 10 CW f (TTGRB)1925 5344 w 10 R f (are as above, and the outputs are the values of)9 1834 1 2250 5344 t 10 S f (\266)4134 5414 w 10 B f (w)4191 5414 w 10 S f (\266)4142 5284 w 10 B f (b)4199 5284 w 10 S1 f (_ ___)1 159 1 4119 5314 t 10 R f (, etc.)1 191 1 4288 5344 t (Example 4 in Appendix 4 shows)5 1302 1 970 5550 t 10 CW f (TTGRB)2297 5550 w 10 R f (,)2597 5550 w 10 CW f (TTGRG)2647 5550 w 10 R f (,)2947 5550 w 10 CW f (TTGRU)2997 5550 w 10 R f (, as well as)3 438 1 3297 5550 t 10 CW f (BTMAP)3760 5550 w 10 R f (, hard at work.)3 579 1 4060 5550 t 10 B f ( for the Sporting User)4 932(6. Advice)1 419 2 720 5790 t 10 R f ( ways of overriding the default options)6 1572(This section describes, in gory detail, the various "knobs" and)9 2498 2 970 5946 t (and subprograms used in)3 1011 1 720 6066 t 10 CW f (TTGR)1761 6066 w 10 R f ( examples of knob twiddling that can increase the speed of)10 2390(. Several)1 379 2 2001 6066 t 10 CW f (TTGR)4800 6066 w 10 R f (considerably are also given.)3 1114 1 720 6186 t (An exceedingly brief outline of the organization of)7 2032 1 970 6342 t 10 CW f (TTGR)3027 6342 w 10 R f (is, in pseudo-English,)2 867 1 3292 6342 t cleartomark showpage saveobj restore end %%EndPage: 21 21 %%Page: 23 22 DpostDict begin /saveobj save def mark 22 pagesetup 10 R f (- 23 -)2 216 1 2772 480 t 10 CW f (t0 = tstart)2 660 1 1080 900 t ( Time-step loop.)2 960( #)1 540(While \( t0 != tstop \))5 1260 3 1080 1020 t ({)1320 1140 w ( Loop to build extrapolation tableau.)5 2220( #)1 540(Do m = 1, ..., mmax)5 1140 3 1320 1260 t ({)1560 1380 w ( Sub-steps loop.)2 960( #)1 540(Do istep = 1, ..., N\(m\))5 1380 3 1560 1500 t ({)1800 1620 w ( Newton loop.)2 780( #)1 540(Do iter = 1, ..., maxit)5 1380 3 1800 1740 t ({)2040 1860 w (Solve the)1 540 1 2040 1980 t 10 I f ( d)1 0( ed)1 50( ze)1 44( iz)1 39( ri)1 28( ar)1 39( ea)1 50( ne)1 44(l li in)2 106 9 2640 1980 t 10 CW f (Galerkin Equations; Update solution)3 2100 1 3100 1980 t (Check ERROR for "convergence"; Check Convergence Rate)6 3180 1 2040 2100 t (})2040 2220 w (})1800 2340 w (Extrapolate and check the ERROR.)4 1920 1 1560 2460 t (})1560 2580 w (Get_Optimal_dt for next time-step.)3 2040 1 1320 2700 t (Output the results for this time-step \( HANDLE \))8 2880 1 1320 2820 t (t0 = t0 + dt)4 720 1 1320 2940 t (})1320 3060 w 10 R f ( procedures names, and the linearized Galerkin solution is)8 2323(where the capitalized items in parentheses refer to)7 1997 2 720 3240 t (obtained by)1 469 1 720 3360 t (Get integrals for matrix \()4 998 1 1130 3540 t 10 CW f (AF)2153 3540 w 10 R f (\), mesh interval by mesh interval,)5 1340 1 2298 3540 t (Get the)1 291 1 1130 3660 t 10 B f (bc)1446 3660 w 10 R f (s \()1 97 1 1546 3660 t 10 CW f (BC)1668 3660 w 10 R f (\).)1813 3660 w (Get)1130 3780 w 10 B f (pde)1299 3780 w 10 R f (\()1480 3780 w 10 CW f (AF)1538 3780 w 10 R f (\) on the boundary.)3 732 1 1683 3780 t (Solve the)1 375 1 1130 3900 t 10 B f (pde)1530 3900 w 10 R f (system.)1711 3900 w ( These)1 303( many variables and procedures that define the above algorithm.)9 2696(This section describes the)3 1071 3 970 4116 t (include the maximum number of levels of extrapolation to allow \()10 2636 1 720 4236 t 10 CW f (mmax)3381 4236 w 10 R f (\), the sequence of sub-steps to take)6 1394 1 3646 4236 t (\()720 4356 w 10 CW f (N)784 4356 w 10 R f (\(1\),)844 4356 w 10 CW f (N)1017 4356 w 10 R f (\(2\),)1077 4356 w (. . .)2 125 1 1275 4331 t (\), how to solve the linearized Galerkin equations, and the maximum number of Newton)13 3583 1 1457 4356 t (iterations to allow \()3 780 1 720 4476 t 10 CW f (maxit)1525 4476 w 10 R f (\).)1850 4476 w 10 B f (Twiddling Procedure Knobs.)2 1238 1 720 4716 t 10 R f (Control over the procedure)3 1123 1 970 4872 t 10 CW f (ERROR)2133 4872 w 10 R f (is given by)2 469 1 2473 4872 t 10 CW f (TTGRR)2983 4872 w 10 R f ( routine allows the user to override)6 1488(. This)1 269 2 3283 4872 t (some of the default routines of)5 1223 1 720 4992 t 10 CW f (TTGR)1968 4992 w 10 R f (.)2208 4992 w 10 CW f (TTGRR)2353 4992 w 10 R f (is invoked by)2 539 1 2678 4992 t 10 CW f (Call TTGRR\(U,Nu,kx,x,nx, ky,y,ny,)2 1980 1 1200 5172 t (tstart,tstop,dt,)1860 5292 w (AF,BC,)1860 5412 w (ERROR,errpar,)1860 5532 w (HANDLE\))1860 5652 w 10 R f (The extra argument)2 787 1 720 5832 t 10 CW f (ERROR)1535 5832 w 10 R f ( direct user control over the accuracy of the integra-)9 2104(in this subroutine provides)3 1073 2 1863 5832 t (tion process.)1 505 1 720 5952 t (The default routine, used by)4 1123 1 970 6108 t 10 CW f (TTGR)2118 6108 w 10 R f (, is)1 117 1 2358 6108 t 10 CW f (TTGRE)2500 6108 w 10 R f (for)2825 6108 w 10 CW f (ERROR)2966 6108 w 10 R f (.)3266 6108 w 10 B f (Error Options)1 614 1 720 6348 t 10 R f ( the user, via the subpro-)5 1047( First,)1 271( error specification.)2 802(There are several possible options available for)6 1950 4 970 6504 t (gram)720 6624 w 10 CW f (ERROR)955 6624 w 10 R f ( option has not)3 604( This)1 232(, may specify literally any accuracy requirement desired for the solution.)10 2949 3 1255 6624 t (yet been implemented because of using)5 1622 1 720 6744 t 10 CW f (IODE)2377 6744 w 10 R f ( can roll their own)4 772(raw. Users)1 461 2 2652 6744 t 10 CW f (ERROR)3920 6744 w 10 R f (routines, but)1 510 1 4255 6744 t 10 CW f (TTGR)4800 6744 w 10 R f ( there are several popular methods of error control which are con-)11 2732( Second,)1 379( one.)1 204(will use its own internal)4 1005 4 720 6864 t (trolled by the switch)3 819 1 720 6984 t 10 CW f (erputs)1564 6984 w 10 R f (, and implemented by the subprogram)5 1515 1 1924 6984 t 10 CW f (TTGRE)3464 6984 w 10 R f (.)3764 6984 w (The error control provided in the subroutine)6 1836 1 970 7140 t 10 CW f (TTGRE)2879 7140 w 10 R f ( of the variables.)3 707(is based on the local value)5 1116 2 3217 7140 t (That is, the error acceptable in)5 1213 1 720 7260 t 10 I f (u u)1 50 1 1958 7260 t 7 I f (i i)1 20 1 2019 7280 t 10 R f (\()2055 7260 w 10 I f (t t)1 28 1 2096 7260 t 10 R f (,)2132 7260 w 10 I f (x x)1 44 1 2165 7260 t 10 R f (,)2217 7260 w 10 I f (y y)1 44 1 2250 7260 t 10 R f (\) is)1 125 1 2302 7260 t cleartomark showpage saveobj restore end %%EndPage: 23 22 %%Page: 24 23 DpostDict begin /saveobj save def mark 23 pagesetup 10 R f (- 24 -)2 216 1 2772 480 t 10 CW f (errpar)1080 900 w 10 R f (\(1\) *)1 191 1 1440 900 t 10 S f ( \357)1 0( \357)1 24(\357 \357)1 49 3 1656 917 t 10 B f (U)1761 900 w 7 I f (. .)1 18 1 1844 920 t 7 R f (.)1862 920 w 7 I f (i i)1 20 1 1885 920 t 10 S f ( \357)1 0( \357)1 24(\357 \357)1 49 3 1946 917 t 7 S f (\245)2008 920 w 10 R f (+)2100 900 w 10 CW f (errpar)2156 900 w 10 R f (\(2\))2516 900 w (where)720 1080 w 10 B f (U)992 1080 w 7 I f (. .)1 18 1 1075 1100 t 7 R f (.)1093 1100 w 7 I f (i i)1 20 1 1116 1100 t 10 R f ( block of B-spline coefficients for)5 1375(denotes the)1 456 2 1173 1080 t 10 I f (u u)1 50 1 3034 1080 t 7 I f (i i)1 20 1 3095 1100 t 10 R f (, which depends only upon the current value of)8 1917 1 3123 1080 t 10 B f (u)720 1200 w 10 R f (.)776 1200 w ( in the subroutine)3 715(The error control provided)3 1076 2 970 1356 t 10 CW f (TTGRE)2792 1356 w 10 R f (is an)1 192 1 3123 1356 t 10 I f ( p)1 0( ep)1 50( te)1 44( st)1 28( -s)1 39( e-)1 33( me)1 44( ti im)2 100( t)1 59( r)1 0( er)1 39( pe)1 44( p)1 81( r)1 0( or)1 39( rr ro)2 89(e er)1 83 17 3346 1356 t 10 R f ( can be)2 294(criterion. This)1 597 2 4149 1356 t ( many time-steps will be taken)5 1229(rather bad if the time-steps taken during the solution process get very small -)13 3091 2 720 1476 t ( use an)2 287( error option is to)4 710( Another)1 381(and the errors may pile up in unacceptable amounts.)8 2114 4 720 1596 t 10 I f ( -)1 0( e-)1 33( me)1 44( ti im)2 100( -t)1 28( it t-)2 61( un ni)2 78( u)1 80( r)1 0( er)1 39( pe)1 44( p)1 80( r)1 0( or)1 39( rr ro)2 89(e er)1 83 16 4242 1596 t ( p)1 0( ep)1 50( te)1 44(s st)1 67 4 720 1716 t 10 R f ( making the error tolerance in)5 1183(criterion. By)1 530 2 906 1716 t 10 I f (u u)1 50 1 2644 1716 t 7 I f (i i)1 20 1 2705 1736 t 10 R f (\()2741 1716 w 10 I f (t t)1 28 1 2782 1716 t 10 R f (,)2818 1716 w 10 I f (x x)1 44 1 2851 1716 t 10 R f (,)2903 1716 w 10 I f (y y)1 44 1 2936 1716 t 10 R f (\) look like)2 411 1 2988 1716 t 10 S f (\357 \357)1 49 1 1105 1913 t 10 CW f (dt)1178 1896 w 10 S f (\357 \357)1 49 1 1323 1913 t 10 R f (* \()1 108 1 1421 1896 t 10 CW f (errpar)1554 1896 w 10 R f (\(1\) *)1 191 1 1914 1896 t 10 S f ( \357)1 0( \357)1 24(\357 \357)1 49 3 2130 1913 t 10 B f (U)2235 1896 w 7 I f (. .)1 18 1 2318 1916 t 7 R f (.)2336 1916 w 7 I f (i i)1 20 1 2359 1916 t 10 S f ( \357)1 0( \357)1 24(\357 \357)1 49 3 2420 1913 t 7 S f (\245)2482 1916 w 10 R f (+)2574 1896 w 10 CW f (errpar)2630 1896 w 10 R f (\(2\) \) ,)2 224 1 2990 1896 t ( when using the error per unit-)6 1304( However,)1 455( the error requirement.)3 942(when the time-step gets small, so does)6 1619 4 720 2076 t ( argument holds - when)4 967(time-step criterion, the reverse)3 1240 2 720 2196 t 10 CW f (dt)2958 2196 w 10 R f ( error per)2 382( The)1 211(is large, so is the error tolerance.)6 1338 3 3109 2196 t (time-step versus unit-time-step option is controlled by the switch)8 2599 1 720 2316 t 10 CW f (erputs)3344 2316 w 10 R f (as described below.)2 784 1 3729 2316 t (The output from)2 655 1 970 2472 t 10 CW f (TTGRR)1650 2472 w 10 R f (is the same as that for)5 868 1 1975 2472 t 10 CW f (TTGR)2868 2472 w 10 R f ( that the user may)4 714(with the additional possibility)3 1193 2 3133 2472 t (alter)720 2592 w 10 CW f (errpar)922 2592 w 10 R f (through his subprogram)2 955 1 1307 2592 t 10 CW f (ERROR)2287 2592 w 10 R f (.)2587 2592 w (The double precision version of)4 1278 1 970 2748 t 10 CW f (TTGRR)2275 2748 w 10 R f (is)2602 2748 w 10 CW f (DTTGRR)2697 2748 w 10 R f (, with all Real arguments typed double precision,)7 1983 1 3057 2748 t (except)720 2868 w 10 CW f (errpar)1005 2868 w 10 R f ( for)1 141( Similarly)1 423(, which remains Real.)3 868 3 1365 2868 t 10 CW f (TTGRE)2822 2868 w 10 R f (and)3147 2868 w 10 CW f (TTGRP)3316 2868 w 10 R f (.)3616 2868 w 10 B f (Twiddling non-Procedure Knobs.)2 1433 1 720 3108 t 10 R f (The main knob-twiddling routine is)4 1430 1 970 3264 t 10 CW f (TTGRV)2427 3264 w 10 R f ( internal variable of)3 793( the user wants to tinker some)6 1205(. When)1 315 3 2727 3264 t 10 CW f (TTGR)720 3384 w 10 R f (the)985 3384 w 10 CW f (Call TTGRV\(j,f,r,i,l\))1 1260 1 1200 3564 t 10 R f (just before calling)2 731 1 720 3744 t 10 CW f (TTGR)1481 3744 w 10 R f ( input to)2 344( The)1 210(will do the trick.)3 676 3 1751 3744 t 10 CW f (TTGRV)3012 3744 w 10 R f (is an index \()3 509 1 3343 3744 t 10 CW f (j)3918 3744 w 10 R f (\) identifying the knob to)4 996 1 4044 3744 t (be turned, and its value \( one of)7 1271 1 720 3864 t 10 CW f (f)2017 3864 w 10 R f (,)2077 3864 w 10 CW f (r)2128 3864 w 10 R f (,)2188 3864 w 10 CW f (i)2239 3864 w 10 R f (or)2325 3864 w 10 CW f (l)2434 3864 w 10 R f ( any value of)3 521(\). For)1 248 2 2520 3864 t 10 CW f (j)3315 3864 w 10 R f (, only)1 229 1 3375 3864 t 10 B f (one)3630 3864 w 10 R f (of)3805 3864 w 10 CW f (f)3913 3864 w 10 R f (,)3973 3864 w 10 CW f (r)4023 3864 w 10 R f (, i or)2 186 1 4083 3864 t 10 CW f (l)4294 3864 w 10 R f (is used to set the)4 661 1 4379 3864 t (knob, the other variables may be set to anything, like)9 2144 1 720 3984 t 10 CW f (0d0)2892 3984 w 10 R f (,)3072 3984 w 10 CW f (0e0)3125 3984 w 10 R f (,)3305 3984 w 10 CW f (0)3393 3984 w 10 R f (or)3482 3984 w 10 CW f (.true.)3594 3984 w 10 R f ( number of variables)3 835(. Any)1 251 2 3954 3984 t (can be set by calling)4 815 1 720 4104 t 10 CW f (TTGRV)1560 4104 w 10 R f ( input to)2 334( The)1 205(any number of times.)3 849 3 1885 4104 t 10 CW f (TTGRV)3298 4104 w 10 R f (is)3623 4104 w 10 CW f (j)770 4260 w 10 R f ( If)1 120( index of the variable to be set.)7 1259(- The)1 305 3 1120 4260 t 10 CW f (j)2833 4260 w 10 R f ( then)1 202(= 0,)1 160 2 2922 4260 t 10 B f (all)3314 4260 w 10 R f (variables are set to their default values.)6 1590 1 3450 4260 t ( not have to use)4 695(You should)1 482 2 1270 4380 t 10 CW f (j = 0)2 334 1 2489 4380 t 10 R f (unless you have already set a few parameters using)8 2175 1 2865 4380 t 10 CW f (TTGRV)1270 4500 w 10 R f (and you now want to reset the default values.)8 1805 1 1595 4500 t 10 CW f (f)770 4656 w 10 R f ( be set is a working-precision item,)6 1425( the variable to)3 605(- If)1 216 3 1120 4656 t 10 CW f (f)3395 4656 w 10 R f ( preci-)1 261( Working)1 409(is to be its new value.)5 886 3 3484 4656 t (sion is Real for)3 608 1 1270 4776 t 10 CW f (TTGR)1903 4776 w 10 R f (and Double Precision for)3 1001 1 2168 4776 t 10 CW f (DTTGR)3194 4776 w 10 R f (.)3494 4776 w 10 CW f (r)770 4932 w 10 R f ( the variable to be set is a Real item \()10 1481(- If)1 216 2 1120 4932 t 10 CW f (hfract)2842 4932 w 10 R f (,)3202 4932 w 10 CW f (egive)3252 4932 w 10 R f (\),)3577 4932 w 10 CW f (r)3660 4932 w 10 R f (is to be its new value.)5 866 1 3745 4932 t 10 CW f (i)770 5088 w 10 R f ( the variable to be set is an Integer item,)9 1597(- If)1 216 2 1120 5088 t 10 CW f (i)2958 5088 w 10 R f (is to be its new value.)5 866 1 3043 5088 t 10 CW f (l)770 5244 w 10 R f ( the variable to be set is a Logical item,)9 1570(- If)1 216 2 1120 5244 t 10 CW f (l)2931 5244 w 10 R f (is to be its new value.)5 866 1 3016 5244 t (The output of)2 544 1 720 5400 t 10 CW f (TTGRV)1289 5400 w 10 R f (is the internal knowledge of the new value of the variable that has been set.)14 3002 1 1614 5400 t (The following list gives the items that can be set using)10 2177 1 970 5556 t 10 CW f (TTGRV)3172 5556 w 10 R f (.)3472 5556 w 10 CW f (theta)770 5712 w 10 R f ( When)1 296( time-discretization parameter, see section 3 and Appendix 2.)8 2509(- The)1 305 3 1270 5712 t 10 CW f (theta)4413 5712 w 10 R f (= 1 the)2 294 1 4746 5712 t ( For)1 189( formula is used.)3 666(extremely stable, first order accurate Backwards-Euler)5 2181 3 1420 5832 t 10 CW f (theta)4481 5832 w 10 R f (= 1/2,)1 234 1 4806 5832 t ( If)1 119(the second-order Crank-Nicholson scheme is used.)5 2044 2 1420 5952 t 10 CW f (theta)3611 5952 w 10 S f (\271)3939 5952 w 10 R f (1/2, then)1 353 1 4022 5952 t 10 CW f (N)4403 5952 w 10 S f (= =)1 55 1 4492 5952 t 10 R f ({ 1, 2, 3, 4,)4 464 1 4576 5952 t (6,)1420 6112 w (. . .)2 125 1 1551 6087 t (} and)1 223 1 1732 6112 t 10 CW f (gamma)1986 6112 w 10 S f (= =)1 55 1 2317 6112 t 10 R f (1. If)1 196 1 2403 6112 t 10 CW f (theta)2629 6112 w 10 S f (= =)1 55 1 2959 6112 t 10 R f (2)3088 6182 w (1)3088 6052 w 10 S1 f (_ _)1 80 1 3073 6082 t 10 R f (, then)1 227 1 3163 6112 t 10 CW f (N)3420 6112 w 10 S f (= =)1 55 1 3510 6112 t 10 R f ({ 2, 4, 6,)3 363 1 3595 6112 t (. . .)2 125 1 4013 6087 t (} and)1 222 1 4193 6112 t 10 CW f (gamma)4445 6112 w 10 S f (= =)1 55 1 4775 6112 t 10 R f (2. 0)1 180 1 4860 6112 t 10 S f (<)1420 6282 w 10 R f (=)1475 6282 w 10 CW f (theta)1556 6282 w 10 S f (<)1881 6282 w 10 R f ( Default:)1 377(= 1 is required.)3 605 2 1936 6282 t 10 CW f (theta)2943 6282 w 10 R f (= 1.)1 156 1 3268 6282 t 10 CW f (j)3544 6282 w 10 R f (= 1.)1 156 1 3629 6282 t 10 CW f (theta)3810 6282 w 10 R f (=)4135 6282 w 10 CW f (f)4216 6282 w 10 R f (.)4276 6282 w 10 CW f (beta)770 6438 w 10 R f ( to \()2 215( error in the discretization scheme is proportional)7 2148(- The)1 305 3 1270 6438 t 10 CW f (t1-t0)3938 6438 w 10 R f (\)**)4238 6438 w 10 CW f (beta)4371 6438 w 10 R f (. Default:)1 429 1 4611 6438 t 10 CW f (beta)1420 6558 w 10 R f (= 1.)1 156 1 1685 6558 t 10 CW f (j)1961 6558 w 10 R f (= 2.)1 156 1 2046 6558 t 10 CW f (beta)2227 6558 w 10 R f (=)2492 6558 w 10 CW f (f)2573 6558 w 10 R f (.)2633 6558 w 10 CW f (gamma)770 6714 w 10 R f ( is proportional to)3 726( error in the discretization scheme)5 1370(- The)1 305 3 1270 6714 t 10 CW f (dt)3700 6714 w 10 R f (**)3820 6714 w 10 CW f (gamma)3920 6714 w 10 R f (. Default:)1 406 1 4220 6714 t 10 CW f (gamma)4655 6714 w 10 R f (=)4984 6714 w (1.)1420 6834 w 10 CW f (j)1615 6834 w 10 R f (= 3.)1 156 1 1700 6834 t 10 CW f (gamma)1881 6834 w 10 R f (=)2206 6834 w 10 CW f (f)2287 6834 w 10 R f (.)2347 6834 w cleartomark showpage saveobj restore end %%EndPage: 24 23 %%Page: 25 24 DpostDict begin /saveobj save def mark 24 pagesetup 10 R f (- 25 -)2 216 1 2772 480 t 10 CW f (delta)770 840 w 10 R f ( error request is proportional to)5 1245(- The)1 305 2 1270 840 t 10 CW f (dt)2845 840 w 10 R f (**)2965 840 w 10 CW f (delta)3065 840 w 10 R f (. Default:)1 402 1 3365 840 t 10 CW f (delta)3792 840 w 10 R f (= 0.)1 156 1 4117 840 t 10 CW f (j)4393 840 w 10 R f (= 4.)1 156 1 4478 840 t 10 CW f (delta)4659 840 w 10 R f (=)4984 840 w 10 CW f (f)1420 960 w 10 R f (.)1480 960 w 10 CW f (hfract)770 1116 w 10 R f ( the user will take relative to)6 1293( Real variable indicating how small a time-step)7 2059(- A)1 222 3 1270 1116 t 10 CW f (dt)4895 1116 w 10 R f (.)5015 1116 w (Default:)1420 1236 w 10 CW f (hfract)1772 1236 w 10 R f (= 1.)1 156 1 2157 1236 t 10 CW f (j)2433 1236 w 10 R f (= 1001.)1 306 1 2518 1236 t 10 CW f (hfract)2849 1236 w 10 R f (=)3234 1236 w 10 CW f (r)3315 1236 w 10 R f (.)3375 1236 w 10 CW f (egive)770 1392 w 10 R f ( try to solve the nonlin-)5 956( Real variable controlling how accurately Newton's method will)8 2592(- A)1 222 3 1270 1392 t ( to the)2 260(ear equations, relative to the user's accuracy request for the solution)10 2778 2 1420 1512 t 10 B f (pde)4488 1512 w 10 R f (. Specifi-)1 396 1 4644 1512 t ( the user's error request)4 945( \()1 84( solve the nonlinear equations to a tolerance of)8 1871(cally, it will try to)4 720 4 1420 1632 t (to)1420 1752 w 10 CW f (TTGR)1534 1752 w 10 R f (\) /)1 97 1 1810 1752 t 10 CW f (egive)1943 1752 w 10 R f (.)2243 1752 w 10 CW f (egive)2399 1752 w 10 R f ( Default:)1 388(should thus be greater than 1.)5 1231 2 2735 1752 t 10 CW f (egive)4389 1752 w 10 R f (= 1e+2.)1 316 1 4724 1752 t 10 CW f (j)1420 1872 w 10 R f (= 1002.)1 306 1 1505 1872 t 10 CW f (egive)1836 1872 w 10 R f (=)2161 1872 w 10 CW f (r)2242 1872 w 10 R f (.)2302 1872 w 10 CW f (kj)770 2028 w 10 R f ( Such)1 269( the Jacobian is re-computed.)4 1243( variable that controls the frequency with which)7 2036(- A)1 222 4 1270 2028 t (evaluations can be very costly and)5 1407 1 1420 2148 t 10 CW f (kj)2859 2148 w 10 R f ( the follow-)2 478(\(for "KeepJacobian"\) controls them in)4 1551 2 3011 2148 t (ing way:)1 347 1 1420 2268 t (0 - New Jacobian every Newton iteration.)6 1665 1 1780 2448 t (Very safe and stable, expensive.)4 1285 1 1880 2568 t (1 - New Jacobian every time sub-step.)6 1527 1 1780 2688 t (Less safe, stable and expensive.)4 1269 1 1880 2808 t (2 - New Jacobian for each time-step.)6 1465 1 1780 2928 t (Not very safe, stable or cheap, except for nearly linear problems.)10 2584 1 1880 3048 t (3 - New Jacobian whenever there is a re-start.)8 1824 1 1780 3168 t (Mostly used for linear or nearly linear problems.)7 1941 1 1880 3288 t (Cheap but flaky.)2 663 1 1880 3408 t (4 - New Jacobian whenever Newton iteration fails to converge.)9 2521 1 1780 3528 t (Only updates Jacobian if it appears out-of-date.)6 1893 1 1880 3648 t (Ditto rest of discussion of 3 above.)6 1396 1 1880 3768 t (5 - Only computes the Jacobian ONCE.)6 1584 1 1780 3888 t (Use only for nearly linear problems.)5 1447 1 1880 4008 t (Exceedingly cheap, when it works.)4 1397 1 1880 4128 t (Default:)1420 4308 w 10 CW f (kj)1772 4308 w 10 R f (= 0.)1 156 1 1917 4308 t 10 CW f (j)2193 4308 w 10 R f (= 2001.)1 306 1 2278 4308 t 10 CW f (kj)2609 4308 w 10 R f (=)2754 4308 w 10 CW f (i)2835 4308 w 10 R f (.)2895 4308 w 10 CW f (minit)770 4464 w 10 R f ( minimum number of Newton iterations to go before checking that the convergence)12 3465(- The)1 305 2 1270 4464 t ( Default:)1 377(rate is reasonable.)2 717 2 1420 4584 t 10 CW f (minit)2539 4584 w 10 R f (= 10.)1 206 1 2864 4584 t 10 CW f (j)3190 4584 w 10 R f (= 2002.)1 306 1 3275 4584 t 10 CW f (minit)3606 4584 w 10 R f (=)3931 4584 w 10 CW f (i)4012 4584 w 10 R f (.)4072 4584 w 10 CW f (maxit)770 4740 w 10 R f ( Default:)1 386( maximum number of Newton iterations to use.)7 1956(- The)1 305 3 1270 4740 t 10 CW f (maxit)3951 4740 w 10 R f (= 50.)1 215 1 4285 4740 t 10 CW f (j)4629 4740 w 10 R f (= 2003.)1 316 1 4724 4740 t 10 CW f (maxit)1420 4860 w 10 R f (=)1745 4860 w 10 CW f (i)1826 4860 w 10 R f (.)1886 4860 w 10 CW f (kmax)770 5016 w 10 R f ( maximal)1 391( The)1 221( in the extrapolation tableau.)4 1204( maximum number of columns allowed)5 1649(- The)1 305 5 1270 5016 t (order that)1 398 1 1420 5136 t 10 CW f (TTGR)1856 5136 w 10 R f (can achieve is then 2*)4 933 1 2134 5136 t 10 CW f (kmax)3067 5136 w 10 R f (if)3344 5136 w 10 CW f (theta)3442 5136 w 10 R f (= 0.5e0, or)2 457 1 3779 5136 t 10 CW f (kmax)4273 5136 w 10 R f (if)4550 5136 w 10 CW f (theta)4648 5136 w 10 S f (\271)4985 5136 w 10 R f (0.5e0. Default:)1 621 1 1420 5256 t 10 CW f (kmax)2066 5256 w 10 R f (= 10.)1 206 1 2331 5256 t 10 CW f (j)2657 5256 w 10 R f (= 2004.)1 306 1 2742 5256 t 10 CW f (kinit)770 5412 w 10 R f ( can allow)2 412( This)1 229( initial level of extrapolation to use for the first time-step.)10 2295(- The)1 305 4 1270 5412 t 10 CW f (TTGR)4537 5412 w 10 R f (to use)1 237 1 4803 5412 t ( kinit = 2.)3 390( Default:)1 377(a higher-order scheme from the start.)5 1479 3 1420 5532 t 10 CW f (j)3786 5532 w 10 R f (= 2005.)1 306 1 3871 5532 t 10 CW f (kinit)4202 5532 w 10 R f (=)4527 5532 w 10 CW f (i)4608 5532 w 10 R f (.)4668 5532 w 10 CW f (mmax)770 5688 w 10 R f ( maximum number of levels of extrapolation permitted.)7 2346(- The)1 305 2 1270 5688 t 10 CW f (mmax)4059 5688 w 10 S f (>)4342 5688 w 10 R f (=)4397 5688 w 10 CW f (kmax)4496 5688 w 10 R f (+ 2 is)2 261 1 4779 5688 t (required and)1 514 1 1420 5808 t 10 CW f (mmax)1972 5808 w 10 S f (>)2250 5808 w 10 R f (=)2305 5808 w 10 CW f (kmax)2399 5808 w 10 R f ( Default:)1 390(+ 4 is a good idea.)5 798 2 2677 5808 t 10 CW f ( 15)1 192(mmax =)1 373 2 3903 5808 t 10 R f (.)4468 5808 w 10 CW f (j)4625 5808 w 10 R f (= 2006.)1 318 1 4722 5808 t 10 CW f (mmax)1420 5928 w 10 R f (=)1685 5928 w 10 CW f (i)1766 5928 w 10 R f (.)1826 5928 w 10 CW f (mxq)770 6084 w 10 R f ( number of)2 442(- The)1 305 2 1270 6084 t 10 I f (x x)1 44 1 2044 6084 t 10 R f ( integrals,)1 397(Gaussian quadrature points to be used to compute the Galerkin)9 2528 2 2115 6084 t (see section 3.)2 535 1 1420 6204 t 10 CW f (mxq)2075 6204 w 10 S f (>)2280 6204 w 10 R f (=)2335 6204 w 10 CW f (kx-1)2416 6204 w 10 R f ( Default:)1 377(is required.)1 449 2 2681 6204 t 10 CW f (mxq)3532 6204 w 10 R f (=)3737 6204 w 10 CW f (kx)3818 6204 w 10 R f (.)3938 6204 w 10 CW f (j)4083 6204 w 10 R f (= 2008.)1 306 1 4168 6204 t 10 CW f (mxq)4499 6204 w 10 R f (=)4704 6204 w 10 CW f (i)4785 6204 w 10 R f (.)4845 6204 w 10 CW f (myq)770 6360 w 10 R f ( number of)2 442(- The)1 305 2 1270 6360 t 10 I f (y y)1 44 1 2044 6360 t 10 R f ( integrals,)1 397(Gaussian quadrature points to be used to compute the Galerkin)9 2528 2 2115 6360 t (see section 3.)2 535 1 1420 6480 t 10 CW f (myq)2075 6480 w 10 S f (>)2280 6480 w 10 R f (=)2335 6480 w 10 CW f (ky-1)2416 6480 w 10 R f ( Default:)1 377(is required.)1 449 2 2681 6480 t 10 CW f (myq)3532 6480 w 10 R f (=)3737 6480 w 10 CW f (ky)3818 6480 w 10 R f (.)3938 6480 w 10 CW f (j)4083 6480 w 10 R f (= 2009.)1 306 1 4168 6480 t 10 CW f (myq)4499 6480 w 10 R f (=)4704 6480 w 10 CW f (i)4785 6480 w 10 R f (.)4845 6480 w 10 CW f (LA)770 6636 w 10 R f ( Integer variable indicating how the matrix equations should be solved.)10 2838(- An)1 272 2 1270 6636 t 10 S f (-)1780 6816 w 10 R f (1 - non-pivoting banded solve.)4 1224 1 1835 6816 t (+1 - pivoting banded solve. Default; see \(6.1\) below for reasons.)10 2576 1 1780 6936 t 10 S f (-)1780 7056 w 10 R f (2 - non-pivoting sparse solve.)4 1185 1 1835 7056 t (+2 - pivoting sparse solve.)4 1058 1 1780 7176 t cleartomark showpage saveobj restore end %%EndPage: 25 24 %%Page: 26 25 DpostDict begin /saveobj save def mark 25 pagesetup 10 R f (- 26 -)2 216 1 2772 480 t (Default:)1420 840 w 10 CW f (LA)1772 840 w 10 R f (= +1.)1 212 1 1917 840 t 10 CW f (j)2249 840 w 10 R f (= 2010.)1 306 1 2334 840 t 10 CW f (LA)2665 840 w 10 R f (=)2810 840 w 10 CW f (i)2891 840 w 10 R f (.)2951 840 w 10 CW f (Pieces)770 996 w 10 R f ( Integer indicating how much of the Jacobian should be used.)10 2449(- An)1 272 2 1270 996 t (+2 - the whole thing. Default.)5 1185 1 1780 1176 t (0 - only the diagonal)4 827 1 1854 1296 t 10 B f (pde)2706 1296 w 10 R f (terms.)2887 1296 w 10 S f (-)1780 1416 w 10 R f (1 - only the lower triangular terms.)6 1395 1 1835 1416 t (+1 - the upper triangular terms.)5 1248 1 1780 1536 t (Default:)1420 1716 w 10 CW f (Pieces)1772 1716 w 10 R f (= +2.)1 212 1 2157 1716 t 10 CW f (j)2489 1716 w 10 R f (= 2011.)1 306 1 2574 1716 t 10 CW f (Pieces)2905 1716 w 10 R f (=)3290 1716 w 10 CW f (i)3371 1716 w 10 R f (.)3431 1716 w 10 CW f (PC)770 1872 w 10 R f ( used when solving the matrix)5 1292( Integer variable indicating the pre-conditioner to be)7 2206(- An)1 272 3 1270 1872 t (equations iteratively.)1 838 1 1420 1992 t (0 - none. Default - using a direct solve.)8 1558 1 1780 2172 t (1 - incomplete LU.)3 760 1 1780 2292 t (2 - modified incomplete LU. See [20].)6 1531 1 1780 2412 t (Default:)1420 2592 w 10 CW f (PC)1772 2592 w 10 R f (= 0.)1 156 1 1917 2592 t 10 CW f (j)2193 2592 w 10 R f (= 2012.)1 306 1 2278 2592 t 10 CW f (PC)2609 2592 w 10 R f (=)2754 2592 w 10 CW f (i)2835 2592 w 10 R f (.)2895 2592 w 10 CW f (Accel)770 2748 w 10 R f ( variable indicating the accelerator to be used when solving the matrix equa-)12 3181( Integer)1 317(- An)1 272 3 1270 2748 t (tions iteratively.)1 650 1 1420 2868 t (0 - none. Default - doing a direct solve.)8 1569 1 1780 3048 t (1 - conjugate gradient on the normal equations.)7 1886 1 1780 3168 t (2 - Orthomin. See [11].)4 932 1 1780 3288 t (Default:)1420 3468 w 10 CW f (Accel)1772 3468 w 10 R f (= 0.)1 156 1 2097 3468 t 10 CW f (j)2373 3468 w 10 R f (= 2013.)1 306 1 2458 3468 t 10 CW f (Accel)2789 3468 w 10 R f (=)3114 3468 w 10 CW f (i)3195 3468 w 10 R f (.)3255 3468 w 10 CW f (xpoly)770 3624 w 10 R f ( variable indicating whether polynomial \(True\) or rational \(False\) extrapolation)9 3214( Logical)1 334(- A)1 222 3 1270 3624 t ( Default:)1 377(is to be used.)3 522 2 1420 3744 t 10 CW f (xpoly)2344 3744 w 10 R f (= False.)1 317 1 2669 3744 t 10 CW f (j)3106 3744 w 10 R f (= 3001.)1 306 1 3191 3744 t 10 CW f (xpoly)3522 3744 w 10 R f (=)3847 3744 w 10 CW f (l)3928 3744 w 10 R f (.)3988 3744 w 10 CW f (erputs)770 3900 w 10 R f ( then use per-)3 597( True,)1 256( If)1 134( variable controlling the use of error-per-unit-time-step.)6 2328(- Logical)1 455 5 1270 3900 t ( Default:)1 385( criterion.)1 396( use the per-step)3 678( Otherwise,)1 494(unit-time-step error criterion.)2 1185 5 1420 4020 t 10 CW f (erputs)4591 4020 w 10 R f (=)4984 4020 w (False.)1420 4140 w 10 CW f (j)1776 4140 w 10 R f (= 3002.)1 306 1 1861 4140 t 10 CW f (erputs)2192 4140 w 10 R f (=)2577 4140 w 10 CW f (l)2658 4140 w 10 R f (.)2718 4140 w 10 CW f (N)770 4296 w 10 R f ( of length)2 385( Integer array)2 536(- An)1 272 3 1270 4296 t 10 CW f (mmax)2489 4296 w 10 R f (giving the number of sub-steps to be used in the extrapo-)10 2285 1 2755 4296 t (lation.)1420 4416 w 10 CW f (N)1795 4416 w 10 R f (must be strictly monotone increasing and positive.)6 2025 1 1882 4416 t 10 CW f (N)4029 4416 w 10 R f (\(i\) is set by)3 450 1 4089 4416 t 10 CW f (j)4565 4416 w 10 S f (= =)1 55 1 4651 4416 t 10 R f (4000)4732 4416 w 10 S f (+ +)1 55 1 4932 4416 t 10 R f (i.)4987 4416 w (If)1420 4536 w 10 CW f (N)1514 4536 w 10 R f (\(i\) is set, then)3 553 1 1574 4536 t 10 CW f (N)2155 4536 w 10 R f (\(i+1\) is set to 0 by default; so be sure to set)11 1757 1 2215 4536 t 10 CW f (N)4001 4536 w 10 R f (in increasing order of i.)4 950 1 4090 4536 t (For any)1 315 1 1420 4656 t 10 CW f (N)1767 4656 w 10 R f ( only)1 209( rule is that if)4 561( The)1 212(\(i\) which is 0, a default value is computed.)8 1759 4 1827 4656 t 10 CW f (N)4599 4656 w 10 R f (\(1\) is set,)2 381 1 4659 4656 t (then the user wants)3 804 1 1420 4782 t 10 CW f (N)2261 4782 w 10 R f (\(i\))2321 4782 w 10 S f (\272)2452 4782 w 10 R f (\()2544 4782 w 11 S f (\326` `)1 127 1 2609 4782 t 10 R f (2 \))1 131 1 2678 4782 t 7 I f (i i)1 20 1 2820 4712 t 7 S f (- -)1 39 1 2856 4712 t 7 R f (1)2906 4712 w 10 CW f (N)2986 4782 w 10 R f ( only)1 216(\(1\). If)1 269 2 3046 4782 t 10 CW f (N)3569 4782 w 10 R f (\(1\) and)1 298 1 3629 4782 t 10 CW f (N)3965 4782 w 10 R f (\(2\) are set, then)3 659 1 4025 4782 t 10 CW f (N)4722 4782 w 10 R f (\(i\))4782 4782 w 10 S f (= =)1 55 1 4914 4782 t 10 R f (\()5007 4782 w 10 CW f (N)1420 4902 w 10 R f (\(2\)/)1480 4902 w 10 CW f (N)1624 4902 w 10 R f (\(1\) \))1 175 1 1684 4902 t 10 CW f (N)1885 4902 w 10 R f (\(i-1\) for any)2 489 1 1945 4902 t 10 CW f (N)2460 4902 w 10 R f (\(i\))2520 4902 w 10 S f (= =)1 55 1 2640 4902 t 10 R f (0. If)1 192 1 2744 4902 t 10 CW f (N)2961 4902 w 10 R f (\(3\) is also set, then)4 752 1 3021 4902 t 10 CW f (N)3798 4902 w 10 R f (\(i\) = 2 *)3 325 1 3858 4902 t 10 CW f (N)4208 4902 w 10 R f (\(i-2\), for any)2 512 1 4268 4902 t 10 CW f (N)4805 4902 w 10 R f (\(i\) =)1 175 1 4865 4902 t ( the)1 147(0. See)1 269 2 1420 5022 t 10 CW f (theta)1861 5022 w 10 R f (description above for some default)4 1386 1 2186 5022 t 10 CW f (N)3597 5022 w 10 R f (values.)3682 5022 w 10 CW f (j)4082 5022 w 10 R f (= 4000 + i.)3 440 1 4167 5022 t 10 CW f (N)4632 5022 w 10 R f (\(i\) =)1 175 1 4692 5022 t 10 CW f (i)4892 5022 w 10 R f (.)4952 5022 w (The following table summarizes the values that can be set by)10 2435 1 720 5178 t 10 CW f (TTGRV)3180 5178 w 10 S f (_ __________________________________)1 1725 1 2017 5258 t 10 R f ( to)1 103( Set)1 279( Default)1 577(Name j)1 605 4 2128 5378 t 10 S f (_ __________________________________)1 1725 1 2017 5398 t (_ __________________________________)1 1725 1 2017 5418 t 10 CW f (theta)2097 5598 w 10 R f (1 1)1 492 1 2694 5598 t 10 CW f (f)3546 5598 w (beta)2127 5718 w 10 R f (2 1)1 492 1 2694 5718 t 10 CW f (f)3546 5718 w (gamma)2097 5838 w 10 R f (3 1)1 492 1 2694 5838 t 10 CW f (f)3546 5838 w (delta)2097 5958 w 10 R f (4 0)1 492 1 2694 5958 t 10 CW f (f)3546 5958 w 10 S f (_ __________________________________)1 1725 1 2017 5978 t 10 CW f (hfract)2067 6098 w 10 R f (1001 1)1 567 1 2619 6098 t 10 CW f (r)3546 6098 w (egive)2097 6218 w 10 R f (1002 100)1 617 1 2619 6218 t 10 CW f (r)3546 6218 w 10 S f (_ __________________________________)1 1725 1 2017 6238 t 10 CW f (kj)2187 6358 w 10 R f (2001 0)1 567 1 2619 6358 t 10 CW f (i)3546 6358 w (minit)2097 6478 w 10 R f (2002 10)1 592 1 2619 6478 t 10 CW f (i)3546 6478 w (maxit)2097 6598 w 10 R f (2003 50)1 592 1 2619 6598 t 10 CW f (i)3546 6598 w (kmax)2127 6718 w 10 R f (2004 10)1 592 1 2619 6718 t 10 CW f (i)3546 6718 w (kinit)2097 6838 w 10 R f (2005 2)1 567 1 2619 6838 t 10 CW f (i)3546 6838 w (mmax)2127 6958 w 10 R f (2006 15)1 592 1 2619 6958 t 10 CW f (i)3546 6958 w 10 S f ( \347)1 -1725(_ __________________________________)1 1725 2 2017 6980 t (\347)2017 6958 w (\347)2017 6858 w (\347)2017 6758 w (\347)2017 6658 w (\347)2017 6558 w (\347)2017 6458 w (\347)2017 6358 w (\347)2017 6258 w (\347)2017 6158 w (\347)2017 6058 w (\347)2017 5958 w (\347)2017 5858 w (\347)2017 5758 w (\347)2017 5658 w (\347)2017 5558 w (\347)2017 5458 w (\347)2017 5358 w (\347)2502 6980 w (\347)2502 6958 w (\347)2502 6858 w (\347)2502 6758 w (\347)2502 6658 w (\347)2502 6558 w (\347)2502 6458 w (\347)2502 6358 w (\347)2502 6258 w (\347)2502 6158 w (\347)2502 6058 w (\347)2502 5958 w (\347)2502 5858 w (\347)2502 5758 w (\347)2502 5658 w (\347)2502 5558 w (\347)2502 5458 w (\347)2502 5358 w (\347)2936 6980 w (\347)2936 6958 w (\347)2936 6858 w (\347)2936 6758 w (\347)2936 6658 w (\347)2936 6558 w (\347)2936 6458 w (\347)2936 6358 w (\347)2936 6258 w (\347)2936 6158 w (\347)2936 6058 w (\347)2936 5958 w (\347)2936 5858 w (\347)2936 5758 w (\347)2936 5658 w (\347)2936 5558 w (\347)2936 5458 w (\347)2936 5358 w (\347)3386 6980 w (\347)3386 6958 w (\347)3386 6858 w (\347)3386 6758 w (\347)3386 6658 w (\347)3386 6558 w (\347)3386 6458 w (\347)3386 6358 w (\347)3386 6258 w (\347)3386 6158 w (\347)3386 6058 w (\347)3386 5958 w (\347)3386 5858 w (\347)3386 5758 w (\347)3386 5658 w (\347)3386 5558 w (\347)3386 5458 w (\347)3386 5358 w (\347)3742 6980 w (\347)3742 6958 w (\347)3742 6858 w (\347)3742 6758 w (\347)3742 6658 w (\347)3742 6558 w (\347)3742 6458 w (\347)3742 6358 w (\347)3742 6258 w (\347)3742 6158 w (\347)3742 6058 w (\347)3742 5958 w (\347)3742 5858 w (\347)3742 5758 w (\347)3742 5658 w (\347)3742 5558 w (\347)3742 5458 w (\347)3742 5358 w cleartomark showpage saveobj restore end %%EndPage: 26 25 %%Page: 27 26 DpostDict begin /saveobj save def mark 26 pagesetup 10 R f (- 27 -)2 216 1 2772 480 t 10 CW f (mxq)2157 840 w 10 R f (2008)2619 840 w 10 CW f (0 i)1 475 1 3131 840 t (myq)2157 960 w 10 R f (2009)2619 960 w 10 CW f (0 i)1 475 1 3131 960 t (la)2187 1080 w 10 R f (2010 1)1 567 1 2619 1080 t 10 CW f (i)3546 1080 w (pieces)2067 1200 w 10 R f (2011 +2)1 595 1 2619 1200 t 10 CW f (i)3546 1200 w (pc)2187 1320 w 10 R f (2012 0)1 567 1 2619 1320 t 10 CW f (i)3546 1320 w (accel)2097 1440 w 10 R f (2013 0)1 567 1 2619 1440 t 10 CW f (i)3546 1440 w 10 S f (_ __________________________________)1 1725 1 2017 1460 t 10 CW f (xpoly)2097 1580 w 10 R f (3001)2619 1580 w 10 CW f (False l)1 595 1 3011 1580 t (erputs)2067 1700 w 10 R f (3002)2619 1700 w 10 CW f (False l)1 595 1 3011 1700 t 10 S f (_ __________________________________)1 1725 1 2017 1720 t 10 CW f (N)2170 1840 w 10 R f (\(i\) 4000+i)1 631 1 2230 1840 t 10 S f (- -)1 55 1 3133 1840 t 10 CW f (i)3546 1840 w 10 S f ( \347)1 -1725(_ __________________________________)1 1725 2 2017 1860 t (\347)2017 1820 w (\347)2017 1720 w (\347)2017 1620 w (\347)2017 1520 w (\347)2017 1420 w (\347)2017 1320 w (\347)2017 1220 w (\347)2017 1120 w (\347)2017 1020 w (\347)2017 920 w (\347)2017 820 w (\347)2502 1860 w (\347)2502 1820 w (\347)2502 1720 w (\347)2502 1620 w (\347)2502 1520 w (\347)2502 1420 w (\347)2502 1320 w (\347)2502 1220 w (\347)2502 1120 w (\347)2502 1020 w (\347)2502 920 w (\347)2502 820 w (\347)2936 1860 w (\347)2936 1820 w (\347)2936 1720 w (\347)2936 1620 w (\347)2936 1520 w (\347)2936 1420 w (\347)2936 1320 w (\347)2936 1220 w (\347)2936 1120 w (\347)2936 1020 w (\347)2936 920 w (\347)2936 820 w (\347)3386 1860 w (\347)3386 1820 w (\347)3386 1720 w (\347)3386 1620 w (\347)3386 1520 w (\347)3386 1420 w (\347)3386 1320 w (\347)3386 1220 w (\347)3386 1120 w (\347)3386 1020 w (\347)3386 920 w (\347)3386 820 w (\347)3742 1860 w (\347)3742 1820 w (\347)3742 1720 w (\347)3742 1620 w (\347)3742 1520 w (\347)3742 1420 w (\347)3742 1320 w (\347)3742 1220 w (\347)3742 1120 w (\347)3742 1020 w (\347)3742 920 w (\347)3742 820 w 10 R f (Note that)1 369 1 720 2076 t 10 CW f (mxq)1114 2076 w 10 S f (= =)1 55 1 1319 2076 t 10 R f (0 means that)2 505 1 1414 2076 t 10 CW f (kx)1944 2076 w 10 R f (points will be used in)4 856 1 2089 2076 t 10 I f (x x)1 44 1 2970 2076 t 10 R f ( for)1 141( Similarly)1 423(, by default.)2 477 3 3014 2076 t 10 CW f (myq)4080 2076 w 10 R f (.)4260 2076 w ( of)1 119(The Double precision version)3 1214 2 970 2232 t 10 CW f (TTGRV)2339 2232 w 10 R f (is)2675 2232 w 10 CW f (DTTGRV)2778 2232 w 10 R f (, with all Real arguments typed Double preci-)7 1902 1 3138 2232 t (sion, except)1 477 1 720 2352 t 10 CW f (hfract)1222 2352 w 10 R f (and)1607 2352 w 10 CW f (egive)1776 2352 w 10 R f (which remain Real.)2 779 1 2101 2352 t 10 B f (Space Used)1 486 1 720 2592 t 10 R f (The space used by the various knob settings is summarized in the following table)13 3244 1 970 2748 t 10 S f (_________________________________________)1856 2828 w 10 R f (Space Required)1 629 1 2565 2948 t 10 S f (_________________________________________)1856 2968 w 10 R f (Method Words)1 1338 1 2046 3088 t 10 S f (_________________________________________)1856 3108 w (_________________________________________)1856 3128 w 10 R f (No pivot/Band)1 592 1 1906 3248 t 10 I f (n n)1 50 1 2931 3248 t 7 I f (u u)1 35 1 2992 3268 t 10 I f (n n)1 50 1 3067 3248 t 10 R f (\( 2)1 91 1 3149 3248 t 10 I f (H H)1 72 1 3272 3248 t 10 S f (- -)1 55 1 3384 3248 t 10 R f (1 \))1 91 1 3479 3248 t 10 S f (_________________________________________)1856 3268 w 10 R f (Pivot/Band)1976 3388 w 10 I f (n n)1 50 1 2931 3388 t 7 I f (u u)1 35 1 2992 3408 t 10 I f (n n)1 50 1 3067 3388 t 10 R f (\( 3)1 91 1 3149 3388 t 10 I f (H H)1 72 1 3272 3388 t 10 S f (- -)1 55 1 3384 3388 t 10 R f (1 \))1 91 1 3479 3388 t 10 S f (_________________________________________)1856 3408 w 10 R f (Sparse \262)1 341 1 2031 3528 t 10 I f (O O)1 72 1 2767 3528 t 10 R f (\()2847 3528 w 10 I f (n n)1 50 1 2888 3528 t 7 I f (u u)1 35 1 2943 3547 t 7 R f (2)2943 3488 w 10 I f (k k)1 44 1 3018 3528 t 7 I f (x x)1 31 1 3073 3548 t 10 I f (k k)1 44 1 3144 3528 t 7 I f (y y)1 31 1 3199 3548 t 10 I f (n n)1 50 1 3270 3528 t 7 I f (x x)1 31 1 3331 3548 t 10 I f (n n)1 50 1 3402 3528 t 10 R f (log)3484 3528 w 10 I f (n n)1 50 1 3644 3528 t 10 R f (\))3702 3528 w 10 S f (_________________________________________)1856 3548 w 10 R f ( \()1 65(MIC 3)1 591 2 2107 3668 t 10 I f (n n)1 50 1 2771 3668 t 7 I f (u u)1 35 1 2826 3687 t 7 R f (2)2826 3628 w 10 I f (n n)1 50 1 2901 3668 t 10 R f (\( 2)1 91 1 2983 3668 t 10 I f (k k)1 44 1 3106 3668 t 7 I f (x x)1 31 1 3161 3688 t 10 S f (- -)1 55 1 3216 3668 t 10 R f ( 2)1 58( \()1 73(1 \))1 91 3 3287 3668 t 10 I f (k k)1 44 1 3541 3668 t 7 I f (y y)1 31 1 3596 3688 t 10 S f (- -)1 55 1 3651 3668 t 10 R f (1 \) \))2 132 1 3722 3668 t 10 S f (_________________________________________ \347)1 0 1 1856 3688 t (\347)1856 3628 w (\347)1856 3528 w (\347)1856 3428 w (\347)1856 3328 w (\347)1856 3228 w (\347)1856 3128 w (\347)1856 3028 w (\347)1856 2928 w (\347)2573 3688 w (\347)2573 3668 w (\347)2573 3568 w (\347)2573 3468 w (\347)2573 3368 w (\347)2573 3268 w (\347)2573 3168 w (\347)2573 3068 w (\347)3904 3688 w (\347)3904 3628 w (\347)3904 3528 w (\347)3904 3428 w (\347)3904 3328 w (\347)3904 3228 w (\347)3904 3128 w (\347)3904 3028 w (\347)3904 2928 w 10 R f (where)720 3868 w 10 I f (H H)1 72 1 992 3868 t 10 S f (\272)1105 3868 w 10 I f (n n)1 50 1 1225 3868 t 7 I f (u u)1 35 1 1286 3888 t 10 R f (\()1361 3868 w 10 I f (k k)1 44 1 1426 3868 t 7 I f (x x)1 31 1 1481 3888 t 10 S f (+ +)1 55 1 1560 3868 t 10 R f (\()1655 3868 w 10 I f (n n)1 50 1 1720 3868 t 7 I f (x x)1 31 1 1781 3888 t 10 S f (- -)1 55 1 1860 3868 t 10 I f (k k)1 44 1 1955 3868 t 7 I f (x x)1 31 1 2010 3888 t 10 R f (\) \()1 106 1 2081 3868 t 10 I f (k k)1 44 1 2195 3868 t 7 I f (y y)1 31 1 2250 3888 t 10 S f (- -)1 55 1 2329 3868 t 10 R f ( and)1 173( \),)1 98(1 \))1 115 3 2424 3868 t 10 I f (n n)1 50 1 2839 3868 t 10 S f (\272)2930 3868 w 10 I f (n n)1 50 1 3026 3868 t 7 I f (x x)1 31 1 3087 3888 t 10 I f (n n)1 50 1 3158 3868 t 7 I f (y y)1 31 1 3219 3888 t 10 R f (. When)1 317 1 3258 3868 t 10 CW f (Pieces)3604 3868 w 10 R f (is 0, a factor of)4 621 1 3993 3868 t 10 I f (n n)1 50 1 4644 3868 t 7 I f (u u)1 35 1 4705 3888 t 10 R f (can be)1 262 1 4778 3868 t (removed from the figures.)3 1042 1 720 3988 t (\262 - This figure is)4 666 1 720 4144 t 10 I f ( y)1 0( ll ly)2 72( ua al)2 78( su)1 50(u us)1 89 5 1411 4144 t 10 R f ( discussion of the)3 697( See)1 194(accurate, but not always.)3 989 3 1725 4144 t 10 B f (pde)3630 4144 w 10 I f (u u)1 50 1 3811 4144 t 7 I f (x xy y)2 62 1 3872 4164 t 10 S f (= =)1 55 1 3991 4144 t 10 R f (0 below.)1 344 1 4095 4144 t 10 B f (Time Used.)1 483 1 720 4384 t 10 R f ( used by the various knob settings is summarized in the following)11 2679(The floating-point operation count)3 1391 2 970 4540 t (table)720 4660 w 10 S f (_ ___________________________________________)1 2193 1 1783 4740 t 10 R f (Operation Count)1 669 1 2545 4860 t 10 S f (_ ___________________________________________)1 2193 1 1783 4880 t 10 R f (Method Multiplies)1 1483 1 1973 5000 t 10 S f (_ ___________________________________________)1 2193 1 1783 5020 t (_ ___________________________________________)1 2193 1 1783 5040 t 10 R f (No pivot/Band)1 592 1 1833 5160 t 10 I f (n n)1 50 1 2948 5160 t 7 I f (u u)1 35 1 3009 5180 t 10 I f (n n)1 50 1 3084 5160 t 10 R f (\()3166 5160 w 10 I f (H H)1 72 1 3207 5160 t 10 S f (- -)1 55 1 3319 5160 t 10 R f (1 \))1 91 1 3414 5160 t 7 R f (2)3510 5120 w 10 S f (_ ___________________________________________)1 2193 1 1783 5180 t 10 R f (Pivot/Band 2)1 1054 1 1903 5300 t 10 I f (n n)1 50 1 2989 5300 t 7 I f (u u)1 35 1 3050 5320 t 10 I f (n n)1 50 1 3125 5300 t 10 R f (\()3207 5300 w 10 I f (H H)1 72 1 3248 5300 t 10 S f (- -)1 55 1 3360 5300 t 10 R f (1 \))1 91 1 3455 5300 t 7 R f (2)3551 5260 w 10 S f (_ ___________________________________________)1 2193 1 1783 5320 t 10 R f (Sparse \262)1 341 1 1958 5440 t 10 I f (O O)1 72 1 2887 5440 t 10 R f (\()2967 5440 w 10 I f (n n)1 50 1 3008 5440 t 7 I f (u u)1 35 1 3063 5459 t 7 R f (2)3063 5400 w 10 I f (k k)1 44 1 3138 5440 t 7 I f (x x)1 31 1 3193 5460 t 10 I f (k k)1 44 1 3264 5440 t 7 I f (y y)1 31 1 3319 5460 t 10 I f ( n)1 0( n)1 82(n n)1 50 3 3390 5440 t 7 I f (x x)1 31 1 3533 5460 t 10 R f (\))3580 5440 w 10 S f (_ ___________________________________________)1 2193 1 1783 5460 t 10 R f (MIC \262\262)1 314 1 1972 5580 t 10 I f (O O)1 72 1 2575 5580 t 10 R f (\()2655 5580 w 10 I f (n n)1 50 1 2696 5580 t 7 I f (u u)1 35 1 2751 5599 t 7 R f (2)2751 5540 w 10 I f (n n)1 50 1 2826 5580 t 7 R f (1. 25)1 128 1 2887 5540 t 10 R f (\( 2)1 91 1 3055 5580 t 10 I f (k k)1 44 1 3178 5580 t 7 I f (x x)1 31 1 3233 5600 t 10 S f (- -)1 55 1 3288 5580 t 10 R f ( 2)1 58( \()1 73(1 \))1 91 3 3359 5580 t 10 I f (k k)1 44 1 3613 5580 t 7 I f (y y)1 31 1 3668 5600 t 10 S f (- -)1 55 1 3723 5580 t 10 R f (1 \) \))2 132 1 3794 5580 t 10 S f ( \347)1 -2193(_ ___________________________________________)1 2193 2 1783 5600 t (\347)1783 5540 w (\347)1783 5440 w (\347)1783 5340 w (\347)1783 5240 w (\347)1783 5140 w (\347)1783 5040 w (\347)1783 4940 w (\347)1783 4840 w (\347)2500 5600 w (\347)2500 5580 w (\347)2500 5480 w (\347)2500 5380 w (\347)2500 5280 w (\347)2500 5180 w (\347)2500 5080 w (\347)2500 4980 w (\347)3976 5600 w (\347)3976 5540 w (\347)3976 5440 w (\347)3976 5340 w (\347)3976 5240 w (\347)3976 5140 w (\347)3976 5040 w (\347)3976 4940 w (\347)3976 4840 w 10 R f (where)720 5780 w 10 I f (H H)1 72 1 992 5780 t 10 S f (\272)1105 5780 w 10 I f (n n)1 50 1 1225 5780 t 7 I f (u u)1 35 1 1286 5800 t 10 R f (\()1361 5780 w 10 I f (k k)1 44 1 1426 5780 t 7 I f (x x)1 31 1 1481 5800 t 10 S f (+ +)1 55 1 1560 5780 t 10 R f (\()1655 5780 w 10 I f (n n)1 50 1 1720 5780 t 7 I f (x x)1 31 1 1781 5800 t 10 S f (- -)1 55 1 1860 5780 t 10 I f (k k)1 44 1 1955 5780 t 7 I f (x x)1 31 1 2010 5800 t 10 R f (\) \()1 106 1 2081 5780 t 10 I f (k k)1 44 1 2195 5780 t 7 I f (y y)1 31 1 2250 5800 t 10 S f (- -)1 55 1 2329 5780 t 10 R f ( and)1 173( \),)1 98(1 \))1 115 3 2424 5780 t 10 I f (n n)1 50 1 2839 5780 t 10 S f (\272)2930 5780 w 10 I f (n n)1 50 1 3026 5780 t 7 I f (x x)1 31 1 3087 5800 t 10 I f (n n)1 50 1 3158 5780 t 7 I f (y y)1 31 1 3219 5800 t 10 R f (. When)1 317 1 3258 5780 t 10 CW f (Pieces)3604 5780 w 10 R f (is 0, a factor of)4 617 1 3993 5780 t 10 I f (n n)1 50 1 4639 5780 t 7 I f (u u)1 35 1 4700 5800 t 10 R f (should)4773 5780 w (be removed from the figures.)4 1161 1 720 5900 t (\262 - This figure is usually accurate, but see the discussion of the)12 2510 1 720 6056 t 10 B f (pde)3255 6056 w 10 I f (u u)1 50 1 3436 6056 t 7 I f (x xy y)2 62 1 3497 6076 t 10 S f (= =)1 55 1 3616 6056 t 10 R f (0 below.)1 344 1 3720 6056 t (\262\262 - Modified incomplete factorization schemes)5 1916 1 720 6212 t 10 I f ( y)1 0( ll ly)2 72( ua al)2 78( su)1 50(u us)1 89 5 2661 6212 t 10 R f (converge in)1 468 1 2975 6212 t 10 I f (O O)1 72 1 3468 6212 t 10 R f (\()3548 6212 w 10 I f (n n)1 50 1 3589 6212 t 7 I f (. .)1 18 1 3650 6172 t 7 R f (25)3673 6172 w 10 R f ( For)1 190( but not always.)3 636(\) iterations,)1 455 3 3759 6212 t (example, problems like)2 935 1 720 6332 t 10 I f (u u)1 50 1 1680 6332 t 7 I f (x x)1 31 1 1741 6352 t 10 S f (+ +)1 55 1 1829 6332 t 10 I f (u u)1 50 1 1933 6332 t 7 I f (y y)1 31 1 1994 6352 t 10 S f (= =)1 55 1 2082 6332 t 10 R f ( also [12] for examples.)4 945( See)1 194(0 will cause MIC to fail.)5 977 3 2186 6332 t ( and thus on the Cray X-MP may take less time even though they)13 2736(Note that the banded options vectorize)5 1584 2 720 6488 t (require more operations than the sparse options.)6 1916 1 720 6608 t cleartomark showpage saveobj restore end %%EndPage: 27 26 %%Page: 28 27 DpostDict begin /saveobj save def mark 27 pagesetup 10 R f (- 28 -)2 216 1 2772 480 t 10 B f (A Bad Example)2 673 1 720 840 t 10 R f (Consider the)1 508 1 970 996 t 10 B f (pde)1503 996 w 10 I f (u u)1 50 1 1220 1176 t 7 I f (x xy y)2 62 1 1281 1196 t 10 S f (= =)1 55 1 1400 1176 t 10 R f (0 \(6.1\))1 3536 1 1504 1176 t ( 0 , 1 ])4 190(on the domain [)3 630 2 720 1356 t 10 S f (\264)1556 1356 w 10 R f ( and bottom sides of the domain.)6 1314( Dirichlet boundary conditions on the left)6 1654( Use)1 205([ 0 , 1 ].)4 248 4 1619 1356 t ( means that any non-)4 900( This)1 245( of whose diagonal elements are 0.)6 1485(This problem generates a Jacobian most)5 1690 4 720 1476 t ( sparse matrix option is used, the)6 1325( the pivoting)2 508( If)1 117(pivoting LU factorization scheme will fail on this problem.)8 2370 4 720 1596 t ( ordering used there, which implicitly assumes that the diagonal elements are non-zero,)12 3627(minimum degree)1 693 2 720 1716 t (causes the LU factorization to)4 1209 1 720 1836 t 10 B f (fill-in)1957 1836 w 10 R f ( the pivoting sparse option uses)5 1278( is,)1 120(completely. That)1 705 3 2219 1836 t 10 I f (O O)1 72 1 4351 1836 t 10 R f (\()4431 1836 w 10 I f (n n)1 50 1 4472 1836 t 7 R f (4)4533 1796 w 10 R f (\) space and)2 456 1 4584 1836 t 10 I f (O O)1 72 1 720 1956 t 10 R f (\()800 1956 w 10 I f (n n)1 50 1 841 1956 t 7 R f (6)902 1916 w 10 R f (\) work on an)3 507 1 953 1956 t 10 I f (n n)1 50 1 1485 1956 t 10 R f (by)1560 1956 w 10 I f (n n)1 50 1 1685 1956 t 10 R f ( pivoting banded solver has no trouble with this problem at all.)11 2510(grid. The)1 391 2 1760 1956 t ( is slower on scalar machines and uses more space than the)11 2496(In general, the pivoting banded solver)5 1574 2 970 2112 t ( the other options have the)5 1112( all of)2 257( However,)1 452(other options, for sufficiently large numbers of mesh points.)8 2499 4 720 2232 t (property that their)2 739 1 720 2352 t 10 B f (worst-case)1493 2352 w 10 R f ( than that of pivoting band)5 1108(behavior, on the above problem, is much worse)7 1957 2 1975 2352 t (solution, which always uses roughly the same space and run-time.)9 2638 1 720 2472 t ( as the default solver - it is predictable, dependable and)10 2248(This explains why we chose pivoting banded)6 1822 2 970 2628 t ( also turns out to be fast on vector machines for most practical grids.)13 2735( It)1 111(never blows up on nice problems.)5 1348 3 720 2748 t 10 B f (Some Knob Twiddling)2 964 1 720 2988 t 10 R f (We now show how twiddling some of the knobs in)9 2080 1 970 3144 t 10 CW f (TTGRV)3080 3144 w 10 R f (can affect the run-time for example 2 in)7 1629 1 3411 3144 t ( knobs considered are)3 867( The)1 205(section 4.)1 383 3 720 3264 t 10 CW f (maxit)2200 3264 w 10 R f (,)2500 3264 w 10 CW f (mxq)2550 3264 w 10 R f (,)2730 3264 w 10 CW f (myq)2780 3264 w 10 R f (,)2960 3264 w 10 CW f (LA)3010 3264 w 10 R f (,)3130 3264 w 10 CW f (Pieces)3180 3264 w 10 R f (,)3540 3264 w 10 CW f (PC)3590 3264 w 10 R f (, and)1 194 1 3710 3264 t 10 CW f (Accel)3929 3264 w 10 R f (.)4229 3264 w (The default call to)3 777 1 970 3420 t 10 CW f (TTGR)1789 3420 w 10 R f (uses a full Newton iteration scheme to solve the nonlinear equations \()11 2969 1 2071 3420 t 10 CW f (maxit)720 3540 w 10 S f (= =)1 55 1 1047 3540 t 10 R f ( in)1 104( run-times reported)2 769( The)1 207( will be referred to as NBE for Nonlinear Backwards Euler.)10 2394( This)1 230(50 \).)1 185 6 1151 3540 t (Appendix 4 for the examples are for the default NBE scheme.)10 2469 1 720 3660 t (The second scheme considered is the same as NBE but with)10 2416 1 970 3816 t 10 CW f (maxit)3413 3816 w 10 R f (set to 1. This may seem strange,)6 1300 1 3740 3816 t (in that it does only)4 753 1 720 3936 t 10 I f ( e)1 0(o on ne)2 144 2 1500 3936 t 10 R f ( it works quite well on)5 902( However,)1 441( nonlinear equation.)2 798(Newton iteration to solve each)4 1228 4 1671 3936 t (most problems and it possesses the asymptotic expansion needed for extrapolation to work, see Appendix 2.)15 4320 1 720 4056 t ( user tells)2 387( The)1 207( Backwards Euler.)2 738(This scheme will be denoted by LBE, for Linearized)8 2123 4 720 4176 t 10 CW f (TTGR)4202 4176 w 10 R f (that)4469 4176 w 10 CW f (maxit)4646 4176 w 10 R f (is)4973 4176 w (to be 1 by inserting the)5 919 1 720 4296 t 10 CW f (call ttgrv\(+2003,0d0,0e0,1,.true.\))1 2040 1 1080 4476 t 10 R f (just before the call to)4 843 1 720 4656 t 10 CW f (TTGR)1588 4656 w 10 R f (.)1828 4656 w (The third scheme will be the LBE scheme above with)9 2162 1 970 4812 t 10 CW f (mxq)3159 4812 w 10 S f (= =)1 55 1 3366 4812 t 10 CW f (kx-1)3448 4812 w 10 R f (and)3715 4812 w 10 CW f (myq)3921 4812 w 10 S f (= =)1 55 1 4128 4812 t 10 CW f (ky-1)4210 4812 w 10 R f ( usu-)1 200(, which is)2 390 2 4450 4812 t ( little care has to be used here however; if)9 1683( A)1 124( partial spatial derivative in them.)5 1353(ally safe for problems with a)5 1160 4 720 4932 t (the time step)2 513 1 720 5052 t 10 CW f (dt)1259 5052 w 10 R f ( approach a mul-)3 679(gets exceedingly small, the Jacobian will become singular, because it will)10 2956 2 1405 5052 t (tiple of the Jacobian for the problem)6 1495 1 720 5172 t 10 I f (u u)1 50 1 2247 5172 t 10 S f (= =)1 55 1 2346 5172 t 10 R f ( Jacobian is singular unless)4 1116(0. That)1 315 2 2450 5172 t 10 I f (k k)1 44 1 3913 5172 t 7 I f (x x)1 31 1 3968 5192 t 10 R f (and)4038 5172 w 10 I f (k k)1 44 1 4213 5172 t 7 I f (y y)1 31 1 4268 5192 t 10 R f (quadrature points)1 702 1 4338 5172 t ( user tells)2 401( The)1 214( denoted by LBE.q for LBE with special Quadrature.)8 2185( scheme will be)3 648( This)1 236(are used.)1 362 6 720 5292 t 10 CW f (TTGR)4800 5292 w 10 R f (that)720 5412 w 10 CW f (mxq)895 5412 w 10 R f (is to be)2 289 1 1100 5412 t 10 CW f (kx-1)1414 5412 w 10 R f (, as well as setting)4 730 1 1654 5412 t 10 CW f (myq)2409 5412 w 10 R f (, by inserting)2 525 1 2589 5412 t 10 CW f (call ttgrv\(+2008,0e0,0e0,kx-1,.true.\))1 2220 1 1080 5592 t (call ttgrv\(+2009,0e0,0e0,ky-1,.true.\))1 2220 1 1080 5712 t 10 R f (just before the call to)4 843 1 720 5892 t 10 CW f (TTGR)1588 5892 w 10 R f (.)1828 5892 w ( above LBE.q with)3 788(The fourth scheme will be the)5 1245 2 970 6048 t 10 CW f (Pieces = 0)2 622 1 3039 6048 t 10 R f ( scheme trades a few more)5 1115(. This)1 264 2 3661 6048 t ( main assumption here is that the)6 1334( The)1 208(time-steps for much cheaper Jacobian calculation and solution times.)8 2778 3 720 6168 t ("big" \( like)2 471 1 720 6288 t 10 B f (u)1230 6288 w 7 I f (t t)1 20 1 1297 6308 t 10 R f (and)1364 6288 w 10 B f (u)1547 6288 w 7 I f (x xx x)2 62 1 1614 6308 t 10 R f ( are on the block diagonal of the)7 1394(\) terms)1 294 2 1723 6288 t 10 B f (pde)3451 6288 w 10 R f ( scheme will be denoted by)5 1165(. This)1 268 2 3607 6288 t ( long as all the)4 587(LBE.q0. So)1 496 2 720 6408 t 10 B f (u)1829 6408 w 7 I f (t t)1 20 1 1896 6428 t 10 R f ( to converge.)2 518(terms are on the diagonal, this option is theoretically guaranteed)9 2572 2 1950 6408 t ( user tells)2 389( The)1 207(The issue is whether it is faster than dealing with the whole Jacobian.)12 2795 3 720 6528 t 10 CW f (TTGR)4139 6528 w 10 R f (that)4407 6528 w 10 CW f (Pieces)4585 6528 w 10 R f (is)4973 6528 w (to be)1 197 1 720 6648 t 10 CW f (0)942 6648 w 10 R f (by inserting)1 475 1 1027 6648 t 10 CW f (call ttgrv\(+2011,0e0,0e0,0,.true.\))1 2040 1 1080 6828 t 10 R f (just before the call to)4 843 1 720 7008 t 10 CW f (TTGR)1588 7008 w 10 R f (.)1828 7008 w ( with)1 216(Finally, we use the above LBE.q0)5 1417 2 970 7164 t 10 CW f (LA)2641 7164 w 10 S f ( -)1 0( -)1 112(= =)1 55 3 2799 7164 t 10 R f (2,)2982 7164 w 10 CW f (PC)3095 7164 w 10 S f (= =)1 55 1 3253 7164 t 10 R f (2 and)1 232 1 3357 7164 t 10 CW f (Accel)3662 7164 w 10 S f (= =)1 55 1 4000 7164 t 10 R f ( uses Modified)2 620(2. This)1 316 2 4104 7164 t ( scheme will be denoted by)5 1195( This)1 249(incomplete factorizations and Orthomin to solve the linear systems.)8 2876 3 720 7284 t cleartomark showpage saveobj restore end %%EndPage: 28 27 %%Page: 29 28 DpostDict begin /saveobj save def mark 28 pagesetup 10 R f (- 29 -)2 216 1 2772 480 t (LBE.q0m.)720 840 w (The user tells)2 538 1 970 996 t 10 CW f (TTGR)1533 996 w 10 R f (that)1798 996 w 10 CW f (LA)1973 996 w 10 R f (is to be)2 289 1 2118 996 t 10 S f (- -)1 55 1 2432 996 t 10 R f (2, etc by inserting the calls)5 1071 1 2503 996 t 10 CW f (call ttgrv\(+2010,0d0,0e0,-2,.true.\))1 2100 1 1440 1176 t (call ttgrv\(+2012,0d0,0e0,2,.true.\))1 2040 1 1440 1296 t (call ttgrv\(+2013,0d0,0e0,2,.true.\))1 2040 1 1440 1416 t 10 R f (just before the call to)4 843 1 720 1596 t 10 CW f (TTGR)1588 1596 w 10 R f (.)1828 1596 w (We summarize the methods used in the table)7 1789 1 970 1752 t 10 S f (_ ____________________________________________________)1 2626 1 1567 1832 t 10 R f (Description of Runs)2 805 1 2477 1952 t 10 S f (_ ____________________________________________________)1 2626 1 1567 1972 t 10 R f ( Method)1 988( Mesh)1 436(Label Order)1 624 3 1631 2092 t 10 S f (_ ____________________________________________________)1 2626 1 1567 2112 t (_ ____________________________________________________)1 2626 1 1567 2132 t 10 R f ( NBE)1 869( by 21)2 250( 21)1 341(2n 2)1 469 4 1695 2252 t 10 S f (_ ____________________________________________________)1 2626 1 1567 2272 t 10 R f ( LBE)1 863( by 21)2 250( 21)1 341(2l 2)1 458 4 1706 2392 t 10 S f (_ ____________________________________________________)1 2626 1 1567 2412 t 10 R f ( LBE.q)1 901( by 21)2 250( 21)1 341(2lq 2)1 483 4 1681 2532 t 10 S f (_ ____________________________________________________)1 2626 1 1567 2552 t 10 R f ( LBE.q0)1 926( by 21)2 250( 21)1 341(2lq0 2)1 508 4 1656 2672 t 10 S f (_ ____________________________________________________)1 2626 1 1567 2692 t 10 R f ( with Orthomin and MIC)4 1000( above)1 388( by 21)2 250(2lq0m 2 21)2 888 4 1617 2812 t 10 S f (_ ____________________________________________________)1 2626 1 1567 2832 t (_ ____________________________________________________)1 2626 1 1567 2852 t 10 R f ( NBE)1 919( by 3)2 200( 3)1 341(n 4)1 444 4 1720 2972 t 10 S f (_ ____________________________________________________)1 2626 1 1567 2992 t 10 R f ( LBE)1 913( by 3)2 200( 3)1 341(l 4)1 433 4 1731 3112 t 10 S f (_ ____________________________________________________)1 2626 1 1567 3132 t 10 R f ( LBE.q)1 951( by 3)2 200( 3)1 341(lq 4)1 458 4 1706 3252 t 10 S f (_ ____________________________________________________)1 2626 1 1567 3272 t 10 R f ( LBE.q0)1 976( by 3)2 200( 3)1 341(lq0 4)1 483 4 1681 3392 t 10 S f (_ ____________________________________________________)1 2626 1 1567 3412 t 10 R f ( with Orthomin and MIC)4 1000( above)1 438( by 3)2 200( 3)1 341(lq0m 4)1 522 5 1642 3532 t 10 S f ( \347)1 -2626(_ ____________________________________________________)1 2626 2 1567 3552 t (\347)1567 3532 w (\347)1567 3432 w (\347)1567 3332 w (\347)1567 3232 w (\347)1567 3132 w (\347)1567 3032 w (\347)1567 2932 w (\347)1567 2832 w (\347)1567 2732 w (\347)1567 2632 w (\347)1567 2532 w (\347)1567 2432 w (\347)1567 2332 w (\347)1567 2232 w (\347)1567 2132 w (\347)1567 2032 w (\347)1567 1932 w (\347)1948 3552 w (\347)1948 3472 w (\347)1948 3372 w (\347)1948 3272 w (\347)1948 3172 w (\347)1948 3072 w (\347)1948 2972 w (\347)1948 2872 w (\347)1948 2772 w (\347)1948 2672 w (\347)1948 2572 w (\347)1948 2472 w (\347)1948 2372 w (\347)1948 2272 w (\347)1948 2172 w (\347)1948 2072 w (\347)2330 3552 w (\347)2330 3472 w (\347)2330 3372 w (\347)2330 3272 w (\347)2330 3172 w (\347)2330 3072 w (\347)2330 2972 w (\347)2330 2872 w (\347)2330 2772 w (\347)2330 2672 w (\347)2330 2572 w (\347)2330 2472 w (\347)2330 2372 w (\347)2330 2272 w (\347)2330 2172 w (\347)2330 2072 w (\347)2830 3552 w (\347)2830 3472 w (\347)2830 3372 w (\347)2830 3272 w (\347)2830 3172 w (\347)2830 3072 w (\347)2830 2972 w (\347)2830 2872 w (\347)2830 2772 w (\347)2830 2672 w (\347)2830 2572 w (\347)2830 2472 w (\347)2830 2372 w (\347)2830 2272 w (\347)2830 2172 w (\347)2830 2072 w (\347)4193 3552 w (\347)4193 3532 w (\347)4193 3432 w (\347)4193 3332 w (\347)4193 3232 w (\347)4193 3132 w (\347)4193 3032 w (\347)4193 2932 w (\347)4193 2832 w (\347)4193 2732 w (\347)4193 2632 w (\347)4193 2532 w (\347)4193 2432 w (\347)4193 2332 w (\347)4193 2232 w (\347)4193 2132 w (\347)4193 2032 w (\347)4193 1932 w 10 R f ( get the same accuracy as the order 4 solution on a 3 by 3)14 2298(The order 2 grid was taken to be 21 by 21 to)11 1772 2 970 3768 t (grid.)720 3888 w (The results for the five methods described above are summarized in the table:)12 3099 1 970 4044 t 10 S f (_ ___________________________________________)1 2161 1 1799 4124 t 10 R f (Effect of Different Solution Methods on Example 2)7 2061 1 1849 4244 t 10 S f (_ ___________________________________________)1 2161 1 1799 4264 t 10 R f ( Jacobians)1 586( \(secs\))1 257( Time)1 440(Method Space)1 754 4 1849 4384 t 10 S f (_ ___________________________________________)1 2161 1 1799 4404 t (_ ___________________________________________)1 2161 1 1799 4424 t 10 R f ( 24)1 539( 5219.2)1 569(2n 134008)1 680 3 1954 4544 t 10 S f (_ ___________________________________________)1 2161 1 1799 4564 t 10 R f ( 9)1 539( 1972.9)1 569(2l 134008)1 669 3 1965 4684 t 10 S f (_ ___________________________________________)1 2161 1 1799 4704 t 10 R f ( 9)1 539( 1603.1)1 569(2lq 133812)1 694 3 1940 4824 t 10 S f (_ ___________________________________________)1 2161 1 1799 4844 t 10 R f ( 12)1 539( 812.8)1 569(2lq0 72954)1 719 3 1915 4964 t 10 S f (_ ___________________________________________)1 2161 1 1799 4984 t 10 R f ( 12)1 539( 506.7)1 569(2lq0m 44743)1 758 3 1876 5104 t 10 S f (_ ___________________________________________)1 2161 1 1799 5124 t (_ ___________________________________________)1 2161 1 1799 5144 t 10 R f ( 30)1 539( 443.6)1 569(n 9398)1 655 3 1979 5264 t 10 S f (_ ___________________________________________)1 2161 1 1799 5284 t 10 R f ( 9)1 539( 133.0)1 569(l 9398)1 644 3 1990 5404 t 10 S f (_ ___________________________________________)1 2161 1 1799 5424 t 10 R f ( 9)1 539( 108.6)1 569(lq 7990)1 669 3 1965 5544 t 10 S f (_ ___________________________________________)1 2161 1 1799 5564 t 10 R f ( 15)1 539( 111.1)1 569(lq0 6090)1 694 3 1940 5684 t 10 S f (_ ___________________________________________)1 2161 1 1799 5704 t 10 R f ( 15)1 539( 111.8)1 569(lq0m 9251)1 733 3 1901 5824 t 10 S f ( \347)1 -2161(_ ___________________________________________)1 2161 2 1799 5844 t (\347)1799 5824 w (\347)1799 5724 w (\347)1799 5624 w (\347)1799 5524 w (\347)1799 5424 w (\347)1799 5324 w (\347)1799 5224 w (\347)1799 5124 w (\347)1799 5024 w (\347)1799 4924 w (\347)1799 4824 w (\347)1799 4724 w (\347)1799 4624 w (\347)1799 4524 w (\347)1799 4424 w (\347)1799 4324 w (\347)1799 4224 w (\347)2235 5844 w (\347)2235 5764 w (\347)2235 5664 w (\347)2235 5564 w (\347)2235 5464 w (\347)2235 5364 w (\347)2235 5264 w (\347)2235 5164 w (\347)2235 5064 w (\347)2235 4964 w (\347)2235 4864 w (\347)2235 4764 w (\347)2235 4664 w (\347)2235 4564 w (\347)2235 4464 w (\347)2235 4364 w (\347)2733 5844 w (\347)2733 5764 w (\347)2733 5664 w (\347)2733 5564 w (\347)2733 5464 w (\347)2733 5364 w (\347)2733 5264 w (\347)2733 5164 w (\347)2733 5064 w (\347)2733 4964 w (\347)2733 4864 w (\347)2733 4764 w (\347)2733 4664 w (\347)2733 4564 w (\347)2733 4464 w (\347)2733 4364 w (\347)3399 5844 w (\347)3399 5764 w (\347)3399 5664 w (\347)3399 5564 w (\347)3399 5464 w (\347)3399 5364 w (\347)3399 5264 w (\347)3399 5164 w (\347)3399 5064 w (\347)3399 4964 w (\347)3399 4864 w (\347)3399 4764 w (\347)3399 4664 w (\347)3399 4564 w (\347)3399 4464 w (\347)3399 4364 w (\347)3960 5844 w (\347)3960 5824 w (\347)3960 5724 w (\347)3960 5624 w (\347)3960 5524 w (\347)3960 5424 w (\347)3960 5324 w (\347)3960 5224 w (\347)3960 5124 w (\347)3960 5024 w (\347)3960 4924 w (\347)3960 4824 w (\347)3960 4724 w (\347)3960 4624 w (\347)3960 4524 w (\347)3960 4424 w (\347)3960 4324 w (\347)3960 4224 w 10 R f ( single precision on a Vax 11/750, equipped with a floating-point accel-)11 2901(All of the above runs were made in)7 1419 2 720 6060 t (erator, under the)2 656 1 720 6180 t 9 R f (UNIX)1399 6180 w 10 S f (\322)1624 6180 w 10 R f (operating system, Research Eighth Edition.)4 1732 1 1728 6180 t (The first 5 runs above refer to using)7 1496 1 970 6336 t 10 I f (k k)1 44 1 2500 6336 t 7 I f (x x)1 31 1 2555 6356 t 10 S f (= =)1 55 1 2643 6336 t 10 R f (2)2747 6336 w 10 S f (= =)1 55 1 2846 6336 t 10 I f (k k)1 44 1 2950 6336 t 7 I f (y y)1 31 1 3005 6356 t 10 R f ( to using)2 365( last five refer)3 583( The)1 214(on a 21 by 21 grid.)5 800 4 3078 6336 t ( table shows that LBE saves the cost of several iterations per time)12 2636( The)1 206(fourth order splines on a 3 by 3 grid.)8 1478 3 720 6456 t ( results in a big savings for linear)7 1448( LBE.q0)1 381( saves even more time.)4 980( fewer quadrature points)3 1020(step. Using)1 491 5 720 6576 t ( by 800 for linear, but less than 100)8 1434( reason is that the Jacobian is 800)7 1361( The)1 208(splines, but not much for cubics.)5 1317 4 720 6696 t ( also that)2 375( Note)1 251( for cubics.)2 460(by 100)1 281 4 720 6816 t 10 CW f (Pieces=0)2119 6816 w 10 R f (used more time-steps but still ran faster for the order 2 run)11 2409 1 2631 6816 t ( a lot for linear splines, and nothing at)8 1566( Orthomin saves)2 669( Finally,)1 366(because each step was significantly faster.)5 1719 4 720 6936 t (all for cubics.)2 546 1 720 7056 t (Note that the)2 532 1 970 7212 t 10 I f ( t)1 0( st)1 28( es)1 39( we)1 44( ow)1 67( lo)1 50(s sl)1 67 7 1535 7212 t 10 R f (run with)1 344 1 1863 7212 t 10 I f (k k)1 44 1 2240 7212 t 10 S f (= =)1 55 1 2333 7212 t 10 R f (4 is)1 150 1 2437 7212 t 10 I f ( r)1 0( er)1 39( te)1 44( st)1 28( as)1 39(f fa)1 78 6 2620 7212 t 10 R f ( fastest)1 289(than the)1 327 2 2881 7212 t 10 I f (k k)1 44 1 3531 7212 t 10 S f (= =)1 55 1 3624 7212 t 10 R f (2 solution setting, and also uses)5 1312 1 3728 7212 t cleartomark showpage saveobj restore end %%EndPage: 29 28 %%Page: 30 29 DpostDict begin /saveobj save def mark 29 pagesetup 10 R f (- 30 -)2 216 1 2772 480 t ( even more important than twiddling knobs in)7 1858( higher order schemes is)4 990( Using)1 294(less space.)1 426 4 720 840 t 10 CW f (TTGR)4317 840 w 10 R f (, at least for)3 483 1 4557 840 t (this example.)1 533 1 720 960 t 10 B f (The Right Stuff.)2 692 1 720 1200 t 10 R f ( method for)2 468(The discussion in [26] shows why NBE was chosen as the default)11 2627 2 970 1356 t 10 CW f (TTGR)4091 1356 w 10 R f ( is very robust)3 572(. It)1 137 2 4331 1356 t ( speed can be achieved, but only by sacrificing robustness.)9 2337( Additional)1 478(and reasonably fast.)2 795 3 720 1476 t (If the user wishes to speed)5 1066 1 970 1632 t 10 CW f (TTGR)2063 1632 w 10 R f ( usually works)2 589( It)1 114( simplest and safest thing to do is try LBE.)9 1733(up, the)1 274 4 2330 1632 t ( user should make sure to compare at least a few LBE runs with NBE, to make)16 3174( The)1 207( than NBE.)2 451(and is faster)2 488 4 720 1752 t ( has been)2 375( the utility of LBE for the class of problems at hand)11 2073( Once)1 261(sure it is correct and running efficiently.)6 1611 4 720 1872 t (ensured by a few comparisons, the production runs can be done with LBE.)12 2978 1 720 1992 t (Clearly, the others knobs of)4 1139 1 970 2148 t 10 CW f (TTGR)2142 2148 w 10 R f ( their tuning is highly)4 898( But)1 204( a lot faster.)3 498(can also make things run)4 1025 4 2415 2148 t ( appreciate hearing from users what settings of knobs they find give)11 2720( authors would)2 598( The)1 207(problem dependent.)1 795 4 720 2268 t (useful, or awful, results.)3 965 1 720 2388 t 10 B f (Run-time Statistics.)1 838 1 720 2628 t 10 R f (A subroutine is provided to print run-time statistics for)8 2188 1 970 2784 t 10 CW f (TTGR)3183 2784 w 10 R f ( statement)1 408(. The)1 230 2 3423 2784 t 10 CW f (Call TTGRX)1 600 1 1440 2964 t 10 R f (will print a line of the form:)6 1116 1 720 3144 t 10 CW f ( 0 0 0)3 900( 130)1 300( 15 76)2 600( 130 130)2 600(ttgr\(j,f,ts,ss,nit,nd,nf,r\) =)1 1740 5 1140 3324 t 10 R f (The fields of this line refer to)6 1170 1 720 3504 t 10 CW f (j)770 3660 w 10 R f ( number of Jacobian evaluations.)4 1317(- The)1 305 2 1020 3660 t 10 CW f (f)770 3816 w 10 R f ( number of factorizations of the Jacobian.)6 1660(- The)1 305 2 1020 3816 t 10 CW f (ts)770 3972 w 10 R f ( number of time-steps.)3 899(- The)1 305 2 1020 3972 t 10 CW f (ss)770 4128 w 10 R f ( number of sub-steps.)3 860(- The)1 305 2 1020 4128 t 10 CW f (nit)770 4284 w 10 R f ( number of Newton iterations.)4 1201(- The)1 305 2 1020 4284 t 10 CW f (nd)770 4440 w 10 R f ( number of predicted Newton failures \( error increasing \).)9 2293(- The)1 305 2 1020 4440 t 10 CW f (nf)770 4596 w 10 R f ( number of Newton failures \( more than maxit iterations \).)10 2321(- The)1 305 2 1020 4596 t 10 CW f (r)770 4752 w 10 R f ( number of restarts.)3 776(- The)1 305 2 1020 4752 t (If)970 4908 w 10 CW f (TTGRX)1063 4908 w 10 R f (is invoked by the user while inside)6 1400 1 1390 4908 t 10 CW f (TTGR)2818 4908 w 10 R f (, the statistics reported will be the current values.)8 1982 1 3058 4908 t (If invoked outside)2 727 1 720 5028 t 10 CW f (TTGR)1472 5028 w 10 R f (, the statistics will be those of the last call to)10 1769 1 1712 5028 t 10 CW f (TTGR)3506 5028 w 10 R f (.)3746 5028 w 10 B f (Another way Into)2 755 1 720 5268 t 10 CW f (TTGR)1500 5268 w 10 B f (.)1740 5268 w 10 R f ( use is strongly discouraged,)4 1175( Their)1 276( subprograms are described for historical reasons.)6 2041(The following)1 578 4 970 5424 t (but possible.)1 506 1 720 5544 t (For those who do not wish \( or like \) to use)11 1790 1 970 5700 t 10 CW f (TTGR)2791 5700 w 10 R f (/)3031 5700 w 10 CW f (TTGRV)3059 5700 w 10 R f (/)3359 5700 w 10 CW f (TTGRR)3387 5700 w 10 R f ( entering)1 359(, there is another way of)5 994 2 3687 5700 t 10 CW f (TTGR)720 5820 w (Call TTGRS\(U,Nu,kx,x,nx, ky,y,ny,)2 1980 1 1080 6000 t (tstart,tstop,dt,)1740 6120 w (AF,BC,)1740 6240 w (errpar,)1740 6360 w (HANDLE\))1740 6480 w 10 R f (with further control over the error of the solution given by)10 2324 1 720 6660 t cleartomark showpage saveobj restore end %%EndPage: 30 29 %%Page: 31 30 DpostDict begin /saveobj save def mark 30 pagesetup 10 R f (- 31 -)2 216 1 2772 480 t 10 CW f (Call TTGR1\(U,Nu,kx,x,nx, ky,y,ny,)2 1980 1 1080 900 t (tstart,tstop,dt,)1740 1020 w (mxq,myq,)1740 1140 w (AF,BC,)1740 1260 w (ERROR,errpar,)1740 1380 w (HANDLE\))1740 1500 w 10 R f ( keep them upwards compatible with their)6 1687( is to)2 195( This)1 228(where the default method for both of the above is LBE.)10 2210 4 720 1680 t ( that NBE is the default for)6 1106( Note)1 249(previous versions.)1 732 3 720 1800 t 10 CW f (TTGR)2836 1800 w 10 R f (,)3076 1800 w 10 I f ( t)1 0(n no ot)2 128 2 3130 1800 t 10 R f ( the default method for)4 931(LBE. Thus,)1 493 2 3287 1800 t 10 CW f (TTGRS)4740 1800 w 10 R f (and)720 1920 w 10 CW f (TTGR1)889 1920 w 10 R f (is different from that for)4 970 1 1214 1920 t 10 CW f (TTGR)2209 1920 w 10 R f (Even more control over the integration process is given by)9 2338 1 970 2076 t 10 CW f (Call TTGR2\(U,Nu,kx,x,nx, ky,y,ny,)2 1980 1 1080 2256 t (tstart,tstop,dt,)1740 2376 w (mxq,myq,)1740 2496 w (LA,Pieces,PC,Accel,)1740 2616 w (AF,BC,)1740 2736 w (ERROR,errpar,)1740 2856 w (HANDLE\))1740 2976 w cleartomark showpage saveobj restore end %%EndPage: 31 30 %%Page: 32 31 DpostDict begin /saveobj save def mark 31 pagesetup 10 R f (- 32 -)2 216 1 2772 480 t 10 B f (Acknowledgements)720 840 w 10 R f ( Edelson and Leonilda Farrow, chemists all, were the reason for this package)12 3119(Georgia Fisanick, Dave)2 951 2 970 996 t ( badgering for something that "works" without worrying about being "opti-)10 3104( constant)1 369( Their)1 277(being created.)1 570 4 720 1116 t ( doing pre-)2 452( Jack Dongarra of Argonne shared his knowledge and code for)10 2558( Also,)1 270(mal" has been successful.)3 1040 4 720 1236 t ( that leg-up from him, we might still)7 1463( Without)1 379( systems.)1 368(conditioned conjugate-gradient solution of the linear)5 2110 4 720 1356 t (be stuck with sparse-matrix and other much slower methods.)8 2427 1 720 1476 t cleartomark showpage saveobj restore end %%EndPage: 32 31 %%Page: 1 32 DpostDict begin /saveobj save def mark 32 pagesetup 10 B f (Bibliography)2598 840 w 10 R f ( deBoor,)1 344([1] C.)1 342 2 720 1476 t 10 B f (A Practical Guide to Splines,)4 1231 1 1431 1476 t 10 R f (Springer, New York, Applied Math. Sciences 27, 1978.)7 2219 1 2687 1476 t ( de Boor, "On Uniform Approximation by Splines,")7 2073([2] C.)1 342 2 720 1632 t 10 I f ( .)1 0( h.)1 25( Th)1 50( T)1 81( .)1 0( x.)1 25( ox)1 44( ro)1 50( pp pr)2 89( Ap)1 50( A)1 86( .)1 0(J J.)1 69 13 3160 1632 t 10 B f (1,)3779 1632 w 10 R f (219-235\(1968\).)3879 1632 w ( de Boor, "On Calculating with B-splines,")6 1715([3] C.)1 342 2 720 1788 t 10 I f ( .)1 0( h.)1 25( Th)1 50( T)1 81( .)1 0( x.)1 25( ox)1 44( ro)1 50( pp pr)2 89( Ap)1 50( A)1 86( .)1 0(J J.)1 69 13 2802 1788 t 10 B f (6,)3421 1788 w 10 R f (50-62\(1972\).)3521 1788 w ( Ver-)1 210( Bulirsch and J. Stoer, "Fehlerabschatzungen und Extrapolation mit rationalen Funktionen bei)11 3768([4] R.)1 342 3 720 1944 t (fahren vom Richardson-Typus,")2 1286 1 970 2064 t 10 I f ( .)1 0( h.)1 25( th)1 50( at)1 28( Ma)1 50( M)1 108( .)1 0( r.)1 25( er)1 39( me)1 44( um)1 72(N Nu)1 117 12 2281 2064 t 10 B f (6,)2889 2064 w 10 R f (413-427\(1964\).)2989 2064 w ( and J. Stoer, "Numerical Treatment of Ordinary Differential Equations by Extrapolation)11 3609( Bulirsch)1 369([5] R.)1 342 3 720 2220 t (Methods,")970 2340 w 10 I f ( .)1 0( h.)1 25( th)1 50( at)1 28( Ma)1 50( M)1 108( .)1 0( r.)1 25( er)1 39( me)1 44( um)1 72(N Nu)1 117 12 1411 2340 t 10 B f (8,)2019 2340 w 10 R f (1-13\(1966\).)2119 2340 w ( J. Stoer, "Asymptotic Upper and Lower Bounds for Results of Extrapolation Meth-)12 3433( Bulirsch and)2 545([6] R.)1 342 3 720 2496 t (ods,")970 2616 w 10 I f ( .)1 0( h.)1 25( th)1 50( at)1 28( Ma)1 50( M)1 108( .)1 0( r.)1 25( er)1 39( me)1 44( um)1 72(N Nu)1 117 12 1200 2616 t 10 B f (8,)1808 2616 w 10 R f (93-104\(1966\).)1908 2616 w ( Hilbert,)1 350( Courant and D.)3 686([7] R.)1 342 3 720 2772 t 10 B f (Methods of Mathematical Physics,)3 1516 1 2140 2772 t 10 R f (Vol. 1, Interscience, New York,)4 1342 1 3698 2772 t (1966.)970 2892 w ( Fundamental Spline Func-)3 1101( Curry and I.J. Schoenberg, "On Polya Frequency Functions IV: The)10 2780([8] H.B.)1 439 3 720 3048 t (tions and their Limits,")3 925 1 970 3168 t 10 I f ( .)1 0( h.)1 25( th)1 50( at)1 28( Ma)1 50( M)1 108( an nd d)3 100( a)1 75( .)1 0( l.)1 25( na al)2 78( An)1 50( A)1 86( f)1 0( of)1 28( o)1 75( .)1 0(J J.)1 69 18 1920 3168 t 10 B f (17,)2817 3168 w 10 R f (71-107\(1966\).)2967 3168 w ( Dahlquist, "A Special Stability Problem for Linear Multistep Methods,")9 2901([9] G.)1 347 2 720 3324 t 10 I f ( T)1 0( IT)1 56(B BI)1 94 3 3993 3324 t 10 B f (3,)4168 3324 w 10 R f (27-43\(1963\).)4268 3324 w ( Dahlquist, "Stability Questions for Some Numerical Methods for Ordinary Differential Equa-)11 3973([10] G.)1 347 2 720 3480 t (tions,")970 3600 w 10 I f ( .)1 0( h.)1 25( th)1 50( at)1 28( Ma)1 50( M)1 108( d)1 0( ed)1 50( li ie)2 72( pp pl)2 78( Ap)1 50( A)1 86( r)1 0( or)1 39( fo)1 50( f)1 53( .)1 0( p.)1 25( mp)1 50( ym)1 72( Sy)1 44( S)1 75( .)1 0( c.)1 25( oc)1 44( ro)1 50(P Pr)1 100 27 1256 3600 t 10 B f (15,)2530 3600 w 10 R f (147-158\(1963\).)2680 3600 w ( Large, Sparse, Nonsymmetric Systems of Linear Equa-)7 2337( C. Elman, "Iterative Methods for)5 1412([11] Howard)1 571 3 720 3756 t (tions," Yale Research Report YALEU/DCS/RR-229, February, 1982.)6 2777 1 970 3876 t ( Research Report)2 713( C. Elman, "A Stability Analysis of Incomplete LU Factorizations," Yale)10 3036([12] Howard)1 571 3 720 4032 t (YALEU/DCS/RR-365, February, 1985.)2 1591 1 970 4152 t ( T.E. Hull and B. Lindberg, "Comparing Numerical Methods for Stiff Systems of Ordi-)13 3504( Enright,)1 350([13] W.H.)1 466 3 720 4308 t (nary Differential Equations,")2 1158 1 970 4428 t 10 I f ( T)1 0( IT)1 56(B BI)1 94 3 2153 4428 t 10 B f (15,)2328 4428 w 10 R f (10-48\(1975\).)2478 4428 w ( and N.L. Schryer, "The PORT Mathematical Subroutine Library,")8 2787( Fox, A.D. Hall)3 664([14] P.A.)1 428 3 720 4584 t 10 I f ( ,)1 0( S,)1 25( MS)1 50( OM)1 83(T TO)1 128 5 4639 4584 t 10 B f (4,)4965 4584 w 10 R f (104-126\(1978\).)970 4704 w ( Automatic Integration of Ordinary Differential Equations,")6 2438( Gear, "The)2 480([15] C.W.)1 461 3 720 4860 t 10 I f ( M)1 0( CM)1 83( AC)1 67( A)1 95( .)1 0( mm m.)2 97( om)1 72(C Co)1 117 8 4133 4860 t 10 B f (14,)4698 4860 w 10 R f (176-)4857 4860 w (179\(1971\).)970 4980 w ( "Repeated Extrapolation to the Limit in the Numerical Solution of Ordinary Differential)12 3560( Gragg,)1 299([16] W.B.)1 461 3 720 5136 t (Equations," Thesis, UCLA \(1963\).)3 1390 1 970 5256 t ( Gragg, "On Extrapolation Algorithms for Ordinary Initial Value Problems")9 3034([17] W.B.)1 461 2 720 5412 t 10 I f ( .)1 0( l.)1 25( na al)2 78( An)1 50( A)1 87( .)1 0( m.)1 25( um)1 72( Nu)1 50( N)1 92( .)1 0( J.)1 25( J)1 69( M)1 0( AM)1 83( IA)1 61(S SI)1 83 17 4240 5412 t 10 B f (2,)970 5532 w 10 R f (384-403\(1965\).)1070 5532 w ( "Lecture Notes on Extrapolation Methods," presented at the SIAM National Meeting,)11 3551( Gragg,)1 308([18] W.B.)1 461 3 720 5688 t ( Conference on Ordinary Differential Equations, Dundee, Scot-)7 2621(Washington, June 1971, and at the)5 1449 2 970 5808 t (land, August, 1971.)2 786 1 970 5928 t ( H. Grosse, "Tensor Spline Approximation,")5 1865([19] E.)1 336 2 720 6084 t 10 B f (Linear Algebra and its Applications)4 1607 1 2964 6084 t 10 R f (, 34, 29-41)2 469 1 4571 6084 t (\(1980\).)970 6204 w ( Gustafsson, "A Class of First Order Factorization Methods,")8 2441([20] Ivar)1 410 2 720 6360 t 10 B f (BIT)3596 6360 w 10 R f (, 14, 142-156 \(1978\).)3 849 1 3769 6360 t ( a Rational Fortran,")3 914( Kernighan, "RATFOR - A Preprocessor for)6 1954([21] B.W.)1 461 3 720 6516 t 10 I f ( an nd d)3 100( a)1 107( ce e)2 44( ti ic)2 72( ct)1 28( ac)1 44( ra)1 50( Pr)1 39( -P)1 61( e-)1 33( re)1 44( ar)1 39( wa)1 50( ft tw)2 95(S So of)2 128 15 4106 6516 t ( ce e)2 44( nc)1 44( en)1 50( ie)1 44( ri)1 28( er)1 39( pe)1 44( xp)1 50(E Ex)1 105 9 970 6636 t 10 B f (5,)1443 6636 w 10 R f (395-406\(1975\).)1543 6636 w ( Richtmeyer and K.W. Morton,)4 1275([22] R.D.)1 439 2 720 6792 t 10 B f (Difference Methods for Initial Value Problems,)5 2038 1 2465 6792 t 10 R f (Interscience,)4534 6792 w (New York, 1967.)2 693 1 970 6912 t ( Schryer, "An Extrapolation Step-Size and Order Monitor for use in Solving Differential Equa-)13 3887([23] N.L.)1 433 2 720 7068 t (tions," Proceedings ACM National Meeting, San Diego, 1974.)7 2498 1 970 7188 t cleartomark showpage saveobj restore end %%EndPage: 1 32 %%Page: 2 33 DpostDict begin /saveobj save def mark 33 pagesetup 10 R f (\261 B-2 \261)2 300 1 2730 480 t ( Schryer, "An Extrapolation Step-Size and Order Monitor for use in Solving Differential Equa-)13 3887([24] N.L.)1 433 2 720 840 t (tions," in preparation.)2 868 1 970 960 t ( B-splines, for Solving Differential Equa-)5 1749( Schryer, "A Tutorial on Galerkin's Method, using)7 2138([25] N.L.)1 433 3 720 1116 t (tions," Bell Laboratories Computing Science Technical Report #52, 1976.)8 2958 1 970 1236 t ( Differential Equations in One Space Variable", Bell Laboratories Computing)9 3184( Schryer, "Partial)2 703([26] N.L.)1 433 3 720 1392 t (Science Technical Report #115, 1984.)4 1525 1 970 1512 t ( Schultz, ")2 462([27] M.)1 364 2 720 1668 t 10 I f (L L)1 56 1 1546 1668 t 7 S f (\245)1613 1628 w 10 R f (-)1672 1668 w 10 I f ( y)1 0( ry)1 44( or)1 39( eo)1 50( he)1 44( Th)1 50( T)1 104( on n)2 50( ti io)2 78( at)1 28( ma)1 50( im)1 72( xi)1 28( ox)1 44( ro)1 50( pp pr)2 89( Ap)1 50( A)1 109( e)1 0( te)1 44( at)1 28( ia)1 50( ri)1 28( ar)1 39( va)1 50( lt ti iv)3 100( ul)1 28(M Mu)1 133 28 1705 1668 t 10 R f (",)3184 1668 w 10 I f ( ,)1 0( s,)1 25( is)1 39( si)1 28( ys)1 39( ly)1 44( na al)2 78( An)1 50( A)1 110( l)1 0( al)1 28( ca)1 50( ic)1 44( ri)1 28( er)1 39( me)1 44( um)1 72( Nu)1 50( N)1 116( on n)2 50( o)1 98( l)1 0( na al)2 78( rn)1 50( ou ur)2 89( Jo)1 50( J)1 92( M)1 0( AM)1 83( IA)1 61(S SI)1 83 31 3298 1668 t 10 B f (6)4965 1668 w 10 R f (,)5015 1668 w (161-183\(1969\).)970 1788 w ( der Houven, "Algorithm 621. Software with Low Storage Require-)9 2809( P. Sommeijer and P. J. van)6 1169([28] B.)1 342 3 720 1944 t (ments for Two-Dimensional Parabolic Differential Equations,")5 2570 1 970 2064 t 10 I f ( e)1 0( re)1 44( ar)1 39( wa)1 50( ft tw)2 95( So of)2 78( S)1 85( .)1 0( h.)1 25( th)1 50( at)1 28( Ma)1 50( M)1 118( on n)2 50( o)1 86( .)1 0( s.)1 25( an ns)2 89( ra)1 50( Tr)1 39( T)1 92( M)1 0( CM)1 83(A AC)1 128 24 3576 2064 t 10 B f (10,)4915 2064 w 10 R f (378-396\(1985\).)970 2184 w ( Two-Dimensional Nonlinear Partial Differ-)4 1771( K. Melgaard and R.F. Sincovec "General Software for)8 2202([29] D.)1 347 3 720 2340 t (ential Equations,")1 713 1 970 2460 t 10 I f ( e)1 0( re)1 44( ar)1 39( wa)1 50( ft tw)2 95( So of)2 78( S)1 75( .)1 0( h.)1 25( th)1 50( at)1 28( Ma)1 50( M)1 108( on n)2 50( o)1 75( .)1 0( s.)1 25( an ns)2 89( ra)1 50( Tr)1 39( T)1 81( M)1 0( CM)1 83(A AC)1 128 24 1708 2460 t 10 B f (7,)2995 2460 w 10 R f (106-125\(1981\).)3095 2460 w ( Expansions for the Error of Discretization Algorithms for Non-Linear)9 3021( Stetter, "Asymptotic)2 888([30] H.J.)1 411 3 720 2616 t (Functional Equations,")1 919 1 970 2736 t 10 I f ( .)1 0( h.)1 25( th)1 50( at)1 28( Ma)1 50( M)1 108( .)1 0( r.)1 25( er)1 39( me)1 44( um)1 72(N Nu)1 117 12 1914 2736 t 10 B f (7,)2522 2736 w 10 R f (18-31\(1965\).)2622 2736 w ( Strang and G. Fix,)4 761([31] G.)1 347 2 720 2892 t 10 B f (An Analysis of the Finite Element Method,)6 1819 1 1853 2892 t 10 R f (Prentice-Hall, New York, 1973.)3 1275 1 3697 2892 t ( Long)1 253( Wahlbin, "Error Estimates for a Galerkin Method for a Class of Model Equations for)14 3640([32] Lars)1 427 3 720 3048 t (Waves,")970 3168 w 10 I f ( .)1 0( h.)1 25( th)1 50( at)1 28( Ma)1 50( M)1 108( .)1 0( r.)1 25( er)1 39( me)1 44( um)1 72(N Nu)1 117 12 1332 3168 t 10 B f (23,)1940 3168 w 10 R f (289-303\(1975\).)2090 3168 w ( in MOS Devices,")3 811( Wilson, "Numerical Simulation of Gate Oxide Thinning)7 2392([33] L.O.)1 433 3 720 3324 t 10 I f ( .)1 0( m.)1 25( em)1 72( he)1 44( ch)1 50( oc)1 44( ro)1 50( tr)1 39( ec ct)2 72( le)1 44( El)1 28( E)1 104( .)1 0(J J.)1 69 14 4399 3324 t ( ., ,)2 25( c.)1 25(S So oc)2 144 3 970 3444 t 10 B f (129,)1189 3444 w 10 R f (831-837\(1982\).)1389 3444 w ( O. Wilson, "Lateral Epitaxial Growth over Oxide", to be submitted to the)12 3050([34] L.)1 336 2 720 3600 t 10 I f ( .)1 0( m.)1 25( em)1 72( he)1 44( ch)1 50( oc)1 44( ro)1 50( tr)1 39( ec ct)2 72( le)1 44( El)1 28( E)1 95( e)1 0( he)1 44( th)1 50( t)1 62( f)1 0( of)1 28( o)1 84( .)1 0(J J.)1 69 21 4140 3600 t ( .)1 0( c.)1 25(S So oc)2 144 3 970 3720 t cleartomark showpage saveobj restore end %%EndPage: 2 33 %%Page: 1 34 DpostDict begin /saveobj save def mark 34 pagesetup 10 B f (Appendix 1)1 493 1 2633 840 t (B-splines)2685 1200 w 10 R f ( solution process is affected by the representation of the approximate numerical solution.)12 3655(The entire)1 415 2 970 1596 t ( choose a space of functions, from which to obtain the element closest to the)14 3104(Specifically, we would like to)4 1216 2 720 1716 t ( space should have several nice properties, including being \(1\) easy to work with and \(2\))15 3732(solution. This)1 588 2 720 1836 t (capable of approximating the solution accurately.)5 1974 1 720 1956 t ( in B-splines of order)4 893(Such a representation exists - expansion)5 1650 2 970 2112 t 10 I f (k k)1 44 1 3549 2112 t 10 R f ( rest of this brief)4 704([1,2,3,8,27]. The)1 707 2 3629 2112 t ( [27] for a complete discussion of the multi-)8 1807( See)1 200(tutorial on splines will discuss the one-dimensional case.)7 2313 3 720 2232 t ( polynomials, that is, polynomi-)4 1287( is a method for representing functions by piecewise)8 2093( This)1 229(dimensional case.)1 711 4 720 2352 t (als of degree)2 515 1 720 2472 t 10 I f (k k)1 44 1 1263 2472 t 10 S f (- -)1 55 1 1331 2472 t 10 R f ( the integer)2 455( Here)1 246(1 or less over each sub-interval of a mesh or grid.)10 2006 3 1402 2472 t 10 I f (k k)1 44 1 4137 2472 t 10 R f (is any number)2 572 1 4209 2472 t 10 I f (k k)1 44 1 4809 2472 t 10 S f (\263)4894 2472 w 10 R f (2)4990 2472 w ( piecewise polynomial representation is required to satisfy certain continuity restric-)10 3451( The)1 213(the user desires.)2 656 3 720 2592 t ( let)1 168( Specifically,)1 595( mesh sub-interval.)2 849(tions at the end points of each)6 1457 4 720 2712 t 10 S f ( =)1 0(p =)1 150 2 3857 2712 t 10 I f ( x)1 0( x)1 52({ {)1 40 3 4047 2712 t 7 R f (1)4150 2732 w 10 R f (,)4225 2712 w (. . .)2 125 1 4307 2687 t (,)4489 2712 w 10 I f (x x)1 44 1 4546 2712 t 7 I f (N N)1 47 1 4601 2732 t 10 I f (} })1 40 1 4664 2712 t 10 R f (, where)1 336 1 4704 2712 t 10 I f (L L)1 56 1 720 2832 t 10 S f (= =)1 55 1 816 2832 t 10 I f (x x)1 44 1 911 2832 t 7 R f (1)966 2852 w 10 S f (\243)1050 2832 w 10 I f (x x)1 44 1 1146 2832 t 7 R f (2)1201 2852 w 10 S f (\243)1285 2832 w 10 R f (. . .)2 125 1 1406 2807 t 10 S f (\243)1597 2832 w 10 I f (x x)1 44 1 1693 2832 t 7 I f (N N)1 47 1 1748 2852 t 10 S f (= =)1 55 1 1843 2832 t 10 I f (R R)1 61 1 1938 2832 t 10 R f (, be a)2 231 1 1999 2832 t 10 B f (grid)2264 2832 w 10 R f ( Let)1 193(on the interval \(L,R\).)3 873 2 2476 2832 t 10 I f (m m)1 72 1 3577 2832 t 7 I f (i i)1 20 1 3660 2852 t 10 R f (be the)1 251 1 3723 2832 t 10 B f (multiplicity)4009 2832 w 10 R f (of)4539 2832 w 10 I f (x x)1 44 1 4657 2832 t 7 I f (i i)1 20 1 4712 2852 t 10 R f (, or the)2 300 1 4740 2832 t (number of times)2 665 1 720 2952 t 10 I f (x x)1 44 1 1415 2952 t 7 I f (i i)1 20 1 1470 2972 t 10 R f ( the list)2 303(appears in)1 412 2 1528 2952 t 10 S f (p)2272 2952 w 10 R f ( space of B-splines of)4 881(. The)1 234 2 2327 2952 t 10 B f (order)3471 2952 w 10 I f (k k)1 44 1 3738 2952 t 10 R f (on the mesh)2 491 1 3811 2952 t 10 S f (p)4331 2952 w 10 R f (is defined to be)3 625 1 4415 2952 t (the collection of all functions)4 1171 1 720 3072 t 10 I f (f f)1 28 1 1916 3072 t 10 R f ( are polynomials of degree)4 1064(\(A1.1\) that)1 500 2 720 3228 t 10 S f (< <)1 55 1 2309 3228 t 10 I f (k k)1 44 1 2413 3228 t 10 R f (on each interval \()3 695 1 2482 3228 t 10 I f (x x)1 44 1 3209 3228 t 7 I f (i i)1 20 1 3264 3248 t 10 R f (,)3324 3228 w 10 I f (x x)1 44 1 3381 3228 t 7 I f (i i)1 20 1 3436 3248 t 7 S f (+ +)1 39 1 3472 3248 t 7 R f (1)3522 3248 w 10 R f (\) for)1 174 1 3597 3228 t 10 I f (i i)1 28 1 3796 3228 t 10 S f (= =)1 55 1 3848 3228 t 10 R f ( . . . ,)4 200(1 ,)1 83 2 3919 3228 t 10 I f (N N)1 67 1 4226 3228 t 10 S f (- -)1 55 1 4317 3228 t 10 R f (1,)4388 3228 w ( which)1 283(\(A1.2\) for)1 466 2 720 3388 t 10 I f (d d)1 50 1 1509 3388 t 7 I f (k k)1 31 1 1570 3334 t 7 S f (- -)1 39 1 1617 3334 t 7 R f (1)1667 3334 w 7 S f (- -)1 39 1 1713 3334 t 7 I f (m m)1 50 1 1763 3334 t 4 I f (i i)1 11 1 1819 3348 t 10 I f (f f)1 28 1 1859 3388 t 10 R f (\()1903 3388 w 10 I f (x x)1 44 1 1944 3388 t 7 I f (i i)1 20 1 1999 3408 t 10 R f (\))2035 3388 w 10 I f ( x)1 0( dx)1 44( d)1 82(/ /)1 28 4 2108 3388 t 7 I f (k k)1 31 1 2273 3334 t 7 S f (- -)1 39 1 2320 3334 t 7 R f (1)2370 3334 w 7 S f (- -)1 39 1 2416 3334 t 7 I f (m m)1 50 1 2466 3334 t 4 I f (i i)1 11 1 2522 3348 t 10 R f (exists and is continuous at each)5 1332 1 2586 3388 t 10 I f (x x)1 44 1 3958 3388 t 7 I f (i i)1 20 1 4013 3408 t 10 R f (, for)1 181 1 4041 3388 t 10 I f (i i)1 28 1 4262 3388 t 10 S f (= =)1 55 1 4314 3388 t 10 R f ( . . . ,)4 200(1 ,)1 83 2 4385 3388 t 10 I f (N N)1 67 1 4692 3388 t 10 R f (, when)1 281 1 4759 3388 t (viewed as a function defined only on [)7 1533 1 1070 3508 t 10 I f (L L)1 56 1 2611 3508 t 10 R f (,)2675 3508 w 10 I f (R R)1 61 1 2708 3508 t 10 R f (], and)1 227 1 2777 3508 t ( have)1 213(\(A1.3\) that)1 500 2 720 3664 t 10 I f (f f)1 28 1 1458 3664 t 10 S f (\272)1576 3664 w 10 R f (0 outside [)2 422 1 1705 3664 t 10 I f (L L)1 56 1 2135 3664 t 10 R f (,)2199 3664 w 10 I f (R R)1 61 1 2232 3664 t 10 R f (].)2301 3664 w (The multiplicity)1 660 1 970 3820 t 10 I f (m m)1 72 1 1667 3820 t 7 I f (i i)1 20 1 1750 3840 t 10 R f (of a point)2 407 1 1815 3820 t 10 I f (x x)1 44 1 2259 3820 t 7 I f (i i)1 20 1 2314 3840 t 10 R f (is restricted to be in the range 1)7 1340 1 2379 3820 t 10 S f (\243)3760 3820 w 10 I f (m m)1 72 1 3856 3820 t 7 I f (i i)1 20 1 3939 3840 t 10 S f (\243)4008 3820 w 10 I f (k k)1 44 1 4104 3820 t 10 R f (. For)1 226 1 4148 3820 t 10 I f (m m)1 72 1 4412 3820 t 7 I f (i i)1 20 1 4495 3840 t 10 S f (= =)1 55 1 4539 3820 t 10 R f (1 we have)2 430 1 4610 3820 t 10 I f (d d)1 50 1 720 3940 t 7 I f (k k)1 31 1 781 3900 t 7 S f (- -)1 39 1 828 3900 t 7 R f (2)878 3900 w 10 I f ( x)1 0( dx)1 44( d)1 82( /)1 0( /)1 76(f f)1 28 6 937 3940 t 7 I f (k k)1 31 1 1178 3900 t 7 S f (- -)1 39 1 1225 3900 t 7 R f (2)1275 3900 w 10 R f (continuous at)1 542 1 1349 3940 t 10 I f (x x)1 44 1 1922 3940 t 7 I f (i i)1 20 1 1977 3960 t 10 R f ( is the most continuity that can be imposed at)9 1862(. This)1 259 2 2005 3940 t 10 I f (x x)1 44 1 4157 3940 t 7 I f (i i)1 20 1 4212 3960 t 10 R f (without making)1 637 1 4271 3940 t 10 I f (f f)1 28 1 4938 3940 t 10 R f (a)4996 3940 w (polynomial of degree)2 868 1 720 4060 t 10 I f (k k)1 44 1 1620 4060 t 10 S f (- -)1 55 1 1688 4060 t 10 R f (1 on \()2 247 1 1759 4060 t 10 I f (x x)1 44 1 2014 4060 t 7 I f (i i)1 20 1 2069 4080 t 7 S f (- -)1 39 1 2105 4080 t 7 R f (1)2155 4080 w 10 R f (,)2230 4060 w 10 I f (x x)1 44 1 2287 4060 t 7 I f (i i)1 20 1 2342 4080 t 7 S f (+ +)1 39 1 2378 4080 t 7 R f (1)2428 4080 w 10 R f (\). For)1 254 1 2479 4060 t 10 I f (m m)1 72 1 2765 4060 t 7 I f (i i)1 20 1 2848 4080 t 10 S f (= =)1 55 1 2916 4060 t 10 I f (k k)1 44 1 3011 4060 t 10 R f (the condition that)2 716 1 3088 4060 t 10 I f (d d)1 50 1 3837 4060 t 7 S f (- -)1 39 1 3898 4020 t 7 R f (1)3948 4020 w 10 I f ( x)1 0( dx)1 44( d)1 82( /)1 0( /)1 76(f f)1 28 6 4007 4060 t 7 S f (- -)1 39 1 4248 4020 t 7 R f (1)4298 4020 w 10 R f (be continuous is)2 666 1 4374 4060 t (interpreted to mean that)3 993 1 720 4180 t 10 I f (f f)1 28 1 1752 4180 t 10 R f ( left\) at)2 314(is continuous from the right \(but not necessarily from the)9 2410 2 1819 4180 t 10 I f (x x)1 44 1 4581 4180 t 10 S f (= =)1 55 1 4674 4180 t 10 I f (x x)1 44 1 4778 4180 t 7 I f (i i)1 20 1 4833 4200 t 10 R f (, for)1 179 1 4861 4180 t 10 I f (x x)1 44 1 720 4300 t 7 I f (i i)1 20 1 775 4320 t 10 S f (< <)1 55 1 852 4300 t 10 I f (R R)1 61 1 956 4300 t 10 R f ( if)1 88(, and continuous from the left)5 1187 2 1017 4300 t 10 I f (x x)1 44 1 2319 4300 t 7 I f (i i)1 20 1 2374 4320 t 10 S f (= =)1 55 1 2451 4300 t 10 I f (R R)1 61 1 2555 4300 t 10 R f ( means that B-splines are continuous at the end points)9 2169(. This)1 255 2 2616 4300 t ( only on [)3 410(of the mesh when viewed as functions defined)7 1912 2 720 4420 t 10 I f (L L)1 56 1 3050 4420 t 10 R f (,)3114 4420 w 10 I f (R R)1 61 1 3147 4420 t 10 R f ( collection of functions is denoted by)6 1530(]. This)1 294 2 3216 4420 t 10 I f (B B)1 61 1 720 4540 t 7 S f (p)792 4560 w 7 R f (,)836 4560 w 7 I f (k k)1 31 1 859 4560 t 10 R f (. These)1 322 1 898 4540 t 10 I f (B B)1 61 1 1254 4540 t 7 S f (p)1326 4560 w 7 R f (,)1370 4560 w 7 I f (k k)1 31 1 1393 4560 t 10 R f ( summed up by deBoor [1,2], in the case)8 1700(spaces have nice approximation properties, as)5 1874 2 1466 4540 t (when)720 4660 w 10 I f (m m)1 72 1 961 4660 t 7 R f (1)1044 4680 w 10 S f (= =)1 55 1 1136 4660 t 10 I f (k k)1 44 1 1240 4660 t 10 S f (= =)1 55 1 1333 4660 t 10 I f (m m)1 72 1 1437 4660 t 7 I f (N N)1 47 1 1520 4680 t 10 R f (:)1575 4660 w 10 B f (Theorem 1)1 463 1 720 4900 t 10 R f (Let)970 5056 w 10 I f (f f)1 28 1 1128 5056 t 10 R f (be any function with)3 824 1 1181 5056 t 10 I f (f f)1 28 1 2030 5056 t 7 R f (\( 0 \))2 91 1 2080 5016 t 10 R f (through)2204 5056 w 10 I f (f f)1 28 1 2541 5056 t 7 R f (\()2591 5016 w 7 I f (k k)1 31 1 2619 5016 t 7 R f (\))2655 5016 w 10 R f (continuous on [)2 624 1 2712 5056 t 10 I f (L L)1 56 1 3344 5056 t 10 R f (,)3408 5056 w 10 I f (R R)1 61 1 3441 5056 t 10 R f (], where)1 327 1 3510 5056 t 10 I f (f f)1 28 1 3863 5056 t 7 R f (\()3913 5016 w 7 I f (j j)1 20 1 3952 5016 t 7 R f (\))3977 5016 w 10 R f (denotes the)1 453 1 4034 5056 t 10 I f (j j)1 28 1 4513 5056 t 7 I f ( h)1 0(t th)1 55 2 4552 5016 t 10 R f (derivative)4641 5056 w (of)720 5176 w 10 I f (f f)1 28 1 839 5176 t 10 R f (. Let)1 219 1 867 5176 t 10 I f (h h)1 50 1 1122 5176 t 10 S f (= =)1 55 1 1212 5176 t 7 I f (i i)1 20 1 1307 5246 t 7 S f (= =)1 39 1 1343 5246 t 7 R f ( . . . ,)4 144(1 ,)1 58 2 1393 5246 t 7 I f (N N)1 47 1 1611 5246 t 7 S f (- -)1 39 1 1674 5246 t 7 R f (1)1724 5246 w 10 I f ( x)1 0( ax)1 44(M Ma)1 133 3 1444 5176 t 10 S f (\357 \357)1 49 1 1758 5193 t 10 I f (x x)1 44 1 1830 5176 t 7 I f (i i)1 20 1 1885 5196 t 7 S f (+ +)1 39 1 1921 5196 t 7 R f (1)1971 5196 w 10 S f (- -)1 55 1 2054 5176 t 10 I f (x x)1 44 1 2149 5176 t 7 I f (i i)1 20 1 2204 5196 t 10 S f (\357 \357)1 49 1 2256 5193 t 10 R f ( there is an element)4 816( Then)1 265(be the largest mesh interval length.)5 1448 3 2308 5176 t 10 I f (g g)1 50 1 4872 5176 t 10 R f (of)4957 5176 w 10 I f (B B)1 61 1 720 5346 t 7 S f (p)792 5366 w 7 R f (,)836 5366 w 7 I f (k k)1 31 1 859 5366 t 10 R f (so that)1 264 1 923 5346 t 10 S f ( \357)1 0( \357)1 24(\357 \357)1 49 3 1220 5543 t 10 I f (f f)1 28 1 1324 5526 t 7 R f (\()1374 5486 w 7 I f (j j)1 20 1 1413 5486 t 7 R f (\))1438 5486 w 10 R f (\()1477 5526 w 10 I f (x x)1 44 1 1518 5526 t 10 R f (\))1570 5526 w 10 S f (- -)1 55 1 1651 5526 t 10 I f (g g)1 50 1 1746 5526 t 7 R f (\()1807 5486 w 7 I f (j j)1 20 1 1846 5486 t 7 R f (\))1871 5486 w 10 R f (\()1910 5526 w 10 I f (x x)1 44 1 1951 5526 t 10 R f (\))2003 5526 w 10 S f ( \357)1 0( \357)1 24(\357 \357)1 49 3 2068 5543 t (\243)2173 5526 w 10 I f (C C)1 67 1 2269 5526 t 10 R f (\()2344 5526 w 10 I f (k k)1 44 1 2385 5526 t 10 R f (,)2437 5526 w 10 I f (f f)1 28 1 2478 5526 t 10 R f (\))2530 5526 w 10 I f (h h)1 50 1 2612 5526 t 7 I f (k k)1 31 1 2673 5486 t 7 S f (- -)1 39 1 2720 5486 t 7 I f (j j)1 20 1 2770 5486 t 10 R f (for 0)1 191 1 720 5706 t 10 S f (\243)952 5706 w 10 I f (j j)1 28 1 1056 5706 t 10 S f (\243)1125 5706 w 10 I f (k k)1 44 1 1221 5706 t 10 R f (, where)1 293 1 1265 5706 t 10 I f (C C)1 67 1 1583 5706 t 10 R f (\()1658 5706 w 10 I f (k k)1 44 1 1699 5706 t 10 R f (,)1751 5706 w 10 I f (f f)1 28 1 1792 5706 t 10 R f (\) represents a constant that depends only upon)7 1844 1 1844 5706 t 10 I f (k k)1 44 1 3713 5706 t 10 R f (and)3782 5706 w 10 I f (f f)1 28 1 3951 5706 t 10 R f (, but not)2 331 1 3979 5706 t 10 I f (h h)1 50 1 4335 5706 t 10 R f (.)4385 5706 w ( as)1 113(That is,)1 304 2 970 5862 t 10 I f (h h)1 50 1 1417 5862 t 10 S f (\256)1508 5862 w 10 R f (0, the error in the best B-spline approximation to)8 1991 1 1648 5862 t 10 I f (f f)1 28 1 3669 5862 t 10 R f (goes to zero like)3 672 1 3727 5862 t 10 I f (h h)1 50 1 4429 5862 t 7 I f (k k)1 31 1 4490 5822 t 10 R f (; the error in)3 511 1 4529 5862 t (its derivative behaves like)3 1040 1 720 5982 t 10 I f (h h)1 50 1 1785 5982 t 7 I f (k k)1 31 1 1846 5942 t 7 S f (- -)1 39 1 1893 5942 t 7 R f (1)1943 5942 w 10 R f (; etc.)1 194 1 1986 5982 t ( relative spacing of the mesh points of)7 1544(Note that this theorem makes no assumption about the)8 2187 2 970 6138 t 10 S f (p)4729 6138 w 10 R f (to get)1 228 1 4812 6138 t 10 I f (h h)1 50 1 720 6258 t 7 I f (k k)1 31 1 781 6218 t 10 R f ( many problems, the ability to grade the mesh with B-splines and still get)13 2979(error. In)1 355 2 849 6258 t 10 I f (h h)1 50 1 4212 6258 t 7 I f (k k)1 31 1 4273 6218 t 10 R f ( a decided)2 410(error is)1 289 2 4341 6258 t (advantage.)720 6378 w (In practice,)1 454 1 970 6534 t 10 I f (k k)1 44 1 1455 6534 t 10 R f ( or even 10, depending on what the function)8 1817( 8)1 91( ,)1 33( 6)1 91( ,)1 33(is usually taken to be 4)5 954 6 1530 6534 t 10 I f (f f)1 28 1 4581 6534 t 10 R f (looks like)1 399 1 4641 6534 t (and how much accuracy is desired.)5 1416 1 720 6654 t 10 I f (k k)1 44 1 2190 6654 t 10 R f ( to be even due to the natural way in)9 1477(is usually, but not always, taken)5 1300 2 2263 6654 t (which such splines arise and their smoothing properties when used to approximate functions described by)14 4320 1 720 6774 t ( the more accuracy desired, the larger the value of)9 2011( Typically,)1 460(discrete data [2].)2 671 3 720 6894 t 10 I f (k k)1 44 1 3889 6894 t 10 R f ( example, if)2 476( For)1 191(should be.)1 413 3 3960 6894 t 10 I f (k k)1 44 1 720 7014 t 10 S f (= =)1 55 1 788 7014 t 10 R f (8 and the mesh length)4 881 1 859 7014 t 10 I f (h h)1 50 1 1766 7014 t 10 R f ( should decrease by a factor of)6 1230(is halved, then Theorem 1 indicates that the error)8 1968 2 1842 7014 t (2)720 7134 w 7 R f (8)775 7094 w 10 S f (= =)1 55 1 867 7134 t 10 R f ( the work needed to solve a problem using)8 1756(256. However,)1 623 2 971 7134 t 10 CW f (TTGR)3383 7134 w 10 R f (is)3655 7134 w 10 I f (O O)1 72 1 3754 7134 t 10 R f (\()3834 7134 w 10 I f ( k)1 0(N Nk)1 111 2 3875 7134 t 7 R f (4)3997 7094 w 10 R f ( a)1 76(\). Thus,)1 340 2 4048 7134 t 10 I f (k k)1 44 1 4496 7134 t 10 S f (= =)1 55 1 4564 7134 t 10 R f (8 solution)1 405 1 4635 7134 t (will cost 16 times as much as a)7 1332 1 720 7254 t 10 I f (k k)1 44 1 2090 7254 t 10 S f (= =)1 55 1 2158 7254 t 10 R f (4 solution for the)3 725 1 2229 7254 t 10 I f ( e)1 0( me)1 44( am)1 72(s sa)1 89 4 2992 7254 t 10 R f ( if the error is held constant, the)7 1361(mesh. But)1 444 2 3235 7254 t cleartomark showpage saveobj restore end %%EndPage: 1 34 %%Page: 2 35 DpostDict begin /saveobj save def mark 35 pagesetup 10 R f (\261 1-2 \261)2 283 1 2738 480 t ( the optimal)2 514( Hence,)1 347( order.)1 278(number of mesh points needed is less for higher than lower)10 2553 4 720 840 t 10 I f (k k)1 44 1 4455 840 t 10 R f (results from)1 498 1 4542 840 t (minimizing)720 960 w 10 I f (O O)1 72 1 1213 960 t 10 R f (\()1293 960 w 10 I f (N N)1 67 1 1334 960 t 7 I f (k k)1 31 1 1412 980 t 10 I f (k k)1 44 1 1459 960 t 7 R f (4)1514 920 w 10 R f (\), where)1 332 1 1565 960 t 10 I f (N N)1 67 1 1928 960 t 7 I f (k k)1 31 1 2006 980 t 10 R f (is the number of mesh points needed to solve the problem to the desired)13 2963 1 2077 960 t (accuracy using a)2 664 1 720 1080 t 10 I f (k k)1 44 1 1409 1080 t 10 R f ( optimization is highly problem dependent.)5 1722( This)1 228(order B-spline.)1 599 3 1478 1080 t ( for the spaces)3 600(A computationally convenient basis exists)4 1720 2 970 1236 t 10 I f (B B)1 61 1 3324 1236 t 7 S f (p)3396 1256 w 7 R f (,)3440 1256 w 7 I f (k k)1 31 1 3463 1256 t 10 R f ( dimension of)2 568(. The)1 239 2 3502 1236 t 10 I f (B B)1 61 1 4343 1236 t 7 S f (p)4415 1256 w 7 R f (,)4459 1256 w 7 I f (k k)1 31 1 4482 1256 t 10 R f (is)4555 1236 w 10 I f (N N)1 67 1 4656 1236 t 10 S f (- -)1 55 1 4747 1236 t 10 I f (k k)1 44 1 4818 1236 t 10 R f (and)4896 1236 w (the basis consists of elements)4 1221 1 720 1356 t 10 I f (B B)1 61 1 1977 1356 t 7 I f (i i)1 20 1 2049 1376 t 10 R f (\()2085 1356 w 10 I f (x x)1 44 1 2126 1356 t 10 R f (\) ,)1 74 1 2178 1356 t 10 I f (i i)1 28 1 2326 1356 t 10 S f (= =)1 55 1 2378 1356 t 10 R f ( . . . ,)4 200(1 ,)1 83 2 2449 1356 t 10 I f (N N)1 67 1 2756 1356 t 10 S f (- -)1 55 1 2847 1356 t 10 I f (k k)1 44 1 2918 1356 t 10 R f ( complete description of the)4 1159(. A)1 158 2 2962 1356 t 10 I f (B B)1 61 1 4315 1356 t 7 I f (i i)1 20 1 4387 1376 t 10 R f ( in [8])2 264(is given)1 325 2 4451 1356 t ( when the multiplicities of the first and last mesh points are both)12 2571( Briefly,)1 358(and [3].)1 310 3 720 1476 t 10 I f (k k)1 44 1 3984 1476 t 10 R f (, so that)2 314 1 4028 1476 t 10 I f (x x)1 44 1 1220 1656 t 7 R f (1)1275 1676 w 10 S f (= =)1 55 1 1367 1656 t 10 R f (. . .)2 125 1 1496 1631 t 10 S f (= =)1 55 1 1695 1656 t 10 I f (x x)1 44 1 1799 1656 t 7 I f (k k)1 31 1 1854 1676 t 10 R f (and)720 1836 w 10 I f (x x)1 44 1 1220 2016 t 7 I f (N N)1 47 1 1275 2036 t 7 S f (- -)1 39 1 1338 2036 t 7 I f (k k)1 31 1 1388 2036 t 7 S f (+ +)1 39 1 1435 2036 t 7 R f (1)1485 2036 w 10 S f (= =)1 55 1 1577 2016 t 10 R f (. . .)2 125 1 1706 1991 t 10 S f (= =)1 55 1 1905 2016 t 10 I f (x x)1 44 1 2009 2016 t 7 I f (N N)1 47 1 2064 2036 t 10 R f (,)2127 2016 w (the main properties of the)4 1026 1 720 2196 t 10 I f (B B)1 61 1 1771 2196 t 7 I f (i i)1 20 1 1843 2216 t 10 R f (\()1879 2196 w 10 I f (x x)1 44 1 1920 2196 t 10 R f (\) follow)1 319 1 1972 2196 t (\(A1.4\) Each)1 549 1 720 2472 t 10 I f (B B)1 61 1 1329 2472 t 7 I f (i i)1 20 1 1401 2492 t 10 R f (is non-zero only on [)4 972 1 1489 2472 t 10 I f (x x)1 44 1 2469 2472 t 7 I f (i i)1 20 1 2524 2492 t 10 R f (,)2584 2472 w 10 I f (x x)1 44 1 2641 2472 t 7 I f (i i)1 20 1 2696 2492 t 7 S f (+ +)1 39 1 2732 2492 t 7 I f (k k)1 31 1 2782 2492 t 10 R f ( at)1 133(] and is identically zero elsewhere, as well as)8 2078 2 2829 2472 t 10 I f (x x)1 44 1 1070 2592 t 7 R f (1)1125 2612 w 10 R f (, . . . ,)4 225 1 1176 2592 t 10 I f (x x)1 44 1 1425 2592 t 7 I f (i i)1 20 1 1480 2612 t 7 S f (- -)1 39 1 1516 2612 t 7 R f (1)1566 2612 w 10 R f (and)1634 2592 w 10 I f (x x)1 44 1 1803 2592 t 7 I f (i i)1 20 1 1858 2612 t 7 S f (+ +)1 39 1 1894 2612 t 7 I f (k k)1 31 1 1944 2612 t 7 S f (+ +)1 39 1 1991 2612 t 7 R f (1)2041 2612 w 10 R f (, . . . ,)4 225 1 2092 2592 t 10 I f (x x)1 44 1 2341 2592 t 7 I f (N N)1 47 1 2396 2612 t 10 R f (, even if they are in [)6 828 1 2451 2592 t 10 I f (x x)1 44 1 3287 2592 t 7 I f (i i)1 20 1 3342 2612 t 10 R f (,)3402 2592 w 10 I f (x x)1 44 1 3459 2592 t 7 I f (i i)1 20 1 3514 2612 t 7 S f (+ +)1 39 1 3550 2612 t 7 R f (1)3600 2612 w 10 R f (].)3651 2592 w ( sum)1 192(\(A1.5\) The)1 505 2 720 2748 t 10 I f (B B)1 61 1 1442 2748 t 7 R f (1)1514 2768 w 10 R f (\()1565 2748 w 10 I f (x x)1 44 1 1606 2748 t 10 R f (\))1658 2748 w 10 S f (+ +)1 55 1 1707 2748 t 10 R f (. . .)2 125 1 1803 2723 t 10 S f (+ +)1 55 1 1969 2748 t 10 I f (B B)1 61 1 2040 2748 t 7 I f (N N)1 47 1 2112 2768 t 7 S f (- -)1 39 1 2175 2768 t 7 I f (k k)1 31 1 2225 2768 t 10 R f (\()2272 2748 w 10 I f (x x)1 44 1 2313 2748 t 10 R f (\) is identically one.)3 766 1 2365 2748 t (\(A1.6\) Each)1 549 1 720 2904 t 10 I f (B B)1 61 1 1294 2904 t 7 I f (i i)1 20 1 1366 2924 t 10 R f (obeys 0)1 308 1 1419 2904 t 10 S f (\243)1768 2904 w 10 I f (B B)1 61 1 1864 2904 t 7 I f (i i)1 20 1 1936 2924 t 10 R f (\()1972 2904 w 10 I f (x x)1 44 1 2013 2904 t 10 R f (\))2065 2904 w 10 S f (\243)2147 2904 w 10 R f (1 everywhere and possesses only one maximum.)6 1944 1 2243 2904 t ( is independent of the multiplicities)5 1419(The convergence result of Theorem 1)5 1498 2 970 3180 t 10 I f (m m)1 72 1 3913 3180 t 7 I f (i i)1 20 1 3996 3200 t 10 R f (of the interior points)3 822 1 4050 3180 t 10 I f (x x)1 44 1 4898 3180 t 7 I f (i i)1 20 1 4953 3200 t 10 R f (\()5007 3180 w 10 I f (k k)1 44 1 720 3300 t 10 S f (< <)1 55 1 813 3300 t 10 I f (i i)1 28 1 917 3300 t 10 S f (\243)986 3300 w 10 I f (N N)1 67 1 1082 3300 t 10 S f (- -)1 55 1 1173 3300 t 10 I f (k k)1 44 1 1244 3300 t 10 R f ( for smooth functions)3 873( Usually,)1 391(\) of the mesh.)3 564 3 1318 3300 t 10 I f (f f)1 28 1 3176 3300 t 10 R f (,)3204 3300 w 10 I f (m m)1 72 1 3258 3300 t 7 I f (i i)1 20 1 3341 3320 t 10 S f (= =)1 55 1 3385 3300 t 10 R f (1 is taken for all these interior \( that is,)9 1584 1 3456 3300 t (strictly between L and R \) mesh points.)7 1571 1 720 3420 t (The end points of the mesh typically have multiplicity)8 2166 1 970 3576 t 10 I f (k k)1 44 1 3161 3576 t 10 R f (since the function)2 710 1 3230 3576 t 10 I f (f f)1 28 1 3965 3576 t 10 R f (usually has)1 448 1 4019 3576 t 10 I f (f f)1 28 1 4493 3576 t 10 R f (\()4537 3576 w 10 I f (L L)1 56 1 4578 3576 t 10 R f (\))4642 3576 w 10 S f (\271)4724 3576 w 10 R f (0 and)1 220 1 4820 3576 t 10 I f (f f)1 28 1 720 3696 t 10 R f (\()764 3696 w 10 I f (R R)1 61 1 805 3696 t 10 R f (\))874 3696 w 10 S f (\271)956 3696 w 10 R f (0, and the elements of)4 891 1 1052 3696 t 10 I f (B B)1 61 1 1971 3696 t 7 S f (p)2043 3716 w 7 R f (,)2087 3716 w 7 I f (k k)1 31 1 2110 3716 t 10 R f (cannot be non-zero at)3 870 1 2177 3696 t 10 I f (L L)1 56 1 3075 3696 t 10 R f (and)3159 3696 w 10 I f (R R)1 61 1 3330 3696 t 10 R f (, unless)1 302 1 3391 3696 t 10 I f (m m)1 72 1 3720 3696 t 7 R f (1)3803 3716 w 10 S f (= =)1 55 1 3886 3696 t 10 I f (k k)1 44 1 3981 3696 t 10 S f (= =)1 55 1 4065 3696 t 10 I f (m m)1 72 1 4160 3696 t 7 I f (N N)1 47 1 4243 3716 t 10 R f (because of \(A1.2\))2 715 1 4325 3696 t ( relations \(A1.2\)-\(A1.5\) show that the only)6 1750( Indeed,)1 352(and \(A1.3\).)1 463 3 720 3816 t 10 I f (B B)1 61 1 3316 3816 t 7 I f (i i)1 20 1 3388 3836 t 10 R f (that are not zero at)4 766 1 3447 3816 t 10 I f (L L)1 56 1 4244 3816 t 10 R f (and)4331 3816 w 10 I f (R R)1 61 1 4506 3816 t 10 R f (are)4598 3816 w 10 I f (B B)1 61 1 4750 3816 t 7 R f (1)4822 3836 w 10 R f (and)4896 3816 w 10 I f (B B)1 61 1 720 3936 t 7 I f (N N)1 47 1 792 3956 t 7 S f (- -)1 39 1 855 3956 t 7 I f (k k)1 31 1 905 3956 t 10 R f (, and these values are simply)5 1148 1 944 3936 t 10 I f (B B)1 61 1 2117 3936 t 7 R f (1)2189 3956 w 10 R f (\()2240 3936 w 10 I f (L L)1 56 1 2281 3936 t 10 R f (\))2345 3936 w 10 S f (= =)1 55 1 2435 3936 t 10 R f (1 and)1 219 1 2539 3936 t 10 I f (B B)1 61 1 2783 3936 t 7 I f (N N)1 47 1 2855 3956 t 7 S f (- -)1 39 1 2918 3956 t 7 I f (k k)1 31 1 2968 3956 t 10 R f (\()3015 3936 w 10 I f (R R)1 61 1 3056 3936 t 10 R f (\))3125 3936 w 10 S f (= =)1 55 1 3215 3936 t 10 R f (1.)3319 3936 w (If the function)2 581 1 970 4092 t 10 I f (f f)1 28 1 1581 4092 t 10 R f (has a discontinuity in its)4 993 1 1639 4092 t 10 I f (j j)1 28 1 2662 4092 t 7 I f ( h)1 0(t th)1 55 2 2701 4052 t 10 R f ( its \()2 190(derivative, but not)2 740 2 2794 4092 t 10 I f (j j)1 28 1 3748 4092 t 10 S f (- -)1 55 1 3792 4092 t 10 R f (1 \))1 91 1 3863 4092 t 7 I f ( h)1 0(t th)1 55 2 3965 4052 t 10 R f (, at)1 128 1 4028 4092 t 10 I f (x x)1 44 1 4187 4092 t 7 I f (i i)1 20 1 4242 4112 t 10 R f (, then)1 228 1 4270 4092 t 10 I f (m m)1 72 1 4529 4092 t 7 I f (i i)1 20 1 4612 4112 t 10 S f (= =)1 55 1 4680 4092 t 10 I f (k k)1 44 1 4775 4092 t 10 S f (- -)1 55 1 4843 4092 t 10 I f (j j)1 28 1 4914 4092 t 10 R f (is)4973 4092 w (chosen because this allows the elements of)6 1744 1 720 4212 t 10 I f (B B)1 61 1 2495 4212 t 7 S f (p)2567 4232 w 7 R f (,)2611 4232 w 7 I f (k k)1 31 1 2634 4232 t 10 R f ( a smaller multiplicity were)4 1123( If)1 122(to have the same behavior.)4 1091 3 2704 4212 t (chosen, the)1 457 1 720 4332 t 10 I f (j j)1 28 1 1210 4332 t 7 I f ( h)1 0(t th)1 55 2 1249 4292 t 10 R f (derivative of all the elements of)5 1307 1 1345 4332 t 10 I f (B B)1 61 1 2685 4332 t 7 S f (p)2757 4352 w 7 R f (,)2801 4352 w 7 I f (k k)1 31 1 2824 4352 t 10 R f (would be continuous at)3 954 1 2896 4332 t 10 I f (x x)1 44 1 3883 4332 t 7 I f (i i)1 20 1 3938 4352 t 10 R f (, and the best fit to)5 784 1 3966 4332 t 10 I f (f f)1 28 1 4784 4332 t 10 R f (from)4846 4332 w 10 I f (B B)1 61 1 720 4452 t 7 S f (p)792 4472 w 7 R f (,)852 4472 w 7 I f (k k)1 31 1 891 4472 t 10 R f (would not be very good at)5 1046 1 955 4452 t 10 I f (x x)1 44 1 2026 4452 t 7 I f (i i)1 20 1 2081 4472 t 10 R f (.)2109 4452 w (Another important property of B-splines is their numerical stability or)9 2822 1 970 4608 t 10 I f ( .)1 0( on n.)2 75( it ti io)3 106( on nd di)3 128(c co)1 94 5 3820 4608 t 10 R f (Since any B-spline)2 763 1 4277 4608 t 10 I f (f f)1 28 1 720 4778 t 10 R f (is of the form)3 571 1 783 4778 t 10 I f (f f)1 28 1 1389 4778 t 10 S f (= =)1 55 1 1473 4778 t 7 I f (i i)1 20 1 1579 4878 t 7 S f (= =)1 39 1 1615 4878 t 7 R f (1)1665 4878 w 15 S f (S)1595 4808 w 7 I f (N N)1 47 1 1567 4678 t 7 S f (- -)1 39 1 1630 4678 t 7 I f (k k)1 31 1 1680 4678 t 10 I f (a a)1 50 1 1727 4778 t 7 I f (i i)1 20 1 1788 4798 t 10 I f (B B)1 61 1 1848 4778 t 7 I f (i i)1 20 1 1920 4798 t 10 R f (and each)1 361 1 1983 4778 t 10 I f (B B)1 61 1 2379 4778 t 7 I f (i i)1 20 1 2451 4798 t 10 R f (obeys 0)1 318 1 2514 4778 t 10 S f (\243)2873 4778 w 10 I f (B B)1 61 1 2969 4778 t 7 I f (i i)1 20 1 3041 4798 t 10 S f (\243 \243)1 55 1 3118 4778 t 10 R f (1, we see that if)4 669 1 3189 4778 t 10 S f ( \357)1 0( \357)1 24(\357 \357)1 49 3 3893 4795 t 10 I f (f f)1 28 1 3973 4778 t 10 S f ( \357)1 0( \357)1 24(\357 \357)1 49 3 4009 4795 t 10 R f (is small compared with)3 957 1 4083 4778 t 10 S f ( \357)1 0( \357)1 24(\357 \357)1 49 3 720 4995 t 10 B f (a)792 4978 w 10 S f ( \357)1 0( \357)1 24(\357 \357)1 49 3 842 4995 t 10 R f (, then many significant digits are lost when computing)8 2382 1 882 4978 t 10 I f (f f)1 28 1 3315 4978 t 10 R f (from)3394 4978 w 10 I f (a a)1 50 1 3639 4978 t 7 R f (1)3700 4998 w 10 R f (,)3775 4978 w (. . .)2 125 1 3857 4953 t (,)4039 4978 w 10 I f (a a)1 50 1 4096 4978 t 7 I f (N N)1 47 1 4157 4998 t 7 S f (- -)1 39 1 4220 4998 t 7 I f (k k)1 31 1 4270 4998 t 10 R f (in floating-point)1 680 1 4360 4978 t (arithmetic. Specifically,)1 982 1 720 5098 t 10 I f (d d)1 50 1 1220 5328 t 10 S f (\243)1311 5328 w 10 I f ( og g)2 50(L Lo)1 106 2 1407 5328 t 7 R f (10)1574 5348 w 10 R f (\()1660 5328 w 10 S f ( \357)1 0( \357)1 24(\357 \357)1 49 3 1693 5345 t 10 B f (a)1765 5328 w 10 S f ( \357)1 0( \357)1 24(\357 \357)1 49 3 1815 5345 t 10 I f (/ /)1 28 1 1911 5328 t 10 S f ( \357)1 0( \357)1 24(\357 \357)1 49 3 1963 5345 t 7 I f (i i)1 20 1 2046 5428 t 7 S f (= =)1 39 1 2082 5428 t 7 R f (1)2132 5428 w 15 S f (S)2062 5358 w 7 I f (N N)1 47 1 2034 5228 t 7 S f (- -)1 39 1 2097 5228 t 7 I f (k k)1 31 1 2147 5228 t 10 I f (a a)1 50 1 2186 5328 t 7 I f (i i)1 20 1 2247 5348 t 10 I f (B B)1 61 1 2307 5328 t 7 I f (i i)1 20 1 2379 5348 t 10 S f ( \357)1 0( \357)1 24(\357 \357)1 49 3 2407 5345 t 10 R f (\) \(A1.7\))1 2561 1 2479 5328 t (decimal digits are lost, due to cancellation, in evaluating)8 2253 1 720 5588 t 10 I f (f f)1 28 1 2998 5588 t 10 R f ( [2] de Boor shows that)5 935(. In)1 158 2 3026 5588 t 10 S f ( \357)1 0( \357)1 24(\357 \357)1 49 3 1220 5835 t 7 I f (i i)1 20 1 1303 5918 t 7 S f (= =)1 39 1 1339 5918 t 7 R f (1)1389 5918 w 15 S f (S)1319 5848 w 7 I f (N N)1 47 1 1291 5718 t 7 S f (- -)1 39 1 1354 5718 t 7 I f (k k)1 31 1 1404 5718 t 10 I f (a a)1 50 1 1443 5818 t 7 I f (i i)1 20 1 1504 5838 t 10 I f (B B)1 61 1 1564 5818 t 7 I f (i i)1 20 1 1636 5838 t 10 S f ( \357)1 0( \357)1 24(\357 \357)1 49 3 1664 5835 t (\263)1769 5818 w 10 I f (C C)1 67 1 1865 5818 t 7 I f (k k)1 31 1 1943 5838 t 10 S f ( \357)1 0( \357)1 24(\357 \357)1 49 3 1982 5835 t 10 B f (a)2054 5818 w 10 S f ( \357)1 0( \357)1 24(\357 \357)1 49 3 2104 5835 t 10 R f (\(A1.8\))4777 5818 w (where)720 6078 w 10 I f (C C)1 67 1 988 6078 t 7 I f (k k)1 31 1 1066 6098 t 10 R f (is a constant depending)3 935 1 1130 6078 t 10 I f ( y)1 0( ly)1 44(o on nl)2 128 3 2090 6078 t 10 R f (upon)2287 6078 w 10 I f (k k)1 44 1 2512 6078 t 10 R f (, and therefore)2 578 1 2556 6078 t 10 I f (d d)1 50 1 1220 6258 t 10 S f (\243)1302 6258 w 10 I f ( og g)2 50(L Lo)1 106 2 1389 6258 t 7 R f (10)1556 6278 w 10 R f (\()1642 6258 w 10 I f (C C)1 67 1 1683 6258 t 7 I f (k k)1 31 1 1755 6277 t 7 S f (- -)1 39 1 1755 6218 t 7 R f (1)1805 6218 w 10 R f (\).)1856 6258 w (In particular, he shows for a uniform mesh, one where all the mesh intervals have the same length, that)18 4106 1 720 6438 t 10 I f (C C)1 67 1 1220 6618 t 7 I f (k k)1 31 1 1298 6638 t 10 S f (~)1378 6598 w (~)1378 6623 w 10 R f (10)1474 6618 w 7 S f (- -)1 39 1 1585 6578 t 7 I f ( /)1 0( /)1 25(k k)1 31 3 1635 6578 t 7 R f (5)1696 6578 w 10 I f (. .)1 25 1 1771 6618 t 10 R f (\(A1.9\))4777 6618 w (Thus, when evaluating a B-spline defined on a uniform, or nearly uniform, mesh, we would expect to lose)17 4320 1 720 6798 t (no more than about)3 810 1 720 6918 t 10 I f ( /)1 0( /)1 36(k k)1 44 3 1567 6918 t 10 R f ( is a very satisfactory result since it shows that, at least for)12 2459( This)1 239( digits.)1 284(5 decimal)1 403 4 1655 6918 t (uniform meshes, the conditioning of the B-spline basis is independent of the size of the mesh.)15 3744 1 720 7038 t (The interesting spline facts are, in summary,)6 1775 1 970 7194 t cleartomark showpage saveobj restore end %%EndPage: 2 35 %%Page: 3 36 DpostDict begin /saveobj save def mark 36 pagesetup 10 R f (\261 1-3 \261)2 283 1 2738 480 t 10 I f (B B)1 61 1 2736 840 t 7 I f (i i)1 20 1 2808 860 t 10 S f (\263)2877 840 w 10 R f (0)2973 840 w 10 I f (B B)1 61 1 2394 1020 t 7 R f (1)2466 1040 w 10 S f (+ +)1 55 1 2549 1020 t 10 R f (. . .)2 125 1 2669 995 t 10 S f (+ +)1 55 1 2859 1020 t 10 I f (B B)1 61 1 2954 1020 t 7 I f (N N)1 47 1 3026 1040 t 7 S f (- -)1 39 1 3089 1040 t 7 I f (k k)1 31 1 3139 1040 t 10 S f (\272)3219 1020 w 10 R f (1)3315 1020 w 10 I f (B B)1 61 1 2262 1200 t 7 R f (1)2334 1220 w 10 R f (\()2409 1200 w 10 I f (x x)1 44 1 2474 1200 t 7 R f (1)2529 1220 w 10 R f (\))2604 1200 w 10 S f (= =)1 55 1 2694 1200 t 10 R f (1)2798 1200 w 10 S f (= =)1 55 1 2897 1200 t 10 I f (B B)1 61 1 3001 1200 t 7 I f (N N)1 47 1 3073 1220 t 7 S f (- -)1 39 1 3136 1220 t 7 I f (k k)1 31 1 3186 1220 t 10 R f (\()3257 1200 w 10 I f (x x)1 44 1 3322 1200 t 7 I f (N N)1 47 1 3377 1220 t 10 R f (\))3464 1200 w 7 I f (x x)1 31 1 2398 1575 t 4 R f (1)2435 1589 w 14 S f (\362)2394 1485 w 7 I f (x x)1 31 1 2395 1330 t 4 I f (N N)1 27 1 2432 1344 t 10 I f (B B)1 61 1 2505 1475 t 7 I f (i i)1 20 1 2577 1495 t 10 I f ( x)1 0(d dx)1 94 2 2637 1475 t 10 S f (= =)1 55 1 2780 1475 t 10 I f (k k)1 44 1 3086 1545 t (x x)1 44 1 2909 1395 t 7 I f (i i)1 20 1 2964 1415 t 7 S f (+ +)1 39 1 3000 1415 t 7 I f (k k)1 31 1 3050 1415 t 10 S f (- -)1 55 1 3129 1395 t 10 I f (x x)1 44 1 3224 1395 t 7 I f (i i)1 20 1 3279 1415 t 10 S1 f (_ ________)1 428 1 2894 1445 t 10 I f (. .)1 25 1 3340 1475 t 10 R f (Also, if we let)3 566 1 720 1785 t 10 I f (i i)1 28 1 1311 1785 t 10 R f (be such that)2 477 1 1364 1785 t 10 I f (x x)1 44 1 1866 1785 t 10 R f (is in [)2 228 1 1935 1785 t 10 I f (x x)1 44 1 2171 1785 t 7 I f (i i)1 20 1 2226 1805 t 10 R f (,)2262 1785 w 10 I f (x x)1 44 1 2319 1785 t 7 I f (i i)1 20 1 2374 1805 t 7 S f (+ +)1 39 1 2410 1805 t 7 R f (1)2460 1805 w 10 R f (\), or)1 166 1 2511 1785 t 10 I f (i i)1 28 1 2702 1785 t 10 S f (= =)1 55 1 2779 1785 t 10 I f (N N)1 67 1 2883 1785 t 10 S f (- -)1 55 1 2974 1785 t 10 I f (k k)1 44 1 3045 1785 t 10 R f (if)3114 1785 w 10 I f (x x)1 44 1 3200 1785 t 10 S f (= =)1 55 1 3293 1785 t 10 I f (x x)1 44 1 3397 1785 t 7 I f (N N)1 47 1 3452 1805 t 10 R f (, then)1 222 1 3507 1785 t 7 I f (j j)1 20 1 2268 2115 t 7 S f (= =)1 39 1 2299 2115 t 7 R f (1)2349 2115 w 15 S f (S)2281 2045 w 7 I f (N N)1 47 1 2253 1915 t 7 S f (- -)1 39 1 2316 1915 t 7 I f (k k)1 31 1 2366 1915 t 10 I f (a a)1 50 1 2405 2015 t 7 I f (j j)1 20 1 2466 2035 t 10 I f (B B)1 61 1 2502 2015 t 7 I f (j j)1 20 1 2574 2035 t 10 R f (\()2610 2015 w 10 I f (x x)1 44 1 2651 2015 t 10 R f (\))2703 2015 w 10 S f (\272)2785 2015 w 7 I f (j j)1 20 1 2881 2115 t 7 S f (= =)1 39 1 2912 2115 t 7 I f (i i)1 20 1 2962 2115 t 7 S f (+ +)1 39 1 2998 2115 t 7 R f (1)3048 2115 w 7 S f (- -)1 39 1 3094 2115 t 7 I f (k k)1 31 1 3144 2115 t 15 S f (S)2983 2045 w 7 I f (i i)1 20 1 3017 1915 t 10 I f (a a)1 50 1 3174 2015 t 7 I f (j j)1 20 1 3235 2035 t 10 I f (B B)1 61 1 3271 2015 t 7 I f (j j)1 20 1 3343 2035 t 10 R f (\()3379 2015 w 10 I f (x x)1 44 1 3420 2015 t 10 R f (\))3472 2015 w (If we let)2 332 1 720 2275 t 10 I f (f f)1 28 1 2552 2505 t 10 S f (\272)2637 2505 w 7 I f (j j)1 20 1 2747 2605 t 7 S f (= =)1 39 1 2778 2605 t 7 R f (1)2828 2605 w 15 S f (S)2760 2535 w 7 I f (N N)1 47 1 2732 2405 t 7 S f (- -)1 39 1 2795 2405 t 7 I f (k k)1 31 1 2845 2405 t 10 I f (a a)1 50 1 2876 2505 t 7 I f (j j)1 20 1 2937 2525 t 10 I f (B B)1 61 1 2973 2505 t 7 I f (j j)1 20 1 3045 2525 t 10 R f (\()3081 2505 w 10 I f (x x)1 44 1 3122 2505 t 10 R f (\))3174 2505 w (then we have)2 526 1 720 2765 t 10 I f (f f)1 28 1 2178 2985 t 10 S f (\242 \242)1 25 1 2230 2985 t 10 R f (\()2263 2985 w 10 I f (L L)1 56 1 2304 2985 t 10 R f (\))2368 2985 w 10 S f (= =)1 55 1 2458 2985 t 10 I f (x x)1 44 1 2587 3055 t 7 I f (k k)1 31 1 2642 3075 t 7 S f (+ +)1 39 1 2689 3075 t 7 R f (1)2739 3075 w 10 S f (- -)1 55 1 2831 3055 t 10 I f (x x)1 44 1 2935 3055 t 7 R f (1)2990 3075 w 10 I f (k k)1 44 1 2716 2925 t 10 S f (- -)1 55 1 2784 2925 t 10 R f (1)2855 2925 w 10 S1 f (_ _________)1 476 1 2573 2955 t 10 R f (\()3091 2985 w 10 I f (a a)1 50 1 3156 2985 t 7 R f (2)3217 3005 w 10 S f (- -)1 55 1 3309 2985 t 10 I f (a a)1 50 1 3413 2985 t 7 R f (1)3474 3005 w 10 R f (\))3549 2985 w (and)720 3235 w 10 I f (f f)1 28 1 1966 3455 t 10 S f (\242 \242)1 25 1 2018 3455 t 10 R f (\()2051 3455 w 10 I f (R R)1 61 1 2092 3455 t 10 R f (\))2161 3455 w 10 S f (= =)1 55 1 2251 3455 t 10 I f (x x)1 44 1 2380 3525 t 7 I f ( x)1 0(n nx)1 66 2 2435 3545 t 10 S f (- -)1 55 1 2558 3525 t 10 I f (x x)1 44 1 2662 3525 t 7 I f ( x)1 0(n nx)1 66 2 2717 3545 t 7 S f (- -)1 39 1 2799 3545 t 7 I f (k k)1 31 1 2849 3545 t 10 I f (k k)1 44 1 2540 3395 t 10 S f (- -)1 55 1 2608 3395 t 10 R f (1)2679 3395 w 10 S1 f (_ __________)1 538 1 2366 3425 t 10 R f (\()2946 3455 w 10 I f (a a)1 50 1 3011 3455 t 7 I f ( x)1 0(n nx)1 66 2 3072 3475 t 7 S f (- -)1 39 1 3154 3475 t 7 I f (k k)1 31 1 3204 3475 t 10 S f (- -)1 55 1 3292 3455 t 10 I f (a a)1 50 1 3396 3455 t 7 I f ( x)1 0(n nx)1 66 2 3457 3475 t 7 S f (- -)1 39 1 3539 3475 t 7 I f (k k)1 31 1 3589 3475 t 7 S f (- -)1 39 1 3636 3475 t 7 R f (1)3686 3475 w 10 R f (\))3761 3455 w cleartomark showpage saveobj restore end %%EndPage: 3 36 %%Page: 1 37 DpostDict begin /saveobj save def mark 37 pagesetup 10 B f (Appendix 2)1 493 1 2633 840 t (Extrapolation.)2570 1200 w 10 R f ( in [5] and [16,17] is the numerical solution of the canonical form)12 2777(The problem treated)2 833 2 970 1596 t 10 B f (ode)4618 1596 w 10 R f (initial)4806 1596 w (value problem)1 574 1 720 1716 t 10 I f ( t)1 0(d dt)1 78 2 1260 2006 t (d d)1 50 1 1245 1876 t 10 B f (x)1303 1876 w 10 S1 f (_ __)1 138 1 1230 1906 t 10 S f (= =)1 55 1 1427 1936 t 10 B f (f)1531 1936 w 10 R f (\()1572 1936 w 10 I f (t t)1 28 1 1613 1936 t 10 R f (,)1649 1936 w 10 B f (x)1715 1936 w 10 R f (\))1773 1936 w 10 I f (a a)1 50 1 2020 1936 t 10 S f (\243)2111 1936 w 10 I f (t t)1 28 1 2207 1936 t 10 R f (\(A2.1\))4777 2166 w 10 B f (x)1220 2346 w 10 R f (\()1278 2346 w 10 I f (a a)1 50 1 1319 2346 t 10 R f (\))1377 2346 w 10 S f (= =)1 55 1 1467 2346 t 10 B f (x)1571 2346 w 7 I f (a a)1 35 1 1632 2366 t 10 R f (where)720 2526 w 10 B f (f)993 2526 w 10 R f (\()1034 2526 w 10 I f (t t)1 28 1 1075 2526 t 10 R f (,)1111 2526 w 10 B f (x)1177 2526 w 10 R f (\) is some smooth vector-valued function of)6 1756 1 1235 2526 t 10 I f (t t)1 28 1 3022 2526 t 10 R f (and)3081 2526 w 10 B f (x)3256 2526 w 10 R f ( brief outline of the ideas developed in)7 1581(. A)1 153 2 3306 2526 t (those papers follows.)2 846 1 720 2646 t ( solving \(A2.1\), such as Gragg's modified mid-point)7 2157(There are many basic differencing schemes for)6 1913 2 970 2802 t ( of these methods have the)5 1152( Most)1 274( and Crank-Nicholson [13,22].)3 1277(rule [16,17], backwards-Euler methods)3 1617 4 720 2922 t (property [30] that if they take N time-steps)7 1711 1 720 3042 t 10 I f (h h)1 50 1 2456 3042 t 10 S f (= =)1 55 1 2555 3042 t 10 R f (\()2659 3042 w 10 I f (t t)1 28 1 2700 3042 t 7 R f (1)2739 3062 w 10 S f (- -)1 55 1 2822 3042 t 10 I f (t t)1 28 1 2917 3042 t 7 R f (0)2956 3062 w 10 R f (\))3007 3042 w 10 I f ( N)1 0( N)1 75(/ /)1 28 3 3048 3042 t 10 R f (to go from)2 422 1 3176 3042 t 10 I f (t t)1 28 1 3623 3042 t 7 R f (0)3662 3062 w 10 R f (to)3730 3042 w 10 I f (t t)1 28 1 3833 3042 t 7 R f (1)3872 3062 w 10 R f ( an approxima-)2 606(and result in)2 494 2 3940 3042 t (tion to)1 259 1 720 3162 t 10 B f (x)1004 3162 w 10 R f (\()1062 3162 w 10 I f (t t)1 28 1 1103 3162 t 7 R f (1)1142 3182 w 10 R f (\) which we shall denote by)5 1073 1 1193 3162 t 10 B f (T)2291 3162 w 10 R f (\()2366 3162 w 10 I f (h h)1 50 1 2407 3162 t 10 R f (\), then)1 255 1 2465 3162 t 10 B f (T)2330 3392 w 10 R f (\()2405 3392 w 10 I f (h h)1 50 1 2446 3392 t 10 R f (\))2504 3392 w 10 S f (= =)1 55 1 2594 3392 t 10 B f (x)2698 3392 w 10 R f (\()2756 3392 w 10 I f (t t)1 28 1 2797 3392 t 7 R f (1)2836 3412 w 10 R f (\))2887 3392 w 10 S f (+ +)1 55 1 2968 3392 t 7 I f (j j)1 20 1 3065 3492 t 7 S f (= =)1 39 1 3096 3492 t 7 R f (1)3146 3492 w 15 S f (S)3079 3422 w 7 S f (\245)3098 3292 w 10 B f (T)3192 3392 w 7 I f (j j)1 20 1 3270 3412 t 10 I f (h h)1 50 1 3306 3392 t 7 I f (j j)1 20 1 3367 3352 t 7 S f (g)3392 3352 w 10 R f (\(A2.2\))4777 3392 w (where)720 3652 w 10 S f (g)989 3652 w 10 R f ( on the basic difference scheme used, and the)8 1819(is a positive constant depending)4 1281 2 1056 3652 t 10 B f (T)4183 3652 w 7 I f (j j)1 20 1 4261 3672 t 10 R f (are unknown con-)2 724 1 4316 3652 t (stant vectors independent of)3 1162 1 720 3772 t 10 I f (h h)1 50 1 1920 3772 t 10 R f ( rule or Crank-Nicholson)3 1037( Gragg's modified mid-point)3 1191(. For)1 227 3 1970 3772 t 10 S f ( =)1 0(g =)1 145 2 4462 3772 t 10 R f (2 and for)2 384 1 4656 3772 t (backwards-Euler methods)1 1039 1 720 3892 t 10 S f ( =)1 0(g =)1 145 2 1784 3892 t 10 R f (1.)1978 3892 w (Let a sequence of)3 700 1 970 4048 t 10 I f (h h)1 50 1 1695 4048 t 10 R f ('s be defined by)3 640 1 1745 4048 t 10 I f (h h)1 50 1 2148 4228 t 7 I f (i i)1 20 1 2209 4248 t 10 S f (= =)1 55 1 2286 4228 t 10 I f (h h)1 50 1 2390 4228 t 7 R f (0)2451 4248 w 10 I f ( N)1 0( N)1 75(/ /)1 28 3 2502 4228 t 7 I f (i i)1 20 1 2616 4248 t 10 R f (,)2652 4228 w 10 I f (i i)1 28 1 2883 4228 t 10 S f (= =)1 55 1 2960 4228 t 10 R f ( ,)1 33( 3)1 91( ,)1 33( 2)1 91(1 ,)1 83 5 3064 4228 t (. . .)2 125 1 3428 4203 t 10 I f (. .)1 25 1 3586 4228 t 10 R f (\(A2.3\))4777 4228 w (where)720 4408 w 10 I f (h h)1 50 1 1001 4408 t 7 R f (0)1062 4428 w 10 S f (= =)1 55 1 1154 4408 t 10 I f (t t)1 28 1 1258 4408 t 7 R f (1)1297 4428 w 10 S f (- -)1 55 1 1356 4408 t 10 I f (t t)1 28 1 1427 4408 t 7 R f (0)1466 4428 w 10 R f (and the)1 304 1 1547 4408 t 10 I f (N N)1 67 1 1889 4408 t 7 I f (i i)1 20 1 1967 4428 t 10 R f ( and)1 183( Bulirsch)1 403( integers.)1 380(form a monotone increasing sequence of positive)6 2041 4 2033 4408 t ( [4] that given an operator)5 1044(Stoer showed in)2 648 2 720 4528 t 10 B f (T)2438 4528 w 10 R f (\()2513 4528 w 10 I f (h h)1 50 1 2554 4528 t 10 R f (\) satisfying \(A2.2\), and such a sequence)6 1602 1 2612 4528 t 10 I f (h h)1 50 1 4240 4528 t 7 I f (i i)1 20 1 4301 4548 t 10 R f (, then the value at)4 711 1 4329 4528 t 10 I f (h h)1 50 1 720 4648 t 10 S f (= =)1 55 1 794 4648 t 10 R f (0 of the polynomial of degree)5 1234 1 865 4648 t 10 I f (m m)1 72 1 2135 4648 t 10 R f (that interpolates)1 652 1 2243 4648 t 10 B f (T)2931 4648 w 10 R f (\()3006 4648 w 10 I f (h h)1 50 1 3047 4648 t 7 I f (i i)1 20 1 3108 4668 t 10 R f (\) for)1 185 1 3144 4648 t 10 I f (i i)1 28 1 3365 4648 t 10 S f (= =)1 55 1 3417 4648 t 10 R f (0 ,)1 83 1 3488 4648 t (. . .)2 125 1 3604 4623 t (,)3762 4648 w 10 I f (m m)1 72 1 3795 4648 t 10 R f (, is given by)3 522 1 3867 4648 t 10 B f (T)4425 4648 w 7 I f (m m)1 50 1 4497 4667 t 7 R f (0)4497 4608 w 10 R f (determined)4591 4648 w (from the recursion)2 737 1 720 4768 t 10 B f (T)1220 4948 w 7 R f (0)1292 4967 w 7 I f (i i)1 20 1 1292 4908 t 10 S f (= =)1 55 1 1384 4948 t 10 B f (T)1488 4948 w 10 R f (\()1563 4948 w 10 I f (h h)1 50 1 1604 4948 t 7 I f (i i)1 20 1 1665 4968 t 10 R f ( 0)1 190(\) for)1 363 2 1701 4948 t 10 S f (\243)2295 4948 w 10 I f (i i)1 28 1 2391 4948 t 10 S f (\243)2460 4948 w 10 I f (m m)1 72 1 2556 4948 t 10 R f (\(A2.4\))4777 5128 w 10 B f (T)1220 5393 w 7 I f (k k)1 31 1 1292 5412 t (i i)1 20 1 1292 5353 t 10 S f (= =)1 55 1 1380 5393 t 10 B f (T)1484 5393 w 7 I f (k k)1 31 1 1556 5412 t 7 S f (- -)1 39 1 1603 5412 t 7 R f (1)1653 5412 w 7 I f (i i)1 20 1 1556 5353 t 7 S f (+ +)1 39 1 1592 5353 t 7 R f (1)1642 5353 w 10 S f (+ +)1 55 1 1736 5393 t 10 R f (\()1831 5393 w 10 B f (T)1872 5393 w 7 I f (k k)1 31 1 1944 5412 t 7 S f (- -)1 39 1 1991 5412 t 7 R f (1)2041 5412 w 7 I f (i i)1 20 1 1944 5353 t 7 S f (+ +)1 39 1 1980 5353 t 7 R f (1)2030 5353 w 10 S f (- -)1 55 1 2124 5393 t 10 B f (T)2219 5393 w 7 I f (k k)1 31 1 2291 5412 t 7 S f (- -)1 39 1 2338 5412 t 7 R f (1)2388 5412 w 7 I f (i i)1 20 1 2291 5353 t 10 R f (\))2439 5393 w 10 I f (/ /)1 28 1 2480 5393 t 10 S f (\354)2524 5306 w (\355)2524 5406 w (\356)2524 5506 w 10 R f (\()2573 5393 w 10 I f (h h)1 50 1 2614 5393 t 7 I f (i i)1 20 1 2675 5413 t 10 I f ( h)1 0( h)1 58(/ /)1 28 3 2711 5393 t 7 I f (i i)1 20 1 2808 5413 t 7 S f (+ +)1 39 1 2844 5413 t 7 I f (k k)1 31 1 2894 5413 t 10 R f (\))2941 5393 w 7 S f (g)2985 5353 w 10 S f (- -)1 55 1 3062 5393 t 10 R f (1)3157 5393 w 10 S f (\374)3207 5306 w (\375)3207 5406 w (\376)3207 5506 w 10 R f (for 0)1 191 1 720 5678 t 10 S f (\243)919 5678 w 10 I f (i i)1 28 1 982 5678 t 10 S f (\243)1018 5678 w 10 I f (m m)1 72 1 1081 5678 t 10 S f (- -)1 55 1 1177 5678 t 10 I f (k k)1 44 1 1248 5678 t 10 R f (and 1)1 219 1 1317 5678 t 10 S f (\243)1544 5678 w 10 I f (k k)1 44 1 1607 5678 t 10 S f (\243 \243)1 55 1 1675 5678 t 10 I f (m m)1 72 1 1746 5678 t 10 R f ( the)1 147(. If)1 141 2 1818 5678 t 10 I f (T T)1 56 1 2131 5678 t 7 I f (k k)1 31 1 2192 5697 t (i i)1 20 1 2192 5638 t 10 R f (are organized into a)3 789 1 2256 5678 t 10 B f (tableau)3070 5678 w 10 R f (of the form)2 449 1 3412 5678 t 10 B f (T)1220 6668 w 10 R f (\()1295 6668 w 10 I f (h h)1 50 1 1336 6668 t 7 R f (5)1397 6688 w 10 R f (\))1448 6668 w 10 S f (= =)1 55 1 1538 6668 t 10 B f (T)1220 6508 w 10 R f (\()1295 6508 w 10 I f (h h)1 50 1 1336 6508 t 7 R f (4)1397 6528 w 10 R f (\))1448 6508 w 10 S f (= =)1 55 1 1538 6508 t 10 B f (T)1220 6348 w 10 R f (\()1295 6348 w 10 I f (h h)1 50 1 1336 6348 t 7 R f (3)1397 6368 w 10 R f (\))1448 6348 w 10 S f (= =)1 55 1 1538 6348 t 10 B f (T)1220 6188 w 10 R f (\()1295 6188 w 10 I f (h h)1 50 1 1336 6188 t 7 R f (2)1397 6208 w 10 R f (\))1448 6188 w 10 S f (= =)1 55 1 1538 6188 t 10 B f (T)1220 6028 w 10 R f (\()1295 6028 w 10 I f (h h)1 50 1 1336 6028 t 7 R f (1)1397 6048 w 10 R f (\))1448 6028 w 10 S f (= =)1 55 1 1538 6028 t 10 B f (T)1220 5868 w 10 R f (\()1295 5868 w 10 I f (h h)1 50 1 1336 5868 t 7 R f (0)1397 5888 w 10 R f (\))1448 5868 w 10 S f (= =)1 55 1 1538 5868 t 10 B f (T)1675 6698 w 7 R f (0)1747 6717 w (5)1747 6658 w 10 B f (T)1675 6528 w 7 R f (0)1747 6547 w (4)1747 6488 w 10 B f (T)1675 6358 w 7 R f (0)1747 6377 w (3)1747 6318 w 10 B f (T)1675 6188 w 7 R f (0)1747 6207 w (2)1747 6148 w 10 B f (T)1675 6018 w 7 R f (0)1747 6037 w (1)1747 5978 w 10 B f (T)1675 5848 w 7 R f (0)1747 5867 w (0)1747 5808 w 10 B f (T)1897 6638 w 7 R f (1)1969 6657 w (4)1969 6598 w 10 B f (T)1897 6468 w 7 R f (1)1969 6487 w (3)1969 6428 w 10 B f (T)1897 6298 w 7 R f (1)1969 6317 w (2)1969 6258 w 10 B f (T)1897 6128 w 7 R f (1)1969 6147 w (1)1969 6088 w 10 B f (T)1897 5958 w 7 R f (1)1969 5977 w (0)1969 5918 w 10 B f (T)2119 6528 w 7 R f (2)2191 6547 w (3)2191 6488 w 10 B f (T)2119 6358 w 7 R f (2)2191 6377 w (2)2191 6318 w 10 B f (T)2119 6188 w 7 R f (2)2191 6207 w (1)2191 6148 w 10 B f (T)2119 6018 w 7 R f (2)2191 6037 w (0)2191 5978 w 10 B f (T)2341 6468 w 7 R f (3)2413 6487 w (2)2413 6428 w 10 B f (T)2341 6298 w 7 R f (3)2413 6317 w (1)2413 6258 w 10 B f (T)2341 6128 w 7 R f (3)2413 6147 w (0)2413 6088 w 10 B f (T)2563 6358 w 7 R f (4)2635 6377 w (1)2635 6318 w 10 B f (T)2563 6188 w 7 R f (4)2635 6207 w (0)2635 6148 w 10 B f (T)2785 6298 w 7 R f (5)2857 6317 w (0)2857 6258 w 10 R f (then \(A2.4\) expresses each element of the)6 1688 1 720 6878 t 10 I f (k k)1 44 1 2436 6878 t 7 I f ( h)1 0(t th)1 55 2 2491 6838 t 10 R f (column \()1 361 1 2582 6878 t 10 I f (k k)1 44 1 2943 6878 t 10 S f (> >)1 55 1 3011 6878 t 10 R f (0\) in terms of its two neighbors in column)8 1715 1 3082 6878 t 10 I f (k k)1 44 1 4826 6878 t 10 S f (- -)1 55 1 4894 6878 t 10 R f (1.)4965 6878 w (A similar result is established for interpolation by rational functions [4].)10 2878 1 720 6998 t ( sufficiently small)2 734( For)1 195( error in each element of the tableau [6].)8 1651(It is also possible to estimate the)6 1330 4 970 7154 t 10 I f (h h)1 50 1 4911 7154 t 7 R f (0)4972 7174 w 10 R f (,)5015 7154 w ([6] shows that)2 566 1 720 7274 t cleartomark showpage saveobj restore end %%EndPage: 1 37 %%Page: 2 38 DpostDict begin /saveobj save def mark 38 pagesetup 10 R f (\261 2-2 \261)2 283 1 2738 480 t 10 S f (\357 \357)1 49 1 1220 942 t 10 B f (T)1268 925 w 7 I f (k k)1 31 1 1340 944 t (i i)1 20 1 1340 885 t 10 S f (- -)1 55 1 1419 925 t 10 B f (T)1514 925 w 10 S f (\357 \357)1 49 1 1581 942 t (~)1662 905 w (~)1662 930 w (\351)1766 838 w (\357)1766 938 w (\353)1766 1038 w 10 R f (1)1816 925 w 10 S f (+ +)1 55 1 1906 925 t 10 R f (\()2026 1005 w 10 I f (h h)1 50 1 2067 1005 t 7 I f (i i)1 20 1 2128 1025 t 10 I f ( h)1 0( h)1 58(/ /)1 28 3 2164 1005 t 7 I f (i i)1 20 1 2261 1025 t 7 S f (+ +)1 39 1 2297 1025 t 7 I f (k k)1 31 1 2347 1025 t 7 S f (+ +)1 39 1 2394 1025 t 7 R f (1)2444 1025 w 10 R f (\))2495 1005 w 7 S f (g)2539 965 w 10 S f (- -)1 55 1 2616 1005 t 10 R f (1)2711 1005 w (1)2369 865 w 10 S1 f (_ _______________)1 765 1 2012 895 t 10 S f (\371)2787 838 w (\357)2787 938 w (\373)2787 1038 w (\357 \357)1 49 1 2825 942 t 10 B f (T)2873 925 w 7 I f (k k)1 31 1 2945 944 t (i i)1 20 1 2945 885 t 7 S f (+ +)1 39 1 2981 885 t 7 R f (1)3031 885 w 10 S f (- -)1 55 1 3114 925 t 10 B f (T)3209 925 w 7 I f (k k)1 31 1 3281 944 t (i i)1 20 1 3281 885 t 10 S f (\357 \357)1 49 1 3320 942 t 10 R f (\(A2.5\))4777 925 w (and we can estimate the error)5 1171 1 720 1210 t 10 S f (e)1916 1210 w 7 I f (k k)1 31 1 1965 1229 t (i i)1 20 1 1965 1170 t 10 S f (= =)1 55 1 2053 1210 t (\357 \357)1 49 1 2149 1227 t 10 B f (T)2197 1210 w 7 I f (k k)1 31 1 2269 1229 t (i i)1 20 1 2269 1170 t 10 S f (- -)1 55 1 2348 1210 t 10 B f (T)2443 1210 w 10 S f (\357 \357)1 49 1 2510 1227 t 10 R f (in)2551 1210 w 10 B f (T)2654 1210 w 7 I f (k k)1 31 1 2726 1229 t (i i)1 20 1 2726 1170 t 10 R f ( also know from [6] that for sufficiently small)8 1831(. We)1 213 2 2765 1210 t 10 I f (h h)1 50 1 4834 1210 t 7 R f (0)4895 1230 w 10 R f (,)4938 1210 w 10 S f (e)1220 1390 w 7 I f (k k)1 31 1 1269 1409 t (i i)1 20 1 1269 1350 t 10 S f (= =)1 55 1 1357 1390 t 10 I f (h h)1 50 1 1461 1390 t 7 R f (0)1516 1409 w 7 S f (b)1516 1350 w 10 B f (d)1571 1390 w 7 I f (k k)1 31 1 1638 1410 t 10 R f (\()1685 1390 w 10 I f (h h)1 50 1 1726 1390 t 7 I f (i i)1 20 1 1787 1410 t 10 R f (. . .)2 125 1 1848 1365 t 10 I f (h h)1 50 1 2006 1390 t 7 I f (i i)1 20 1 2067 1410 t 7 S f (+ +)1 39 1 2103 1410 t 7 I f (k k)1 31 1 2153 1410 t 10 R f (\))2200 1390 w 7 S f (g)2244 1350 w 10 R f (\(A2.6\))4777 1390 w (where the)1 393 1 720 1570 t 10 B f (d)1166 1570 w 7 I f (k k)1 31 1 1233 1590 t 10 R f (are constant vectors,)2 823 1 1300 1570 t 10 S f (g)2151 1570 w 10 R f ( extrapolated and)2 695(is the order of the basic process being)7 1526 2 2220 1570 t 10 S f (b)4470 1570 w 10 R f (is a positive)2 486 1 4554 1570 t ( extrapolating Gragg's modified mid-point rule or Crank-Nicholson,)7 2856(constant. When)1 664 2 720 1690 t 10 S f ( =)1 0(g =)1 145 2 4283 1690 t 10 R f (2 and)1 237 1 4477 1690 t 10 S f ( =)1 0(b =)1 159 2 4757 1690 t 10 R f (1.)4965 1690 w (When extrapolating a Backwards-Euler time differencing process,)6 2658 1 720 1810 t 10 S f ( =)1 0(g =)1 145 2 3406 1810 t 10 R f (1)3600 1810 w 10 S f ( b)1 104(= =)1 55 2 3699 1810 t 10 R f ( both estimate)2 569( we can)2 310(. Thus,)1 303 3 3858 1810 t (the accuracy of each element in the tableau and tell how rapidly each column in the tableau is converging.)18 4232 1 720 1930 t (In \(A2.4\))1 374 1 970 2086 t 10 I f (m m)1 72 1 1372 2086 t 10 R f (is called the)2 483 1 1472 2086 t 10 B f (level)1983 2086 w 10 R f ( from \(A2.6\) we see that the)6 1146(of extrapolation, while)2 913 2 2205 2086 t 10 B f (order)4293 2086 w 10 R f (in column)1 407 1 4560 2086 t 10 I f (k k)1 44 1 4996 2086 t 10 R f (is \()1 131 1 720 2206 t 10 I f (k k)1 44 1 859 2206 t 10 S f (+ +)1 55 1 927 2206 t 10 R f (1 \))1 91 1 998 2206 t 10 S f (g)1105 2206 w 10 R f ( basic ordinary differential equation solver, a process of)8 2265( by extrapolating the results of a)6 1323(. Thus,)1 306 3 1146 2206 t ( value)1 245( The)1 209( be obtained.)2 521(arbitrarily high order can)3 1009 4 720 2326 t 10 I f (h h)1 50 1 2733 2326 t 7 R f (0)2794 2346 w 10 S f (= =)1 55 1 2886 2326 t 10 I f (t t)1 28 1 2990 2326 t 7 R f (1)3029 2346 w 10 S f (- -)1 55 1 3112 2326 t 10 I f (t t)1 28 1 3207 2326 t 7 R f (0)3246 2346 w 10 R f (is referred to as the)4 780 1 3318 2326 t 10 B f (time-step)4127 2326 w 10 R f (while the)1 373 1 4549 2326 t 10 I f (h h)1 50 1 4951 2326 t 7 I f (i i)1 20 1 5012 2346 t 10 R f (are called)1 406 1 720 2446 t 10 B f (sub-steps.)1172 2446 w 10 R f (Extrapolation approximates the)2 1296 1 1663 2446 t 10 B f (x)3005 2446 w 10 R f (\()3063 2446 w 10 I f (t t)1 28 1 3104 2446 t 7 R f (1)3143 2466 w 10 R f (\) values accurately, but does not accurately)6 1846 1 3194 2446 t (approximate)720 2566 w 10 B f (x)1244 2566 w 10 R f (\()1302 2566 w 10 I f (t t)1 28 1 1343 2566 t 10 S f (+ +)1 55 1 1395 2566 t 10 I f (n nh h)2 100 1 1466 2566 t 7 I f (i i)1 20 1 1577 2586 t 10 R f (\) for 0)2 249 1 1613 2566 t 10 S f (< <)1 55 1 1878 2566 t 10 I f (n n)1 50 1 1949 2566 t 10 S f (< <)1 55 1 2023 2566 t 10 I f (N N)1 67 1 2094 2566 t 10 R f (.)2161 2566 w cleartomark showpage saveobj restore end %%EndPage: 2 38 %%Page: 1 39 DpostDict begin /saveobj save def mark 39 pagesetup 10 B f (Appendix 3)1 493 1 2633 840 t (Wish-List)2668 1200 w 10 R f ( that could be made in)5 910(This section describes several improvements)4 1802 2 970 1596 t 10 CW f (TTGR)3712 1596 w 10 R f ( improvements range)2 853(. The)1 235 2 3952 1596 t (from better human engineering \(easier use\) to making the algorithm more efficient and extending it to solve)16 4320 1 720 1716 t (more general problems.)2 945 1 720 1836 t 10 B f (Periodic Boundary Conditions)2 1302 1 720 2076 t 10 R f (Since the)1 374 1 970 2232 t 10 B f (pde)1374 2232 w 10 R f ( defined on a rectangle, and mapping is sometimes used to work on more general)14 3320(s are)1 190 2 1530 2232 t ( will be done in the next version of)8 1417( This)1 232(domains, it would be good to allow periodic boundary conditions.)9 2671 3 720 2352 t (the package.)1 498 1 720 2472 t 10 B f (Speed)720 2712 w 10 R f (The current assembly phase \(making the Jacobian\) and solution processes have been sped up by fac-)15 4070 1 970 2868 t ( unpolished code will be ready for the next version.)9 2052( This)1 228(tors of 3 and more.)4 757 3 720 2988 t 10 B f (One Dimensional Pde and Ode Coupling)5 1734 1 720 3228 t 10 R f ( one-dimensional)1 698(In the spirit of the)4 740 2 970 3384 t 10 B f (pde)2440 3384 w 10 R f (solver)2628 3384 w 10 CW f (POST)2904 3384 w 10 R f ([26], it would be good if)5 1012 1 3176 3384 t 10 CW f (TTGR)4220 3384 w 10 R f (allowed solu-)1 548 1 4492 3384 t (tion of problems involving one and two dimensional)7 2160 1 720 3504 t 10 B f (pde)2914 3504 w 10 R f (s coupled to)2 501 1 3070 3504 t 10 B f (ode)3604 3504 w 10 R f ( is important for several)4 986(s. This)1 300 2 3754 3504 t ( The)1 212( some real world problems, like those of [33] and [34].)10 2255( first is that it allows solution of)7 1318(reasons. The)1 535 4 720 3624 t ( that can-)2 375(second reason is just plain mathematical completeness: without such coupling, there are problems)12 3945 2 720 3744 t (not even be posed, let alone solved.)6 1420 1 720 3864 t 10 B f (Faster Solution Times)2 939 1 720 4104 t 10 R f (The block sparse-matrix package of Kent Smith offers a factor of)10 2724 1 970 4260 t 10 I f (n n)1 50 1 3731 4260 t 7 I f (u u)1 35 1 3792 4280 t 10 R f ( run-time on)2 520(improvement in)1 648 2 3872 4260 t ( will probably be an option in the next version.)9 1871( This)1 228(vector machines like the Cray.)4 1217 3 720 4380 t ( robustness and applica-)3 977( The)1 210( methods.)1 394(Multi-grid schemes offer optimal run-time and space solution)7 2489 4 970 4536 t ( worth exploring, but will take some time: such codes are complex, but prob-)13 3103(bility of these schemes is well)5 1217 2 720 4656 t (ably worth the work.)3 832 1 720 4776 t cleartomark showpage saveobj restore end %%EndPage: 1 39 %%Page: 1 40 DpostDict begin /saveobj save def mark 40 pagesetup 10 B f (Appendix 4)1 493 1 2633 840 t (Examples - Programs)2 921 1 2419 1200 t 10 R f ( use of)2 276(The program examples given below are intended to both illustrate the)10 2819 2 970 1596 t 10 CW f (TTGR)4095 1596 w 10 R f (and provide pro-)2 675 1 4365 1596 t ( contemplating using)2 843( Anyone)1 368(totypes for a prospective user.)4 1212 3 720 1716 t 10 CW f (TTGR)3170 1716 w 10 R f (would be well advised to pick an exam-)7 1603 1 3437 1716 t (ple program that invokes those capabilities of)6 1874 1 720 1836 t 10 CW f (TTGR)2628 1836 w 10 R f ( type it in \(or)4 562(the intended problem will require, and)5 1576 2 2902 1836 t ( and confirming the cor-)4 998( running the example)3 870( After)1 268(obtain a copy of the example code from the authors\).)9 2184 4 720 1956 t (rectness of the program, the)4 1135 1 720 2076 t 10 CW f (AF)1886 2076 w 10 R f (and)2037 2076 w 10 CW f (BC)2213 2076 w 10 R f (subroutines specifying the)2 1063 1 2365 2076 t 10 B f (pde)3460 2076 w 10 R f (-)3616 2076 w 10 B f (bc)3649 2076 w 10 R f (may simply be altered to solve)5 1259 1 3781 2076 t ( much more likely that the user will easily produce a correct)11 2442( progression makes it)3 867( This)1 233(the user's problem.)2 778 4 720 2196 t (program unit for the problem at hand.)6 1506 1 720 2316 t ( is to make their run-)5 845( This)1 230( require small memory and run-time resources.)6 1883(The examples are chosen to)4 1112 4 970 2472 t (ning on small machines, like Vaxen, not too onerous a chore for folks installing and testing the package.)17 4167 1 720 2592 t ( This)1 234( section 4 where they were formulated and analyzed.)8 2150(The examples are taken in sequence from)6 1686 3 970 2748 t ( The)1 211( in section 4.)3 529(section is only intended to show how to program the solution of the formulations given)14 3580 3 720 2868 t ( section 5 describing the)4 991(user must have read)3 807 2 720 2988 t 10 CW f (TTGR)2548 2988 w 10 R f (software before proceeding in this appendix, otherwise)6 2222 1 2818 2988 t (the reading will be dark and obscure.)6 1477 1 720 3108 t (Before invoking)1 652 1 970 3264 t 10 CW f (TTGR)1647 3264 w 10 R f (the user must)2 533 1 1912 3264 t 10 S f (\267)970 3420 w 10 R f (Make a B-spline mesh.)3 921 1 1041 3420 t 10 S f (\267)970 3576 w 10 R f (Make initial conditions for the B-spline coefficients)6 2070 1 1041 3576 t 10 B f (U)3136 3576 w 10 R f (in \(3.1\).)1 319 1 3233 3576 t 10 S f (\267)970 3732 w 10 R f (Write subroutines)1 713 1 1041 3732 t 10 S f (\267)1220 3888 w 10 CW f (AF)1326 3888 w 10 R f (- to evaluate)2 493 1 1471 3888 t 10 B f (a)1989 3888 w 10 R f (and)2064 3888 w 10 B f (f)2233 3888 w 10 R f (in \(2.1\).)1 319 1 2291 3888 t 10 S f (\267)1220 4044 w 10 CW f (BC)1326 4044 w 10 R f (- to evaluate)2 493 1 1471 4044 t 10 B f (b)1989 4044 w 10 R f (in \(2.2\).)1 319 1 2070 4044 t 10 S f (\267)1220 4200 w 10 CW f (HANDLE)1326 4200 w 10 R f (- to output \(print\) the solution results.)6 1503 1 1711 4200 t ( initial con-)2 463( creation of a mesh and)5 928( The)1 205(The subroutine writing will be amply illustrated later in this section.)10 2724 4 720 4356 t (ditions \()1 331 1 720 4476 t 10 B f (ic)1076 4476 w 10 R f (s \) for)2 238 1 1148 4476 t 10 B f (u)1411 4476 w 10 R f (are now briefly described.)3 1041 1 1492 4476 t 10 B f (Mesh Making)1 592 1 720 4716 t 10 R f ( is)1 103( There)1 293(The PORT Library [14] has several B-spline mesh generation subroutines available.)10 3458 3 970 4872 t 10 CW f (UMB)4860 4872 w 10 R f ( creating B-spline meshes that are the)6 1593( For)1 205( given interval.)2 634(for generating uniform B-spline meshes on a)6 1888 4 720 4992 t ( there are)2 374(union of uniform meshes over basic contiguous intervals)7 2274 2 720 5112 t 10 CW f (LUMB)3395 5112 w 10 R f (and)3662 5112 w 10 CW f (PUMB)3833 5112 w 10 R f (.)4073 5112 w 10 CW f (LUMB)4220 5112 w 10 R f (uses the same)2 553 1 4487 5112 t (number of mesh points in each basic interval and)8 1958 1 720 5232 t 10 CW f (PUMB)2738 5232 w 10 R f (allows that number to vary by interval.)6 1551 1 3003 5232 t ( about)1 257(If you find yourself using more than 50 to 100 mesh points, in any direction, think carefully)16 3813 2 970 5388 t (using)720 5508 w 10 CW f (LUMB)967 5508 w 10 R f ( example,)1 392( For)1 193(to make the mesh more carefully tailored to the solution, or use mesh mapping.)13 3218 3 1237 5508 t ( but it would require roughly)5 1167(example 5 below could be solved on a uniform mesh,)9 2154 2 720 5628 t 10 B f (90,000)4069 5628 w 10 R f (grid points to do)3 668 1 4372 5628 t (it for)1 201 1 720 5748 t 10 I f (k k)1 44 1 950 5748 t 10 S f (= =)1 55 1 1043 5748 t 10 R f ( Remem-)1 397( nicely.)1 297( the non-uniform mesh scheme given for that problem, 625 do the job)12 2831(2. Using)1 368 4 1147 5748 t (ber that the run-time and memory requirements of)7 2083 1 720 5868 t 10 CW f (TTGR)2840 5868 w 10 R f ( the number of mesh points)5 1156(are proportional to)2 767 2 3117 5868 t ( little thought given to mesh construction can save a lot of computer run-time and memory.)15 3641(used. A)1 330 2 720 5988 t 10 B f (Initial Conditions for u.)3 1013 1 720 6228 t 10 R f (There are)1 388 1 970 6384 t 10 B f (ic)1393 6384 w 10 R f (s for)1 190 1 1465 6384 t 10 B f (u)1690 6384 w 10 R f (and these must be converted into)5 1362 1 1781 6384 t 10 B f (ic)3178 6384 w 10 R f (s for)1 191 1 3250 6384 t 10 B f (U)3477 6384 w 10 R f (, the B-spline coefficients for)4 1211 1 3549 6384 t 10 B f (u)4796 6384 w 10 R f (, see)1 188 1 4852 6384 t ( case is when the)4 680( simplest)1 361(\(3.1\). The)1 423 3 720 6504 t 10 B f (ic)2210 6504 w 10 R f (s for)1 181 1 2282 6504 t 10 B f (u)2489 6504 w 10 R f ( simply setting all the spline coefficients to)7 1726( By)1 168(are a constant.)2 575 3 2571 6504 t (that constant, the spline)3 944 1 720 6624 t 10 B f (is)1689 6624 w 10 R f (the constant and we are done.)5 1180 1 1781 6624 t (If the)1 213 1 970 6780 t 10 B f (ic)1208 6780 w 10 R f (s for)1 180 1 1280 6780 t 10 I f (u u)1 50 1 1485 6780 t 10 R f (are not constant, then there is)5 1170 1 1560 6780 t 10 CW f (TSL2W)2755 6780 w 10 R f (available, see section 4.)3 945 1 3080 6780 t cleartomark showpage saveobj restore end %%EndPage: 1 40 %%Page: 2 41 DpostDict begin /saveobj save def mark 41 pagesetup 10 R f (\261 4-2 \261)2 283 1 2738 480 t 10 B f (The Examples)1 609 1 720 840 t 10 R f ( here were run on a VAX 11/750, equipped with a floating-point accelerator,)12 3167(All examples reported)2 903 2 970 996 t (using single-precision arithmetic, under the)4 1734 1 720 1116 t 9 R f (UNIX)2477 1116 w 10 R f (operating system, Research Eighth Edition.)4 1732 1 2727 1116 t 10 B f (Example 1 - A Simple Heat Equation.)6 1604 1 720 1356 t 10 R f (As a simple example of the use of)7 1356 1 970 1512 t 10 CW f (TTGR)2351 1512 w 10 R f (, consider solving the scalar heat equation)6 1672 1 2591 1512 t 10 I f (a a)1 50 1 1220 1692 t 7 R f (\( 1 \))2 91 1 1281 1652 t 10 S f (= =)1 55 1 1437 1692 t 10 I f (u u)1 50 1 1541 1692 t 10 S f (+ +)1 55 1 1631 1692 t 10 I f (u u)1 50 1 1726 1692 t 7 I f (x x)1 31 1 1787 1712 t 10 S f (+ +)1 55 1 1866 1692 t 10 I f (. .)1 25 1 1961 1692 t 10 R f (1)1994 1692 w 10 I f (u u)1 50 1 2076 1692 t 7 I f (y y)1 31 1 2137 1712 t 10 R f (,)2184 1692 w 10 I f (a a)1 50 1 1228 1872 t 7 R f (\( 2 \))2 91 1 1289 1832 t 10 S f (= =)1 55 1 1445 1872 t 10 I f (u u)1 50 1 1549 1872 t 10 S f (+ +)1 55 1 1639 1872 t 10 I f (u u)1 50 1 1734 1872 t 7 I f (y y)1 31 1 1795 1892 t 10 S f (+ +)1 55 1 1874 1872 t 10 I f (. .)1 25 1 1969 1872 t 10 R f (1)2002 1872 w 10 I f (u u)1 50 1 2084 1872 t 7 I f (x x)1 31 1 2145 1892 t 10 R f (, \(A4.1\))1 2848 1 2192 1872 t 10 I f (f f)1 28 1 1236 2052 t 10 S f (= =)1 55 1 1329 2052 t 10 I f (u u)1 50 1 1433 2052 t 7 I f (t t)1 20 1 1494 2072 t 10 S f (+ +)1 55 1 1562 2052 t 10 I f (u u)1 50 1 1657 2052 t 7 I f (x x)1 31 1 1718 2072 t 10 S f (+ +)1 55 1 1797 2052 t 10 I f (u u)1 50 1 1892 2052 t 7 I f (y y)1 31 1 1953 2072 t 10 S f (- -)1 55 1 2032 2052 t 10 I f (g g)1 50 1 2127 2052 t 10 R f (\()2185 2052 w 10 I f (t t)1 28 1 2226 2052 t 10 R f (,)2262 2052 w 10 I f (x x)1 44 1 2295 2052 t 10 R f (,)2347 2052 w 10 I f (y y)1 44 1 2380 2052 t 10 R f (\))2432 2052 w (on the unit square, with)4 941 1 720 2232 t 10 B f (bc)1686 2232 w 10 R f (s \(4.2\))1 255 1 1786 2232 t 10 B f (b)1220 2412 w 10 S f (= =)1 55 1 1325 2412 t 10 I f (u u)1 50 1 1429 2412 t 10 R f (\()1487 2412 w 10 I f (t t)1 28 1 1528 2412 t 10 R f (,)1564 2412 w 10 I f (x x)1 44 1 1597 2412 t 10 R f (,)1649 2412 w 10 I f (y y)1 44 1 1682 2412 t 10 R f (\))1734 2412 w 10 S f (- -)1 55 1 1815 2412 t 10 I f ( .)1 0( .)1 33( y)1 0( y)1 76( x)1 0( x)1 76(t t)1 28 7 1910 2412 t 10 R f (\(A4.2\))4777 2412 w (The solution is)2 595 1 720 2592 t 10 I f (u u)1 50 1 1340 2592 t 10 S f (\272)1431 2592 w 10 I f ( y)1 0( y)1 76( x)1 0( x)1 76(t t)1 28 5 1527 2592 t 10 R f ( initial conditions are taken to be 0.)7 1410( The)1 205(, which can be gotten exactly.)5 1189 3 1707 2592 t ( unit, written in Ratfor [21], solves \(A4.1\)-\(A4.2\) using)8 2278(The following program)2 946 2 970 2748 t 10 CW f (TTGR)4227 2748 w 10 R f (, with a linear)3 573 1 4467 2748 t (B-spline \()1 401 1 720 2868 t 10 I f (k k)1 44 1 1121 2868 t 10 S f (= =)1 55 1 1214 2868 t 10 R f ( with the time)3 562( 0 , 1 \),)4 215( mesh consisting of 3 equally spaced, distinct points on \()10 2293(2\) over a spatial)3 652 4 1318 2868 t (evolution carried out to 10)4 1060 1 720 2988 t 7 S f (- -)1 39 1 1791 2948 t 7 R f (2)1841 2948 w 10 R f (absolute accuracy.)1 736 1 1909 2988 t ( old)1 158(Ratfor is much easier to read and type than standard)9 2112 2 970 3144 t 10 CW f (Fortran)3270 3144 w 10 R f ( we use Ratfor to pre-)5 892(. Although)1 458 2 3690 3144 t (sent the code in the examples, we ship standard)8 1894 1 720 3264 t 10 CW f (Fortran)2639 3264 w 10 R f (.)3059 3264 w ( Specifi-)1 378(Ratfor stands for "Rational Fortran" and provides modern control structures over Fortran.)11 3692 2 970 3420 t ( \()1 66(cally, it allows multiple statements on a line to be separated by semicolons \( ; \), a # begins a comment)20 4254 2 720 3540 t ( line \), and curly braces \( { and })9 1327(hence a comment and an executable statement can be together on the same)12 2993 2 720 3660 t ( for statement labels on Do loops,)6 1382( last feature removes the need)5 1223( This)1 235(\) are used to delimit blocks of code.)7 1480 4 720 3780 t ( is "syntactic sugar" in the form of ")8 1510( There)1 291(for example.)1 512 3 720 3900 t 10 S f (<)3033 3900 w 10 R f ( ")1 75( and)1 203(" for ".lt.", "&" for ".and.")5 1092 3 3088 3900 t 10 S f (\357)4492 3900 w 10 R f (" for ".or.".)2 465 1 4575 3900 t (Finally, if a line ends in one of ", + - * / \( =)14 1724 1 720 4020 t 10 S f (< >)1 135 1 2469 4020 t 10 R f (&)2629 4020 w 10 S f (\357)2732 4020 w 10 R f (", the next line is considered a continuation of it.)9 1942 1 2781 4020 t (The main program uses several PORT [14] library subprograms.)8 2576 1 970 4176 t 10 S f (\267)970 4332 w 10 CW f (ISTKIN)1078 4332 w 10 R f ( the stack to 350,000 double)5 1142( this case, it initializes)4 894( In)1 135(initializes the PORT Library stack.)4 1404 4 1465 4332 t (precision items, consistent with the declaration for)6 2029 1 970 4452 t 10 CW f (Ds)3026 4452 w 10 R f ( precision alias of the stack in the)7 1341(in the double)2 526 2 3173 4452 t (common region)1 630 1 970 4572 t 10 CW f (CSTAK)1625 4572 w 10 R f (.)1925 4572 w 10 S f (\267)970 4728 w 10 CW f (ENTER)1080 4728 w 10 R f (and)1409 4728 w 10 CW f (LEAVE)1582 4728 w 10 R f ( effect is that)3 533( The)1 210(bracket blocks of code in which stack allocations are done.)9 2386 3 1911 4728 t (all allocations made after the)4 1153 1 970 4848 t 10 CW f (ENTER)2148 4848 w 10 R f (but before the)2 554 1 2473 4848 t 10 CW f (LEAVE)3087 4848 w 10 R f (are released by the)3 744 1 3412 4848 t 10 CW f (LEAVE)4181 4848 w 10 R f (.)4481 4848 w 10 S f (\267)970 5004 w 10 CW f (IUMB)1076 5004 w 10 R f ( returns a pointer to the mesh.)6 1190( It)1 111(makes uniformly spaced B-spline meshes on the stack.)7 2186 3 1341 5004 t 10 S f (\267)970 5160 w 10 CW f (ISTKGT)1121 5160 w 10 R f ( the example below,)3 942( In)1 179( stack.)1 301(allocates storage on the Port Library)5 1675 4 1551 5160 t 10 CW f (iU =)1 286 1 4754 5160 t (ISTKGT\(L,3\))970 5280 w 10 R f ( pointer)1 318(sets the)1 308 2 1666 5280 t 10 CW f (iU)2327 5280 w 10 R f (so that locations)2 670 1 2482 5280 t 10 CW f (Ws\(iU\), ..., Ws\(iU+L-1\))2 1400 1 3222 5280 t 10 R f (are avail-)1 383 1 4657 5280 t (able for the B-spline coefficients of the solution.)7 1936 1 970 5400 t 10 S f (\267)970 5556 w 10 CW f (SETR)1080 5556 w 10 R f (sets an array to a given Real constant.)7 1536 1 1349 5556 t 10 CW f (SETR)3010 5556 w 10 R f (provides the constant)2 859 1 3280 5556 t 10 B f (ic)4169 5556 w 10 R f ('s \(A4.3\) via the B-)4 799 1 4241 5556 t ( are equal to a constant, then the B-spline)8 1650(spline coefficients \(3.1\), since if all the B-spline coefficients)8 2420 2 970 5676 t (itself is identically equal to that constant \(see Appendix 1\).)9 2353 1 970 5796 t 10 S f (\267)970 5952 w 10 CW f (WRAPUP)1076 5952 w 10 R f (checks that a run has terminated without errors and prints out the stack space used.)14 3302 1 1461 5952 t (At the end of each time-step the solution is printed out at)11 2273 1 970 6148 t 10 I f (x x)1 44 1 3268 6148 t 10 R f (,)3320 6148 w 10 I f (y y)1 44 1 3353 6148 t 10 S f (= =)1 55 1 3446 6148 t 10 R f (0 ,)1 83 1 3550 6148 t (2)3699 6218 w (1)3699 6088 w 10 S1 f (_ _)1 80 1 3684 6118 t 10 R f ( main program is)3 680( The)1 205(and 1.)1 244 3 3799 6148 t cleartomark showpage saveobj restore end %%EndPage: 2 41 %%Page: 3 42 DpostDict begin /saveobj save def mark 42 pagesetup 10 R f (\261 4-3 \261)2 283 1 2738 480 t 10 CW f (# Main)1 360 1 1080 900 t (# To solve the heat equation with solution u == t*x*y,)10 3240 1 1080 1140 t ( . \( U + Ux + .1 * Uy, U + Uy + .1 * Ux \) = Ut + Ux + Uy +g\(x,t\))24 3840(# grad)1 480 2 1080 1380 t (Real tstart,tstop,dt,Lx,Rx,Ly,Ry)1 1920 1 1200 1620 t (Real errpar\(2\))1 840 1 1200 1740 t (Integer Nu,kx,ix,nx, ky,iy,ny,ISTKGT, IUMB,iU,ndx,ndy)3 3180 1 1200 1860 t (External AF,BC,HANDLE)1 1260 1 1200 1980 t (Common / CSTAK / Ds\(350000\); Double Precision Ds)7 2880 1 1200 2340 t ( The PORT Library stack and its aliases.)7 2400( #)1 300(Real Ws\(1000\))1 780 3 1200 2460 t (Real Rs\(1000\) ; Integer Is\(1000\) ; Complex Cs\(500\) ; Logical Ls\(1000\))10 4140 1 1200 2580 t (Equivalence \( Ds\(1\),Cs\(1\),Ws\(1\),Rs\(1\),Is\(1\),Ls\(1\) \))3 3060 1 1200 2700 t ( Initialize the PORT Library stack length.)6 2520( #)1 300(Call ISTKIN\(350000,4\))1 1260 3 1200 2940 t (Call ENTER\(1\))1 780 1 1200 3180 t (Nu = 1)2 360 1 1200 3420 t (Lx = 0; Rx = 1)5 840 1 1200 3660 t (Ly = 0; Ry = 1)5 840 1 1200 3780 t (kx = 2; ky = 2)5 840 1 1200 4020 t (ndx = 3; ndy = 3)5 960 1 1200 4260 t (tstart = 0; tstop = 1; dt = 1)8 1740 1 1200 4500 t (errpar\(1\) = 1e-2; errpar\(2\) = 1e-4)5 2040 1 1200 4740 t ( Uniform grid.)2 840( #)1 300(ix = IUMB\(Lx,Rx,ndx,kx,nx\))2 1560 3 1200 4980 t ( Uniform grid.)2 840( #)1 300(iy = IUMB\(Ly,Ry,ndy,ky,ny\))2 1560 3 1200 5100 t ( Space for the solution.)4 1440( #)1 300(iU = ISTKGT\(Nu*\(nx-kx\)*\(ny-ky\),3\))2 1980 3 1200 5340 t ( Initial conditions for U.)4 1560( #)1 300(Call SETR\(Nu*\(nx-kx\)*\(ny-ky\),0e0,Ws\(iU\)\))1 2400 3 1200 5460 t (Call TTGR \(Ws\(iU\),Nu,kx,Ws\(ix\),nx, ky,Ws\(iy\),ny, tstart,tstop, dt,)5 3960 1 1200 5700 t (AF,BC,)1860 5820 w (errpar,)1860 5940 w (HANDLE\))1860 6060 w (Call LEAVE)1 600 1 1200 6300 t (Call WRAPUP)1 660 1 1200 6540 t (Stop)1200 6780 w (End)1200 7020 w 10 R f ( the various arguments of the)5 1197(The dimension statements for)3 1203 2 720 7236 t 10 CW f (AF)3152 7236 w 10 R f (,)3272 7236 w 10 CW f (BC)3329 7236 w 10 R f (and)3481 7236 w 10 CW f (HANDLE)3657 7236 w 10 R f (subroutines given below)2 991 1 4049 7236 t cleartomark showpage saveobj restore end %%EndPage: 3 42 %%Page: 4 43 DpostDict begin /saveobj save def mark 43 pagesetup 10 R f (\261 4-4 \261)2 283 1 2738 480 t (are general and thus will not be repeated in subsequent)9 2187 1 720 840 t 10 B f (pde-bc)2932 840 w 10 R f (examples.)3246 840 w (Note that since the arrays)4 1038 1 970 996 t 10 CW f (A)2039 996 w 10 R f (, ... ,)2 187 1 2099 996 t 10 CW f (FUyt)2317 996 w 10 R f (are set to zero by)4 705 1 2588 996 t 10 CW f (TTGR)3324 996 w 10 R f (before it calls)2 555 1 3595 996 t 10 CW f (AF)4181 996 w 10 R f ( active)1 270(, only the)2 387 2 4301 996 t 10 B f (a)4990 996 w 10 R f (and)720 1116 w 10 B f (f)890 1116 w 10 R f (terms and their derivatives need be computed in)7 1923 1 949 1116 t 10 CW f (AF)2898 1116 w 10 R f ( subroutine)1 448(. The)1 231 2 3018 1116 t 10 CW f (AF)3722 1116 w 10 R f (for specifying the)2 704 1 3867 1116 t 10 B f (pde)4596 1116 w 10 R f (\(A4.1\))4777 1116 w (is)720 1236 w 10 CW f (Subroutine AF\(t,x,nx,y,ny,Nu,)1 1740 1 1200 1416 t (U,Ut,Ux,Uy,Uxt,Uyt,)2040 1536 w (a,AU,AUt,AUx,AUy,AUxt,AUyt,)2040 1656 w (f,FU,FUt,FUx,FUy,FUxt,FUyt\))2040 1776 w (Real t,x\(nx\),y\(ny\),)1 1140 1 1200 2016 t (U\(nx,ny,Nu\),Ut\(nx,ny,Nu\),Ux\(nx,ny,Nu\),Uy\(nx,ny,Nu\),)1800 2136 w (Uxt\(nx,ny,Nu\),Uyt\(nx,ny,Nu\),)1800 2256 w ( \(nx,ny,Nu \),)2 1020( f)1 120(a \(nx,ny,Nu ,2\),)2 1200 3 1800 2376 t ( \(nx,ny,Nu,Nu\),)1 960( FU)1 180(AU \(nx,ny,Nu,Nu,2\),)1 1200 3 1800 2496 t (AUt \(nx,ny,Nu,Nu,2\), FUt \(nx,ny,Nu,Nu\),)3 2340 1 1800 2616 t (AUx \(nx,ny,Nu,Nu,2\), FUx \(nx,ny,Nu,Nu\),)3 2340 1 1800 2736 t (AUy \(nx,ny,Nu,Nu,2\), FUy \(nx,ny,Nu,Nu\),)3 2340 1 1800 2856 t (AUxt\(nx,ny,Nu,Nu,2\), FUxt\(nx,ny,Nu,Nu\),)1 2340 1 1800 2976 t (AUyt\(nx,ny,Nu,Nu,2\), FUyt\(nx,ny,Nu,Nu\))1 2280 1 1800 3096 t (Integer nx,ny,Nu)1 960 1 1200 3216 t (Integer p,q,i)1 780 1 1200 3456 t (Do i = 1, Nu)4 720 1 1200 3696 t ({)1320 3816 w (Do q = 1, ny)4 720 1 1320 3936 t ({)1440 4056 w (Do p = 1, nx)4 720 1 1440 4176 t ({)1560 4296 w (a\(p,q,i,1\) = Ux\(p,q,i\) + .1*Uy\(p,q,i\) + U\(p,q,i\))6 2880 1 1560 4416 t (a\(p,q,i,2\) = Uy\(p,q,i\) + .1*Ux\(p,q,i\) + U\(p,q,i\))6 2880 1 1560 4536 t (AUx\(p,q,i,i,1\) = 1; AUy\(p,q,i,i,2\) = 1)5 2280 1 1560 4656 t (AUy\(p,q,i,i,1\) = .1; AUx\(p,q,i,i,2\) = .1)5 2400 1 1560 4776 t (AU \(p,q,i,i,1\) = 1; AU \(p,q,i,i,2\) = 1)7 2280 1 1560 4896 t (f\(p,q,i\) = Ut\(p,q,i\)+Ux\(p,q,i\)+Uy\(p,q,i\))2 2400 1 1560 5136 t (FUt\(p,q,i,i\) = 1; fUx\(p,q,i,i\) = 1; fUy\(p,q,i,i\) = 1)8 3120 1 1560 5256 t (f\(p,q,i\) = f\(p,q,i\) + .2*t-x\(p\)*y\(q\))4 2160 1 1560 5376 t (})1560 5496 w (})1440 5616 w (})1320 5736 w (Return)1200 5976 w (End)1200 6216 w 10 R f (Note that since the arrays)4 1034 1 720 6432 t 10 CW f (B)1784 6432 w 10 R f (, ... ,)2 185 1 1844 6432 t 10 CW f (BUyt)2060 6432 w 10 R f (are set to zero by)4 705 1 2331 6432 t 10 CW f (TTGR)3067 6432 w 10 R f (before it calls)2 555 1 3338 6432 t 10 CW f (BC)3924 6432 w 10 R f (, only the active)3 656 1 4044 6432 t 10 B f (b)4731 6432 w 10 R f (terms)4818 6432 w (and their derivatives need be computed in)6 1669 1 720 6552 t 10 CW f (BC)2414 6552 w 10 R f ( subroutine)1 447(. The)1 230 2 2534 6552 t 10 CW f (BC)3236 6552 w 10 R f (for specifying the)2 704 1 3381 6552 t 10 B f (bc)4110 6552 w 10 R f ('s \(A4.2\) is)2 452 1 4210 6552 t cleartomark showpage saveobj restore end %%EndPage: 4 43 %%Page: 5 44 DpostDict begin /saveobj save def mark 44 pagesetup 10 R f (\261 4-5 \261)2 283 1 2738 480 t 10 CW f (Subroutine BC\(t,x,nx,y,ny,Lx,Rx,Ly,Ry,)1 2280 1 1200 900 t (U,Ut,Ux,Uy,Uxt,Uyt,Nu,)2040 1020 w (b,bu,but,bux,buy,buxt,buyt\))2040 1140 w (Real t,x\(nx\),y\(ny\),Lx,Rx,Ly,Ry,)1 1860 1 1200 1380 t (U\(nx,ny,Nu\),Ut\(nx,ny,Nu\),Ux\(nx,ny,Nu\),Uy\(nx,ny,Nu\),)1800 1500 w (Uxt\(nx,ny,Nu\),Uyt\(nx,ny,Nu\),)1800 1620 w (b\(nx,ny,Nu\),bu\(nx,ny,Nu,Nu\),bux\(nx,ny,Nu,Nu\),buy\(nx,ny,Nu,Nu\),)1800 1740 w (but\(nx,ny,Nu,Nu\),buxt\(nx,ny,Nu,Nu\),buyt\(nx,ny,Nu,Nu\))1800 1860 w (Integer nx,ny,Nu)1 960 1 1200 1980 t (Integer i,j)1 660 1 1200 2220 t (Do j = 1, ny)4 720 1 1200 2460 t ({)1320 2580 w (Do i = 1, nx)4 720 1 1320 2700 t ({)1440 2820 w (bu\(i,j,1,1\) = 1; b\(i,j,1\) = U\(i,j,1\)-t*x\(i\)*y\(j\))5 2880 1 1440 2940 t (})1440 3060 w (})1320 3180 w (Return)1200 3420 w (End)1200 3660 w 10 R f (The following output subroutine simply evaluates and prints)7 2420 1 720 3916 t 10 I f (u u)1 50 1 3166 3916 t 10 R f (\()3224 3916 w 10 I f (t t)1 28 1 3265 3916 t 10 R f (,)3301 3916 w 10 I f (x x)1 44 1 3334 3916 t 10 R f (,)3386 3916 w 10 I f (y y)1 44 1 3419 3916 t 10 R f (\), for)1 201 1 3471 3916 t 10 I f (x x)1 44 1 3699 3916 t 10 R f (,)3751 3916 w 10 I f (y y)1 44 1 3784 3916 t 10 S f (= =)1 55 1 3877 3916 t 10 R f (0 ,)1 83 1 3981 3916 t (2)4130 3986 w (1)4130 3856 w 10 S1 f (_ _)1 80 1 4115 3886 t 10 R f (, and 1, at the end of)6 827 1 4213 3916 t ( is for arbitrary input and)5 1061( dimension statement for the various arguments)6 1976( The)1 218(each successful time-step.)2 1065 4 720 4086 t (thus will not be repeated in subsequent examples.)7 1981 1 720 4206 t (Three PORT Library routines are used)5 1538 1 970 4362 t 10 S f (\267)970 4518 w 10 CW f (I1MACH)1076 4518 w 10 R f (determines the standard output unit number,)5 1765 1 1461 4518 t 10 CW f (I1MACH)3251 4518 w 10 R f (\(2\).)3611 4518 w 10 S f (\267)970 4674 w 10 CW f (TSD1)1076 4674 w 10 R f (evaluates a spline, given the mesh and the coefficients. See section 4.)11 2767 1 1341 4674 t 10 S f (\267)970 4830 w 10 CW f (ILUMD)1079 4830 w 10 R f (generates a list of distinct points from a basic mesh by inserting a given number of points)16 3633 1 1407 4830 t (between the basic points.)3 1004 1 970 4950 t (Also, two)1 389 1 720 5106 t 10 CW f (Common)1134 5106 w 10 R f (regions from)1 513 1 1519 5106 t 10 CW f (TTGR)2057 5106 w 10 R f (are used to provide, magically, the meshes for)7 1838 1 2322 5106 t 10 I f (x x)1 44 1 4186 5106 t 10 R f (and)4256 5106 w 10 I f (y y)1 44 1 4426 5106 t 10 R f (and)4496 5106 w 10 CW f (Nu)4666 5106 w 10 R f (. This)1 254 1 4786 5106 t ( the fixed calling sequence for the output routine from)9 2196(is because)1 412 2 720 5226 t 10 CW f (IODE)3357 5226 w 10 R f (doesn't currently allow the passing)4 1414 1 3626 5226 t ( we pass it under the table.)6 1062( So)1 156(of such extra information.)3 1037 3 720 5346 t cleartomark showpage saveobj restore end %%EndPage: 5 44 %%Page: 6 45 DpostDict begin /saveobj save def mark 45 pagesetup 10 R f (\261 4-6 \261)2 283 1 2738 480 t 10 CW f (Subroutine HANDLE\(t0,U0,t,U,Nv,dt,tstop\))1 2400 1 1200 900 t (Real t0,U0\(Nv\),t,U\(Nv\),dt,tstop)1 1860 1 1200 1140 t (Integer Nv)1 600 1 1200 1260 t (Common / A7TGRM / kx,ix,nx, ky,iy,ny; Integer kx,ix,nx, ky,iy,ny)8 3840 1 1200 1500 t (Common / A7TGRP / errpar\(2\), Nu,mxq,myq; Real errpar; Integer Nu,mxq,myq)9 4320 1 1200 1740 t (If \( t0 == t \))5 840 1 1200 1980 t ({)1320 2100 w (iwunit = 6)2 600 1 1320 2220 t (Write\(iwunit,9000\) t)1 1200 1 1320 2340 t (9000 Format\(" Restart for t =",1p4e10.2\))5 2400 1 1140 2460 t (Return)1320 2580 w (})1320 2700 w ( Print results.)2 900( #)1 240(Call GERR\(kx,ix,nx, ky,iy,ny, U,Nu, t\))4 2280 3 1200 2940 t (Return)1200 3180 w (End)1200 3420 w 10 R f (where the procedure)2 813 1 720 3636 t 10 CW f (GERR)1558 3636 w 10 R f (is used to print out the results)6 1178 1 1823 3636 t cleartomark showpage saveobj restore end %%EndPage: 6 45 %%Page: 7 46 DpostDict begin /saveobj save def mark 46 pagesetup 10 R f (\261 4-7 \261)2 283 1 2738 480 t 10 CW f (Subroutine GERR\(kx,ix,nx, ky,iy,ny, U,Nu, t\))4 2640 1 1200 900 t (# To print the solution at each time-step.)7 2520 1 1080 1140 t ( U\(nx-kx,ny,ky,Nu\).)1 1140( #)1 300(Real U\(1\),t)1 660 3 1200 1380 t (Integer kx,ix,nx, ky,iy,ny,Nu)2 1740 1 1200 1500 t (Common / CSTAK / Ds\(500\); Double Precision Ds)7 2700 1 1200 1740 t ( The PORT Library stack and its aliases.)7 2400( #)1 300(Real Ws\(1000\))1 780 3 1200 1860 t (Real Rs\(1000\) ; Integer Is\(1000\) ; Complex Cs\(500\) ; Logical Ls\(1000\))10 4140 1 1200 1980 t (Equivalence \( Ds\(1\),Cs\(1\),Ws\(1\),Rs\(1\),Is\(1\),Ls\(1\) \))3 3060 1 1200 2100 t (Integer i,I1MACH,ISTKGT,)1 1440 1 1200 2340 t (KA\(2\),ITA\(2\),NTA\(2\),IXA\(2\),NXA\(2\),MA\(2\),iFA,)1680 2460 w (ILUMD,ixs,iys,nxs,nys)1740 2580 w (Call ENTER\(1\))1 780 1 1200 2820 t (# Find the solution at 2 * 2 points / mesh rectangle.)11 3180 1 1080 3060 t ( x search grid.)3 900( #)1 300(ixs = ILUMD\(Ws\(ix\),nx,2,nxs\))2 1680 3 1200 3300 t ( y search grid.)3 900( #)1 300(iys = ILUMD\(Ws\(iy\),ny,2,nys\))2 1680 3 1200 3420 t (KA\(1\) = kx; KA\(2\) = ky; ITA\(1\) = ix; ITA\(2\) = iy; NTA\(1\) = nx; NTA\(2\) = ny)17 4440 1 1200 3660 t (IXA\(1\) = ixs; IXA\(2\) = iys; NXA\(1\) = nxs; NXA\(2\) = nys)11 3240 1 1200 3780 t ( Get solution.)2 840( #)1 300(MA\(1\) = 0; MA\(2\) = 0)5 1200 3 1200 3900 t ( Approximate solution values.)3 1740( #)1 300(iFA = ISTKGT\(nxs*nys,3\))2 1380 3 1200 4140 t ( Evaluate them.)2 900( #)1 300(Call TSD1\(2,KA,Ws,ITA,NTA,U, Ws,IXA,NXA,MA, Ws\(iFA\)\))3 3120 3 1200 4380 t (iwunit = I1MACH\(2\))2 1080 1 1200 4620 t (Write\(iwunit,9001\) t,\(Ws\(i\),i = iFA, iFA+nxs*nys-1 \))5 3120 1 1200 4740 t (9001 Format\(" U\(",1p1e10.2,",.,.\) =",\(\(1p5e10.2/20x,1p4e10.2\)\)\))3 3780 1 1140 4860 t (Call LEAVE)1 600 1 1200 5100 t (Return)1200 5340 w (End)1200 5580 w 10 R f (The output of the above program unit is)7 1590 1 970 5796 t 10 CW f ( 0.00e+00 5.14e-17 5.87e-09 0.00e+00 2.50e-01)5 3000( =)1 120(U\( 1.00e+00,.,.\))1 1020 3 1140 5976 t (5.00e-01 0.00e+00 5.00e-01 1.00e+00)3 2280 1 2400 6096 t ( of the stack allowed.)4 1320( 700000)1 540( /)1 120(used 825)1 780 4 1140 6216 t 10 R f (The run-time for the above was 5.2 seconds.)7 1771 1 720 6396 t ( since)1 237( Well,)1 276( that it is difficult to determine that the above output is in fact exact.)14 2815(A skeptic might observe)3 992 4 720 6552 t ( may also check the accuracy of the numerical solu-)9 2069(the exact solution of the problem is known, the program)9 2251 2 720 6672 t ( procedure)1 423(tion. The)1 386 2 720 6792 t 10 CW f (GERR)1554 6792 w 10 R f (below checks the error rather than just evaluate the solution)9 2381 1 1819 6792 t cleartomark showpage saveobj restore end %%EndPage: 7 46 %%Page: 8 47 DpostDict begin /saveobj save def mark 47 pagesetup 10 R f (\261 4-8 \261)2 283 1 2738 480 t 10 CW f (Subroutine GERR\(kx,ix,nx, ky,iy,ny, U,Nu, t\))4 2640 1 1200 900 t (# To get and print the error at each time-step.)9 2820 1 1080 1140 t ( U\(nx-kx,ny0ky,Nu\).)1 1140( #)1 300(Real U\(1\),t)1 660 3 1200 1380 t (Integer kx,ix,nx, ky,iy,ny,Nu)2 1740 1 1200 1500 t (Common / CSTAK / Ds\(500\); Double Precision Ds)7 2700 1 1200 1740 t ( The PORT Library stack and its aliases.)7 2400( #)1 300(Real Ws\(1000\))1 780 3 1200 1860 t (Real Rs\(1000\) ; Integer Is\(1000\) ; Complex Cs\(500\) ; Logical Ls\(1000\))10 4140 1 1200 1980 t (Equivalence \( Ds\(1\),Cs\(1\),Ws\(1\),Rs\(1\),Is\(1\),Ls\(1\) \))3 3060 1 1200 2100 t (Real errU)1 540 1 1200 2340 t (Integer i,I1MACH,ISTKGT,)1 1440 1 1200 2460 t (KA\(2\),ITA\(2\),NTA\(2\),IXA\(2\),NXA\(2\),MA\(2\),iFA, ILUMD,ixs,iys,nxs,nys,iEwe)1 4260 1 1680 2580 t (Call ENTER\(1\))1 780 1 1200 2820 t (# Find the error in the solution at 2*kx * 2*ky points / mesh rectangle.)14 4320 1 1080 3060 t ( x search grid.)3 900( #)1 300(ixs = ILUMD\(Ws\(ix\),nx,2*kx,nxs\))2 1860 3 1200 3300 t ( y search grid.)3 900( #)1 300(iys = ILUMD\(Ws\(iy\),ny,2*ky,nys\))2 1860 3 1200 3420 t ( U search grid values.)4 1320( #)1 300(iEwe = ISTKGT\(nxs*nys,3\))2 1440 3 1200 3540 t ( The exact solution.)3 1200( #)1 300(Call EWE\(t,Ws\(ixs\),nxs,Ws\(iys\),nys,Ws\(iEwe\),Nu\))1 2820 3 1200 3780 t (KA\(1\) = kx; KA\(2\) = ky; ITA\(1\) = ix; ITA\(2\) = iy; NTA\(1\) = nx; NTA\(2\) = ny)17 4440 1 1200 4020 t (IXA\(1\) = ixs; IXA\(2\) = iys; NXA\(1\) = nxs; NXA\(2\) = nys)11 3240 1 1200 4140 t ( Get solution.)2 840( #)1 300(MA\(1\) = 0; MA\(2\) = 0)5 1200 3 1200 4260 t ( Approximate solution values.)3 1740( #)1 300(iFA = ISTKGT\(nxs*nys,3\))2 1380 3 1200 4500 t ( Evaluate them.)2 900( #)1 300(Call TSD1\(2,KA,Ws,ITA,NTA,U, Ws,IXA,NXA,MA, Ws\(iFA\)\))3 3120 3 1200 4740 t ( Error in solution values.)4 1560( #)1 300(errU = 0)2 480 3 1200 4860 t (Do i = 1, nxs*nys)4 1020 1 1200 4980 t ({)1320 5100 w (errU = Max\(errU,Abs\(Ws\(iEwe+i-1\)-Ws\(iFA+i-1\)\)\))2 2760 1 1320 5220 t (})1320 5340 w (iwunit = I1MACH\(2\))2 1080 1 1200 5580 t (Write\(iwunit,9001\) t, errU)2 1560 1 1200 5700 t (9001 Format\(" error in U\(.,",1p1e10.2,"\) =", 1p4e10.2\))6 3240 1 1140 5820 t (Call LEAVE)1 600 1 1200 6060 t (Return)1200 6300 w (End)1200 6540 w 10 R f (and the procedure)2 714 1 720 6720 t 10 CW f (EWE)1459 6720 w 10 R f (below evaluates the exact solution at any position in time and space.)11 2730 1 1664 6720 t cleartomark showpage saveobj restore end %%EndPage: 8 47 %%Page: 9 48 DpostDict begin /saveobj save def mark 48 pagesetup 10 R f (\261 4-9 \261)2 283 1 2738 480 t 10 CW f (Subroutine EWE\(t,x,nx,y,ny,U,Nu\))1 1920 1 1200 900 t (# The exact solution.)3 1260 1 1080 1140 t (Real t,x\(nx\),y\(ny\),U\(nx,ny,Nu\))1 1800 1 1200 1380 t (Integer nx,ny,Nu)1 960 1 1200 1500 t (Integer p,i,j)1 780 1 1200 1740 t (Do p = 1, Nu)4 720 1 1200 1980 t ({)1320 2100 w (Do i = 1, nx)4 720 1 1320 2220 t ({)1440 2340 w (Do j = 1, ny)4 720 1 1440 2460 t ({)1560 2580 w (U\(i,j,p\) = t*x\(i\)*y\(j\))2 1320 1 1560 2700 t (})1560 2820 w (})1440 2940 w (})1320 3060 w (Return)1200 3300 w (End)1200 3540 w 10 R f (The above program unit gives)4 1198 1 720 3720 t 10 CW f ( 5.87e-09)1 600( =)1 120( 1.00e+00\))1 660(error in U\(.,)2 780 4 1140 3900 t ( of the stack allowed.)4 1320( 700000)1 540( /)1 120(used 825)1 780 4 1140 4020 t 10 R f ( for the)2 306( run-time)1 377( The)1 213(and we can see that \(A4.1\)-\(A4.2\) has indeed been solved to within rounding error.)13 3424 4 720 4200 t (above example was 5.3 seconds.)4 1297 1 720 4320 t 10 B f (Example 2 - A Coupled System of pdes.)7 1678 1 720 4560 t 10 R f (The solution of a coupled system of)6 1432 1 970 4716 t 10 B f (pde)2427 4716 w 10 R f ( The)1 205('s is now illustrated.)3 811 2 2583 4716 t 10 B f (pde)3624 4716 w 10 R f (is)3805 4716 w 10 I f (a a)1 50 1 1220 4896 t 7 R f (1)1275 4915 w (\( 1 \))2 91 1 1275 4856 t 10 S f (= =)1 55 1 1431 4896 t 10 I f (u u)1 50 1 1535 4896 t 7 R f (1)1596 4916 w 7 I f (x x)1 31 1 1636 4916 t 10 R f (,)1683 4896 w 10 I f (a a)1 50 1 1815 4896 t 7 R f (2)1870 4915 w (\( 1 \))2 91 1 1870 4856 t 10 S f (= =)1 55 1 2026 4896 t 10 I f (u u)1 50 1 2130 4896 t 7 R f (2)2191 4916 w 7 I f (x x)1 31 1 2231 4916 t 10 R f (,)2278 4896 w 10 I f (a a)1 50 1 1228 5076 t 7 R f (1)1283 5095 w (\( 2 \))2 91 1 1283 5036 t 10 S f (= =)1 55 1 1439 5076 t 10 I f (u u)1 50 1 1543 5076 t 7 R f (1)1604 5096 w 7 I f (y y)1 31 1 1644 5096 t 10 R f (,)1691 5076 w 10 I f (a a)1 50 1 1823 5076 t 7 R f (2)1878 5095 w (\( 2 \))2 91 1 1878 5036 t 10 S f (= =)1 55 1 2034 5076 t 10 I f (u u)1 50 1 2138 5076 t 7 R f (2)2199 5096 w 7 I f (y y)1 31 1 2239 5096 t 10 R f (, \(A4.3\))1 2754 1 2286 5076 t 10 I f (f f)1 28 1 1236 5256 t 7 R f (1)1275 5276 w 10 S f (= =)1 55 1 1367 5256 t 10 I f (u u)1 50 1 1471 5256 t 7 R f (1)1532 5276 w 7 I f (t t)1 20 1 1572 5276 t 10 S f (+ +)1 55 1 1640 5256 t 10 I f (u u)1 50 1 1735 5256 t 7 R f (1)1796 5276 w 10 I f (u u)1 50 1 1871 5256 t 7 R f (2)1932 5276 w 10 S f (- -)1 55 1 2015 5256 t 10 I f (g g)1 50 1 2110 5256 t 7 R f (1)2171 5276 w 10 R f (,)2222 5256 w 10 I f (f f)1 28 1 2362 5256 t 7 R f (2)2401 5276 w 10 S f (= =)1 55 1 2493 5256 t 10 I f (u u)1 50 1 2597 5256 t 7 R f (2)2658 5276 w 7 I f (t t)1 20 1 2698 5276 t 10 S f (+ +)1 55 1 2766 5256 t 10 I f (u u)1 50 1 2861 5256 t 7 R f (1)2922 5276 w 10 I f (u u)1 50 1 2997 5256 t 7 R f (2)3058 5276 w 10 S f (- -)1 55 1 3141 5256 t 10 I f (g g)1 50 1 3236 5256 t 7 R f (2)3297 5276 w 10 R f (with)720 5436 w 10 B f (bc)923 5436 w 10 R f (s)1023 5436 w 10 I f (b b)1 50 1 1220 5616 t 7 R f (1)1281 5636 w 10 S f (= =)1 55 1 1373 5616 t 10 I f (u u)1 50 1 1477 5616 t 7 R f (1)1538 5636 w 10 R f (\()1589 5616 w 10 I f (t t)1 28 1 1630 5616 t 10 R f (,)1666 5616 w 10 I f (x x)1 44 1 1699 5616 t 10 R f (,)1751 5616 w 10 I f (y y)1 44 1 1784 5616 t 10 R f (\))1836 5616 w 10 S f (- -)1 55 1 1917 5616 t 10 I f (e e)1 44 1 2012 5616 t 7 I f (t t)1 20 1 2067 5576 t 7 R f (\()2092 5576 w 7 I f (x x)1 31 1 2120 5576 t 7 S f (- -)1 39 1 2167 5576 t 7 I f (y y)1 31 1 2217 5576 t 7 R f (\))2253 5576 w 10 R f (and)2498 5616 w 10 I f (b b)1 50 1 2848 5616 t 7 R f (2)2909 5636 w 10 S f (= =)1 55 1 3001 5616 t 10 I f (u u)1 50 1 3105 5616 t 7 R f (2)3166 5636 w 10 R f (\()3217 5616 w 10 I f (t t)1 28 1 3258 5616 t 10 R f (,)3294 5616 w 10 I f (x x)1 44 1 3327 5616 t 10 R f (,)3379 5616 w 10 I f (y y)1 44 1 3412 5616 t 10 R f (\))3464 5616 w 10 S f (- -)1 55 1 3545 5616 t 10 I f (e e)1 44 1 3640 5616 t 7 S f (- -)1 39 1 3695 5576 t 7 I f (t t)1 20 1 3745 5576 t 7 R f (\()3770 5576 w 7 I f (x x)1 31 1 3798 5576 t 7 S f (- -)1 39 1 3845 5576 t 7 I f (y y)1 31 1 3895 5576 t 7 R f (\))3931 5576 w 10 R f (\(A4.4\))4777 5616 w (The solution is)2 595 1 720 5832 t 10 I f (u u)1 50 1 1340 5832 t 7 R f (1)1401 5852 w 10 S f (\272)1468 5832 w 10 I f (e e)1 44 1 1564 5832 t 7 I f (t t)1 20 1 1619 5792 t 7 R f (\()1644 5792 w 7 I f (x x)1 31 1 1672 5792 t 7 S f (- -)1 39 1 1719 5792 t 7 I f (y y)1 31 1 1769 5792 t 7 R f (\))1805 5792 w 10 R f (and)1861 5832 w 10 I f (u u)1 50 1 2030 5832 t 7 R f (2)2091 5852 w 10 S f (\272)2175 5832 w 10 I f (e e)1 44 1 2271 5832 t 7 S f (- -)1 39 1 2326 5792 t 7 I f (t t)1 20 1 2376 5792 t 7 R f (\()2401 5792 w 7 I f (x x)1 31 1 2429 5792 t 7 S f (- -)1 39 1 2476 5792 t 7 I f (y y)1 31 1 2526 5792 t 7 R f (\))2562 5792 w 10 R f (.)2593 5832 w ( using)1 246(The following program solves \(A4.3\)-\(A4.4\))4 1803 2 970 5988 t 10 CW f (TTGR)3048 5988 w 10 R f (, with a cubic B-spline, over a spatial mesh)8 1752 1 3288 5988 t ( with the time-evolution carried out to 10)7 1695( 0 , 1 \),)4 215(consisting of 3 equally spaced, distinct points on \()8 2058 3 720 6108 t 7 S f (- -)1 39 1 4699 6068 t 7 R f (2)4749 6068 w 10 R f (abso-)4824 6108 w ( the accuracy of the computed solution.)6 1602( error at each time-step is printed out to confirm)9 1952( The)1 209(lute accuracy.)1 557 4 720 6228 t (The main program is)3 835 1 720 6348 t cleartomark showpage saveobj restore end %%EndPage: 9 48 %%Page: 10 49 DpostDict begin /saveobj save def mark 49 pagesetup 10 R f (\261 4-10 \261)2 333 1 2713 480 t 10 CW f (# Main)1 360 1 1080 900 t (# To solve two coupled, nonlinear heat equations.)7 2940 1 1080 1140 t ( sub t = div . \( U1x, U1y \) - U1*U2 + g1)13 2400(# U1)1 360 2 1080 1380 t ( sub t = div . \( U2x, U2y \) - U1*U2 + g2)13 2400(# U2)1 360 2 1080 1500 t (Real tstart,tstop,dt,Lx,Rx,Ly,Ry)1 1920 1 1200 1740 t (Real errpar\(2\))1 840 1 1200 1860 t (Integer Nu,kx,ix,nx, ky,iy,ny,ISTKGT, IUMB,iU,ndx,ndy)3 3180 1 1200 1980 t (External AF,BC,HANDLE)1 1260 1 1200 2100 t (Common / CSTAK / Ds\(350000\); Double Precision Ds)7 2880 1 1200 2460 t ( The PORT Library stack and its aliases.)7 2400( #)1 300(Real Ws\(1000\))1 780 3 1200 2580 t (Real Rs\(1000\) ; Integer Is\(1000\) ; Complex Cs\(500\) ; Logical Ls\(1000\))10 4140 1 1200 2700 t (Equivalence \( Ds\(1\),Cs\(1\),Ws\(1\),Rs\(1\),Is\(1\),Ls\(1\) \))3 3060 1 1200 2820 t ( Initialize the PORT Library stack length.)6 2520( #)1 300(Call ISTKIN\(350000,4\))1 1260 3 1200 3060 t (Call ENTER\(1\))1 780 1 1200 3300 t (Nu = 2)2 360 1 1200 3540 t (Lx = 0; Rx = +1)5 900 1 1200 3780 t (Ly = 0; Ry = +1)5 900 1 1200 3900 t (kx = 4; ky = 4)5 840 1 1200 4140 t (ndx = 3; ndy = 3)5 960 1 1200 4380 t (tstart = 0; tstop = 1; dt = 1e-2)8 1920 1 1200 4620 t (errpar\(1\) = 1e-2; errpar\(2\) = 1e-4)5 2040 1 1200 4860 t ( Uniform grid.)2 840( #)1 300(ix = IUMB\(Lx,Rx,ndx,kx,nx\))2 1560 3 1200 5100 t ( Uniform grid.)2 840( #)1 300(iy = IUMB\(Ly,Ry,ndy,ky,ny\))2 1560 3 1200 5220 t ( Space for the solution.)4 1440( #)1 300(iU = ISTKGT\(Nu*\(nx-kx\)*\(ny-ky\),3\))2 1980 3 1200 5460 t (Call SETR\(Nu*\(nx-kx\)*\(ny-ky\),1e0,Ws\(iU\)\))1 2400 1 1200 5580 t (Call TTGR \(Ws\(iU\),Nu,kx,Ws\(ix\),nx, ky,Ws\(iy\),ny, tstart,tstop, dt,)5 3960 1 1200 5820 t (AF,BC,)1860 5940 w (errpar,)1860 6060 w (HANDLE\))1860 6180 w (Call LEAVE)1 600 1 1200 6420 t (Call WRAPUP)1 660 1 1200 6660 t (Stop)1200 6900 w (End)1200 7140 w cleartomark showpage saveobj restore end %%EndPage: 10 49 %%Page: 11 50 DpostDict begin /saveobj save def mark 50 pagesetup 10 R f (\261 4-11 \261)2 333 1 2713 480 t (The only change in the subroutine)5 1362 1 720 840 t 10 CW f (AF)2107 840 w 10 R f (of the last example is the code for specifying the)9 1938 1 2252 840 t 10 B f (pde)4215 840 w 10 CW f (Integer p,q)1 660 1 1200 1020 t (Do q = 1, ny)4 720 1 1200 1260 t ({)1320 1380 w (Do p = 1, nx)4 720 1 1320 1500 t ({)1440 1620 w (a\(p,q,1,1\) = Ux\(p,q,1\); AUx\(p,q,1,1,1\) = 1)5 2520 1 1440 1740 t (a\(p,q,1,2\) = Uy\(p,q,1\); AUy\(p,q,1,1,2\) = 1)5 2520 1 1440 1860 t (f\(p,q,1\) = Ut\(p,q,1\) + U\(p,q,1\)*U\(p,q,2\))4 2400 1 1440 2100 t (FU\(p,q,1,1\) = U\(p,q,2\); FU\(p,q,1,2\) = U\(p,q,1\); FUt\(p,q,1,1\) = 1)8 3840 1 1440 2220 t (a\(p,q,2,1\) = Ux\(p,q,2\); AUx\(p,q,2,2,1\) = 1)5 2520 1 1440 2460 t (a\(p,q,2,2\) = Uy\(p,q,2\); AUy\(p,q,2,2,2\) = 1)5 2520 1 1440 2580 t (f\(p,q,2\) = Ut\(p,q,2\) + U\(p,q,1\)*U\(p,q,2\))4 2400 1 1440 2820 t (FU\(p,q,2,1\) = U\(p,q,2\); FU\(p,q,2,2\) = U\(p,q,1\); FUt\(p,q,2,2\) = 1)8 3840 1 1440 2940 t (f\(p,q,1\) = f\(p,q,1\) - \( Exp\(t*\(x\(p\)-y\(q\)\)\)*\( x\(p\)-y\(q\) - 2*t*t \) + 1 \))12 4200 1 1440 3180 t (f\(p,q,2\) = f\(p,q,2\) - \( Exp\(t*\(y\(q\)-x\(p\)\)\)*\( y\(q\)-x\(p\) - 2*t*t \) + 1 \))12 4200 1 1440 3300 t (})1440 3420 w (})1320 3540 w 10 R f (The only change in the subroutine)5 1362 1 720 3756 t 10 CW f (BC)2107 3756 w 10 R f (of the previous example is the code for specifying the)9 2143 1 2252 3756 t 10 B f (bc)4420 3756 w 10 R f (s)4520 3756 w 10 CW f (Do j = 1, ny)4 720 1 1200 3936 t ({)1320 4056 w (Do i = 1, nx)4 720 1 1320 4176 t ({)1440 4296 w (bu\(i,j,1,1\) = 1; b\(i,j,1\) = U\(i,j,1\)-Exp\(t*\(x\(i\)-y\(j\)\)\))5 3300 1 1440 4416 t (bu\(i,j,2,2\) = 1; b\(i,j,2\) = U\(i,j,2\)-Exp\(t*\(y\(j\)-x\(i\)\)\))5 3300 1 1440 4656 t (})1440 4776 w (})1320 4896 w 10 R f (The)720 5112 w 10 CW f (HANDLE)900 5112 w 10 R f (subroutine simply checks the accuracy of the computed solution,)8 2588 1 1285 5112 t cleartomark showpage saveobj restore end %%EndPage: 11 50 %%Page: 12 51 DpostDict begin /saveobj save def mark 51 pagesetup 10 R f (\261 4-12 \261)2 333 1 2713 480 t 10 CW f (Subroutine HANDLE\(t0,U0,t,U,Nv,dt,tstop\))1 2400 1 1200 900 t (Real t0,U0\(Nv\),t,U\(Nv\),dt,tstop)1 1860 1 1200 1140 t (Integer Nv)1 600 1 1200 1260 t (Common / A7TGRM / kx,ix,nx, ky,iy,ny; Integer kx,ix,nx, ky,iy,ny)8 3840 1 1200 1500 t (Common / A7TGRP / errpar\(2\), Nu,mxq,myq; Real errpar; Integer Nu,mxq,myq)9 4320 1 1200 1740 t (Common / CSTAK / Ds\(500\); Double Precision Ds)7 2700 1 1200 1980 t ( The PORT Library stack and its aliases.)7 2400( #)1 300(Real Ws\(1000\))1 780 3 1200 2100 t (Real Rs\(1000\) ; Integer Is\(1000\) ; Complex Cs\(500\) ; Logical Ls\(1000\))10 4140 1 1200 2220 t (Equivalence \( Ds\(1\),Cs\(1\),Ws\(1\),Rs\(1\),Is\(1\),Ls\(1\) \))3 3060 1 1200 2340 t (Real errU)1 540 1 1200 2580 t (Integer i,I1MACH,ISTKGT,j,)1 1560 1 1200 2700 t (KA\(2\),ITA\(2\),NTA\(2\),IXA\(2\),NXA\(2\),MA\(2\),iFA,)1680 2820 w (ILUMD,ixs,iys,nxs,nys,iEwe)1740 2940 w (If \( t0 == t \))5 840 1 1200 3180 t ({)1320 3300 w (iwunit = 6)2 600 1 1320 3420 t (Write\(iwunit,9000\) t)1 1200 1 1320 3540 t (9000 Format\(" Restart for t =",1p4e10.2\))5 2400 1 1140 3660 t (Return)1320 3780 w (})1320 3900 w (Call ENTER\(1\))1 780 1 1200 4140 t (# Find the error in the solution at 2*kx * 2*ky points / mesh rectangle.)14 4320 1 1080 4380 t ( x search grid.)3 900( #)1 300(ixs = ILUMD\(Ws\(ix\),nx,2*kx,nxs\))2 1860 3 1200 4620 t ( y search grid.)3 900( #)1 300(iys = ILUMD\(Ws\(iy\),ny,2*ky,nys\))2 1860 3 1200 4740 t ( U search grid values.)4 1320( #)1 300(iEwe = ISTKGT\(Nu*nxs*nys,3\))2 1620 3 1200 4860 t ( The exact solution.)3 1200( #)1 300(Call EWE\(t,Ws\(ixs\),nxs,Ws\(iys\),nys,Ws\(iEwe\),Nu\))1 2820 3 1200 5100 t (KA\(1\) = kx; KA\(2\) = ky; ITA\(1\) = ix; ITA\(2\) = iy; NTA\(1\) = nx; NTA\(2\) = ny)17 4440 1 1200 5340 t (IXA\(1\) = ixs; IXA\(2\) = iys; NXA\(1\) = nxs; NXA\(2\) = nys)11 3240 1 1200 5460 t ( Get solution.)2 840( #)1 300(MA\(1\) = 0; MA\(2\) = 0)5 1200 3 1200 5580 t ( Approximate solution values.)3 1740( #)1 300(iFA = ISTKGT\(nxs*nys,3\))2 1380 3 1200 5820 t (Do j = 1, Nu)4 720 1 1200 6060 t ({)1320 6180 w (Call TSD1\(2,KA,Ws,ITA,NTA,U\(1+\(j-1\)*\(nx-kx\)*\(ny-ky\)\),)1 3180 1 1380 6300 t ( Evaluate them.)2 900( #)1 300(Ws,IXA,NXA,MA, Ws\(iFA\)\))1 1380 3 1980 6420 t ( Error in solution values.)4 1560( #)1 300(errU = 0)2 480 3 1320 6540 t (Do i = 1, nxs*nys)4 1020 1 1320 6660 t ({)1440 6780 w (errU = Max\(errU,Abs\(Ws\(iEwe+i-1+\(j-1\)*nxs*nys\)-Ws\(iFA+i-1\)\)\))2 3600 1 1440 6900 t (})1440 7020 w (iwunit = I1MACH\(2\))2 1080 1 1320 7260 t cleartomark showpage saveobj restore end %%EndPage: 12 51 %%Page: 13 52 DpostDict begin /saveobj save def mark 52 pagesetup 10 R f (\261 4-13 \261)2 333 1 2713 480 t 10 CW f (Write\(iwunit,9001\) t,j, errU)2 1680 1 1320 840 t (9001 Format\(" error in U\(.,",1p1e10.2,",",I2,"\) =", 1p4e10.2\))6 3660 1 1140 960 t (})1320 1080 w (Call LEAVE)1 600 1 1200 1320 t (Return)1200 1560 w (End)1200 1800 w 10 R f (where the following body for the subroutine)6 1764 1 720 2016 t 10 CW f (EWE)2509 2016 w 10 R f (computes the exact solution.)3 1138 1 2714 2016 t 10 CW f (Integer p,i,j)1 780 1 1200 2196 t (Do p = 1, Nu)4 720 1 1200 2436 t ({)1320 2556 w (Do i = 1, nx)4 720 1 1320 2676 t ({)1440 2796 w (Do j = 1, ny)4 720 1 1440 2916 t ({)1560 3036 w (U\(i,j,p\) = Exp\( \(-1\)**\(p+1\) * t * \( x\(i\) - y\(j\) \) \))12 3060 1 1560 3156 t (})1560 3276 w (})1440 3396 w (})1320 3516 w 10 R f (The output of the above program unit is)7 1590 1 720 3696 t 10 CW f ( 7.63e-06)1 600( 1\) =)2 300( 1.00e-02,)1 660(error in U\(.,)2 780 4 1140 3876 t ( 7.57e-06)1 600( 2\) =)2 300( 1.00e-02,)1 660(error in U\(.,)2 780 4 1140 3996 t ( 3.85e-04)1 600( 1\) =)2 300( 2.11e-01,)1 660(error in U\(.,)2 780 4 1140 4116 t ( 3.85e-04)1 600( 2\) =)2 300( 2.11e-01,)1 660(error in U\(.,)2 780 4 1140 4236 t ( 1.80e-03)1 600( 1\) =)2 300( 8.76e-01,)1 660(error in U\(.,)2 780 4 1140 4356 t ( 1.80e-03)1 600( 2\) =)2 300( 8.76e-01,)1 660(error in U\(.,)2 780 4 1140 4476 t ( 4.55e-04)1 600( 1\) =)2 300( 1.00e+00,)1 660(error in U\(.,)2 780 4 1140 4596 t ( 4.55e-04)1 600( 2\) =)2 300( 1.00e+00,)1 660(error in U\(.,)2 780 4 1140 4716 t ( of the stack allowed.)4 1320( 700000)1 540( /)1 120(used 9398)1 780 4 1140 4836 t 10 R f (The run-time for the above example was 431.5 seconds.)8 2234 1 720 5052 t ( it is the first non-trivial)5 958( Thus,)1 276( not exactly representable as a spline.)6 1494(Note that the solution of this problem is)7 1592 4 720 5208 t (example use of)2 604 1 720 5328 t 10 CW f (TTGR)1349 5328 w 10 R f (.)1589 5328 w 10 B f (Example 3 - Interfaces.)3 987 1 720 5568 t 10 R f (Consider the)1 508 1 970 5724 t 10 B f (pde)1503 5724 w 10 I f (f f)1 28 1 1220 6224 t 10 S f (= =)1 55 1 1313 6224 t 10 B f (u)1417 6224 w 7 I f (t t)1 20 1 1484 6244 t 10 S f (- -)1 55 1 1552 6224 t 10 I f (g g)1 50 1 1647 6224 t (a a)1 50 1 1220 6064 t 7 R f (\( 2 \))2 91 1 1281 6024 t 10 S f ( k)1 104(= =)1 55 2 1437 6064 t 10 R f (\()1628 6064 w 10 I f (x x)1 44 1 1669 6064 t 10 R f (,)1721 6064 w 10 I f (y y)1 44 1 1754 6064 t 10 R f (\))1806 6064 w 10 B f (u)1879 6064 w 7 I f (y y)1 31 1 1946 6084 t 10 I f (a a)1 50 1 1220 5894 t 7 R f (\( 1 \))2 91 1 1281 5854 t 10 S f ( k)1 104(= =)1 55 2 1437 5894 t 10 R f (\()1628 5894 w 10 I f (x x)1 44 1 1669 5894 t 10 R f (,)1721 5894 w 10 I f (y y)1 44 1 1754 5894 t 10 R f (\))1806 5894 w 10 B f (u)1879 5894 w 7 I f (x x)1 31 1 1946 5914 t 10 R f (\(A4.5\))4777 6064 w (where)720 6404 w 10 S f (k \272)1 151 1 1220 6769 t (\354)1420 6582 w (\357)1420 6682 w (\355)1420 6782 w (\357)1420 6882 w (\356)1420 6982 w 10 R f (1)1535 6919 w 10 I f (/ /)1 28 1 1593 6919 t 10 R f (32)1629 6919 w 10 S f (< <)1 55 1 1745 6919 t 10 I f (y y)1 44 1 1816 6919 t 10 S f (\243)1868 6919 w 10 R f (3)1931 6919 w (1)1535 6779 w 10 I f (/ /)1 28 1 1593 6779 t 10 R f (21)1629 6779 w 10 S f (< <)1 55 1 1745 6779 t 10 I f (y y)1 44 1 1816 6779 t 10 S f (\243)1868 6779 w 10 R f (2)1931 6779 w (10)1535 6639 w 10 S f (\243)1643 6639 w 10 I f (y y)1 44 1 1706 6639 t 10 S f (\243)1758 6639 w 10 R f (1)1821 6639 w (,)1989 6769 w (with)720 7154 w 10 B f (bc)923 7154 w 10 R f (s on the bottom and top)5 942 1 1023 7154 t cleartomark showpage saveobj restore end %%EndPage: 13 52 %%Page: 14 53 DpostDict begin /saveobj save def mark 53 pagesetup 10 R f (\261 4-14 \261)2 333 1 2713 480 t 10 B f (b)1220 840 w 10 S f (= =)1 55 1 1325 840 t 10 I f (u u)1 50 1 1429 840 t 7 I f (y y)1 31 1 1490 860 t 10 R f (\(A4.6a\))4733 840 w (and those on the sides)4 877 1 720 1020 t 10 B f (b)1220 1200 w 10 S f (= =)1 55 1 1325 1200 t 10 B f (u)1429 1200 w 10 S f (- -)1 55 1 1534 1200 t 10 I f (s s)1 39 1 1638 1200 t 10 R f (\()1685 1200 w 10 I f (t t)1 28 1 1726 1200 t 10 R f (,)1762 1200 w 10 I f (x x)1 44 1 1795 1200 t 10 R f (,)1847 1200 w 10 I f (y y)1 44 1 1880 1200 t 10 R f (\) \(A4.6b\))1 3108 1 1932 1200 t (The following program solves \(A4.5\)-\(A4.6\) using)5 2108 1 720 1416 t 10 CW f (TTGR)2868 1416 w 10 R f ( \()1 74(, with a linear B-spline)4 973 2 3108 1416 t 10 I f (k k)1 44 1 4155 1416 t 10 S f (= =)1 55 1 4248 1416 t 10 R f (2\) over a spatial)3 688 1 4352 1416 t ( consisting of 3 equally spaced, distinct points, with the time evolution carried out to roughly)15 3733( 0 , 1 \))4 190( \()1 59(mesh on)1 338 4 720 1536 t (10)720 1656 w 7 S f (- -)1 39 1 831 1616 t 7 R f (2)881 1616 w 10 R f ( at each time-step is printed out to confirm the accuracy of the numerical)13 2954( error)1 221( The)1 208(relative accuracy.)1 705 4 952 1656 t (solution.)720 1776 w (The main program uses two Port Library subroutines)7 2117 1 970 1932 t 10 S f (\267)970 2088 w 10 CW f (ILUMB)1077 2088 w 10 R f (makes a B-spline mesh by putting a uniform mesh on each of a set of contiguous intervals,)16 3637 1 1403 2088 t ( 3 ].)2 172( ,)1 33( 2)1 82( and [)2 227( 2 ])2 147( ,)1 33( 1)1 82( [)1 58( 1 ],)2 172( ,)1 33( 0)1 82(in this case 3 points each on [)7 1179 12 970 2208 t 10 S f (\267)970 2364 w 10 CW f (IMMM)1088 2364 w 10 R f (makes)1365 2364 w 10 S f ( =)1 0(h =)1 164 2 1657 2364 t 10 R f (1 a mesh point of multiplicity)5 1247 1 1870 2364 t 10 CW f (k-1)3154 2364 w 10 R f (, so that)2 338 1 3334 2364 t 10 B f (u)3709 2364 w 10 R f (\()3797 2364 w 10 I f (x x)1 44 1 3862 2364 t 10 R f (,)3914 2364 w 10 I f (y y)1 44 1 3971 2364 t 10 R f (,)4023 2364 w 10 I f (t t)1 28 1 4080 2364 t 10 R f (\) can have its partials)4 900 1 4140 2364 t (with respect to)2 606 1 970 2484 t 10 I f (y y)1 44 1 1610 2484 t 10 R f (discontinuous, see appendix 1.)3 1251 1 1688 2484 t 10 I f (k k)1 44 1 2998 2484 t 10 S f (- -)1 55 1 3066 2484 t 10 R f ( be added to the mesh to accom-)7 1345(2 points must)2 558 2 3137 2484 t ( dimension array can be added to, the mesh is)9 1864( it is dangerous to assume that a fixed)8 1536( Since)1 276(plish this.)1 394 4 970 2604 t (put on the stack by)4 763 1 970 2724 t 10 CW f (ILUMB)1760 2724 w 10 R f ( mesh is put on the Port stack and)8 1352( The)1 206(where it may be safely lengthened.)5 1396 3 2086 2724 t (a pointer to it is returned.)5 1010 1 970 2844 t (The main program is)3 835 1 970 3000 t cleartomark showpage saveobj restore end %%EndPage: 14 53 %%Page: 15 54 DpostDict begin /saveobj save def mark 54 pagesetup 10 R f (\261 4-15 \261)2 333 1 2713 480 t 10 CW f (# Main)1 360 1 1080 900 t (# To solve the layered heat equation, with kappa = 1, 1/2, 1/3,)12 3780 1 1080 1140 t ( . \( kappa\(x,y\) * grad U \) = Ut + g)11 2100(# div)1 420 2 1080 1380 t (Real tstart,tstop,dt,Lx,Rx,yb\(4\))1 1920 1 1200 1620 t (Real errpar\(2\))1 840 1 1200 1740 t (Integer Nu,kx,ix,nx, ky,iy,ny,ISTKGT, IUMB, ILUMB,iU,ndx,ndy,IMMM ,i)5 4080 1 1200 1860 t (External AF,BC,HANDLE)1 1260 1 1200 1980 t (Common / CSTAK / Ds\(350000\); Double Precision Ds)7 2880 1 1200 2340 t ( The PORT Library stack and its aliases.)7 2400( #)1 300(Real Ws\(1000\))1 780 3 1200 2460 t (Real Rs\(1000\) ; Integer Is\(1000\) ; Complex Cs\(500\) ; Logical Ls\(1000\))10 4140 1 1200 2580 t (Equivalence \( Ds\(1\),Cs\(1\),Ws\(1\),Rs\(1\),Is\(1\),Ls\(1\) \))3 3060 1 1200 2700 t ( Initialize the PORT Library stack length.)6 2520( #)1 300(Call ISTKIN\(350000,4\))1 1260 3 1200 2940 t (Call ENTER\(1\))1 780 1 1200 3180 t (Nu = 1)2 360 1 1200 3420 t (Lx = 0; Rx = 1)5 840 1 1200 3660 t (Do i = 1, 4 { yb\(i\) = i-1 })9 1620 1 1200 3780 t (kx = 2; ky = 2)5 840 1 1200 4020 t (ndx = 3; ndy = 3)5 960 1 1200 4260 t (tstart = 0; tstop = 1; dt = 1)8 1740 1 1200 4500 t (errpar\(1\) = 1e-2; errpar\(2\) = 1e-4)5 2040 1 1200 4740 t ( Uniform grid.)2 840( #)1 300(ix = IUMB\(Lx,Rx,ndx,kx,nx\))2 1560 3 1200 4980 t ( Uniform grid.)2 840( #)1 300(iy = ILUMB\(yb,4,ndy,ky,ny\))2 1560 3 1200 5100 t ( Make mult = ky-1.)4 1080( #)1 300(iy = IMMM \(iy,ny,yb\(2\),ky-1\))3 1680 3 1200 5220 t ( Make mult = ky-1.)4 1080( #)1 300(iy = IMMM \(iy,ny,yb\(3\),ky-1\))3 1680 3 1200 5340 t ( Space for the solution.)4 1440( #)1 300(iU = ISTKGT\(Nu*\(nx-kx\)*\(ny-ky\),3\))2 1980 3 1200 5580 t (Call SETR\(Nu*\(nx-kx\)*\(ny-ky\),0e0,Ws\(iU\)\))1 2400 1 1200 5700 t (Call TTGR \(Ws\(iU\),Nu,kx,Ws\(ix\),nx, ky,Ws\(iy\),ny, tstart,tstop, dt,)5 3960 1 1200 5940 t (AF,BC,)1860 6060 w (errpar,)1860 6180 w (HANDLE\))1860 6300 w (Call LEAVE)1 600 1 1200 6540 t (Call WRAPUP)1 660 1 1200 6780 t (Stop)1200 7020 w (End)1200 7260 w cleartomark showpage saveobj restore end %%EndPage: 15 54 %%Page: 16 55 DpostDict begin /saveobj save def mark 55 pagesetup 10 R f (\261 4-16 \261)2 333 1 2713 480 t (The body of the)3 635 1 720 840 t 10 CW f (AF)1380 840 w 10 R f (subroutine for specifying the)3 1151 1 1525 840 t 10 B f (pde)2701 840 w 10 R f (\(A4.5\) is)1 355 1 2882 840 t 10 CW f (Real kappa)1 600 1 1200 1020 t (Integer p,q,i)1 780 1 1200 1140 t (Do i = 1, Nu)4 720 1 1200 1380 t ({)1320 1500 w (Do q = 1, ny)4 720 1 1320 1620 t ({)1440 1740 w (Do p = 1, nx)4 720 1 1440 1860 t ({)1560 1980 w ( y\(q\) < 1 \) { kappa = 1 })9 1500(If \()1 540 2 1560 2100 t (Else If \( y\(q\) < 2 \) { kappa = 0.5 })11 2160 1 1560 2220 t ( kappa = 1/3e0 })4 960(Else {)1 1320 2 1560 2340 t (a\(p,q,i,1\) = kappa*Ux\(p,q,i\); AUx\(p,q,i,i,1\) = kappa)5 3120 1 1560 2580 t (a\(p,q,i,2\) = kappa*Uy\(p,q,i\); AUy\(p,q,i,i,2\) = kappa)5 3120 1 1560 2700 t (f\(p,q,i\) = Ut\(p,q,i\); FUt\(p,q,i,i\) = 1)5 2280 1 1560 2940 t (f\(p,q,i\) = f\(p,q,i\) - y\(q\)/kappa)4 1920 1 1560 3180 t (If \( 1 < y\(q\) & y\(q\) < 2 \) { f\(p,q,i\) = f\(p,q,i\) + 1 })16 3240 1 1560 3420 t (If \( 2 < y\(q\) & y\(q\) < 3 \) { f\(p,q,i\) = f\(p,q,i\) + 3 })16 3240 1 1560 3540 t (})1560 3660 w (})1440 3780 w (})1320 3900 w 10 R f (The body of the)3 635 1 720 4116 t 10 CW f (BC)1380 4116 w 10 R f (subroutine for specifying the)3 1151 1 1525 4116 t 10 B f (bc)2701 4116 w 10 R f (s \(A4.6\) is)2 419 1 2801 4116 t 10 CW f (Do j = 1, ny)4 720 1 1200 4296 t ({)1320 4416 w (Do i = 1, nx)4 720 1 1320 4536 t ({)1440 4656 w ( Left or right.)3 900( #)1 300(If \( x\(i\) == Lx | x\(i\) == Rx \))9 1800 3 1440 4776 t ({)1560 4896 w ( Neumann BCs.)2 780( #)1 300(BUx\(i,j,1,1\) = 1; B\(i,j,1\) = Ux\(i,j,1\))5 2280 3 1560 5016 t (})1560 5136 w ( Bottom.)1 480( #)1 300(Else If \( y\(j\) == Ly \))6 1320 3 1440 5256 t ({)1560 5376 w (B\(i,j,1\) = U\(i,j,1\); BU\(i,j,1,1\) = 1)5 2160 1 1560 5496 t (})1560 5616 w ( Top.)1 300(Else #)1 540 2 1440 5736 t ({)1560 5856 w (B\(i,j,1\) = U\(i,j,1\)-6*t; BU\(i,j,1,1\) = 1)5 2400 1 1560 5976 t (})1560 6096 w (})1440 6216 w (})1320 6336 w 10 R f ( in the)2 260(There is no change)3 768 2 720 6552 t 10 CW f (HANDLE)1778 6552 w 10 R f (or)2168 6552 w 10 CW f (GERR)2281 6552 w 10 R f ( only change in the subroutine)5 1232( The)1 210(subroutines of example 1.)3 1047 3 2551 6552 t 10 CW f (EWE)720 6672 w 10 R f (, for computing)2 619 1 900 6672 t 10 I f (u u)1 50 1 1544 6672 t 10 R f (, of example 1 is the code for computing)8 1617 1 1594 6672 t 10 I f (u u)1 50 1 3236 6672 t 10 CW f ( })1 120( t*y\(j\))1 480( < 1 \) { U\(i,j,p\) =)6 1140( y\(j\))1 600(If \()1 240 5 1560 6852 t (Else If \( y\(j\) < 2 \) { U\(i,j,p\) = 2*t*y\(j\)-t })11 2760 1 1560 6972 t ( U\(i,j,p\) = 3*t*y\(j\)-3*t })4 1560(Else {)1 1320 2 1560 7092 t 10 R f (The output from this program is)5 1280 1 720 7308 t cleartomark showpage saveobj restore end %%EndPage: 16 55 %%Page: 17 56 DpostDict begin /saveobj save def mark 56 pagesetup 10 R f (\261 4-17 \261)2 333 1 2713 480 t 10 CW f ( 4.77e-07)1 600( =)1 120( 1.00e+00\))1 660(error in U\(.,)2 780 4 1140 900 t ( of the stack allowed.)4 1320( 700000)1 540( /)1 120(used 1260)1 780 4 1140 1020 t 10 R f ( run-time for the above exam-)5 1209( The)1 208( been solved to rounding error.)5 1246(and we see that \(A4.5\)-\(A4.6\) has indeed)6 1657 4 720 1200 t (ple was 10.9 seconds.)3 868 1 720 1320 t 10 B f (Example 4 - Non-Rectangular Domains.)4 1702 1 720 1560 t 10 R f (Consider the)1 508 1 970 1716 t 10 B f (pde)1503 1716 w 10 I f (f f)1 28 1 1220 2456 t 10 S f (= =)1 55 1 1313 2456 t 10 B f (u)1417 2456 w 7 I f (t t)1 20 1 1484 2476 t 10 S f (- -)1 55 1 1552 2456 t 10 I f (g g)1 50 1 1647 2456 t (a a)1 50 1 1220 2246 t 7 R f (\( 2 \))2 91 1 1281 2206 t 10 S f (= =)1 55 1 1437 2246 t 10 B f (u)1565 2246 w 7 I f (y y)1 31 1 1632 2266 t 10 S f (- -)1 55 1 1711 2246 t 10 R f (10)1834 2316 w 10 B f (u)1831 2166 w 7 I f (x x)1 31 1 1898 2186 t 10 S1 f (_ __)1 136 1 1816 2216 t 10 I f (a a)1 50 1 1220 1956 t 7 R f (\( 1 \))2 91 1 1281 1916 t 10 S f (= =)1 55 1 1437 1956 t 10 B f (u)1565 1956 w 7 I f (x x)1 31 1 1632 1976 t 10 S f (- -)1 55 1 1687 1956 t 10 R f (10)1786 2026 w 10 B f (u)1783 1876 w 7 I f (y y)1 31 1 1850 1896 t 10 S1 f (_ __)1 136 1 1768 1926 t 10 R f (\(A4.7\))4777 2246 w (on the domain)2 572 1 720 2672 t (y)2021 2895 w (x)3914 4283 w (\(0,-1\))2103 5208 w 2047 5146 2972 4221 Dl 2046 4221 1 1 De 2972 3296 2972 4221 Dl 2971 3296 2046 4221 Dl 2046 4223 2046 4248 Dl 3812 4221 2046 4221 Dl 2046 4248 2046 4253 Dl 2046 4252 2046 4242 Dl 2046 4262 2046 4242 Dl 2046 4263 1 1 De 2046 4262 2046 4221 Dl 2046 4263 2046 2960 Dl (\(0,0\))1783 4283 w (\(1,1\))2961 3274 w (\(1,0\))2961 4199 w 2087 3001 2046 2960 Dl 2046 2961 2005 3002 Dl 3811 4222 3770 4263 Dl 3728 4263 1 1 De 3811 4220 3770 4179 Dl 2046 4246 2046 4221 Dl 2046 4248 1 1 De 2046 4221 1 1 De 2046 5146 2046 4221 Dl (with)720 5446 w 10 B f (bc)923 5446 w 10 R f (s on the bottom and top)5 942 1 1023 5446 t 10 B f (b)1220 5626 w 10 S f (= =)1 55 1 1325 5626 t 10 I f (u u)1 50 1 1429 5626 t 7 I f (N N)1 47 1 1490 5646 t 10 S f (- -)1 55 1 1561 5626 t 10 I f (n n)1 50 1 1632 5626 t 10 R f (\()1690 5626 w 10 I f (x x)1 44 1 1731 5626 t 10 R f (,)1783 5626 w 10 I f (y y)1 44 1 1816 5626 t 10 R f (\) \(A4.8a\))1 3172 1 1868 5626 t (and those on the sides)4 877 1 720 5806 t 10 B f (b)1220 5986 w 10 S f (= =)1 55 1 1325 5986 t 10 B f (u)1429 5986 w 10 S f (- -)1 55 1 1534 5986 t 10 I f (s s)1 39 1 1638 5986 t 10 R f (\()1685 5986 w 10 I f (t t)1 28 1 1726 5986 t 10 R f (,)1762 5986 w 10 I f (x x)1 44 1 1795 5986 t 10 R f (,)1847 5986 w 10 I f (y y)1 44 1 1880 5986 t 10 R f (\) \(A4.8b\))1 3108 1 1932 5986 t (where)720 6166 w 10 I f (g g)1 50 1 988 6166 t 10 R f (,)1038 6166 w 10 I f (n n)1 50 1 1088 6166 t 10 R f (and)1163 6166 w 10 I f (s s)1 39 1 1332 6166 t 10 R f (are chosen so that the solution is)6 1299 1 1396 6166 t 10 I f (u u)1 50 1 2720 6166 t 10 S f (= =)1 55 1 2819 6166 t 10 I f ( y)1 0( y)1 76( x)1 0( x)1 76(t t)1 28 5 2923 6166 t 10 R f (.)3103 6166 w (The following program solves the)4 1390 1 970 6322 t 10 B f (pde)2394 6322 w 10 R f (-)2550 6322 w 10 B f (bc)2583 6322 w 10 R f ( B-splines over a)3 704(combination \(A4.7\)-\(A4.8\), using cubic)3 1619 2 2717 6322 t ( with the time-evolution carried out to 10)7 1695( 0 , 1 \),)4 215( distinct points on \()4 801(mesh consisting of 3 equally spaced,)5 1505 4 720 6442 t 7 S f (- -)1 39 1 4947 6402 t 7 R f (2)4997 6402 w 10 R f ( accuracy of the computed solu-)5 1292( error at each time-step is printed out to confirm the)10 2083( The)1 207(absolute accuracy.)1 738 4 720 6562 t ( main program is)3 680(tion. The)1 386 2 720 6682 t cleartomark showpage saveobj restore end %%EndPage: 17 56 %%Page: 18 57 DpostDict begin /saveobj save def mark 57 pagesetup 10 R f (\261 4-18 \261)2 333 1 2713 480 t 10 CW f (# Main)1 360 1 1080 900 t (# To solve the linear heat equation)6 2100 1 1080 1140 t ( \) = Ut - x*y)5 780( Uy)1 240( . \( Ux - 0.1 * Uy , 0.1*Ux +)10 1740(# grad)1 480 4 1080 1380 t (# with solution u == t*x*y on [0,+1]**2, exact for k = 4,)12 3420 1 1080 1620 t (# with tilted top and bottom, normal BCs there.)8 2820 1 1080 1740 t (Real tstart,tstop,dt,Lx,Rx,Ly,Ry)1 1920 1 1200 1980 t (Real errpar\(2\))1 840 1 1200 2100 t (Integer Nu,kx,ix,nx, ky,iy,ny,ISTKGT, IUMB,iU,ndx,ndy)3 3180 1 1200 2220 t (External AF,BC,HANDLE)1 1260 1 1200 2340 t (Common / CSTAK / Ds\(350000\); Double Precision Ds)7 2880 1 1200 2700 t ( The PORT Library stack and its aliases.)7 2400( #)1 300(Real Ws\(1000\))1 780 3 1200 2820 t (Real Rs\(1000\) ; Integer Is\(1000\) ; Complex Cs\(500\) ; Logical Ls\(1000\))10 4140 1 1200 2940 t (Equivalence \( Ds\(1\),Cs\(1\),Ws\(1\),Rs\(1\),Is\(1\),Ls\(1\) \))3 3060 1 1200 3060 t ( Initialize the PORT Library stack length.)6 2520( #)1 300(Call ISTKIN\(350000,4\))1 1260 3 1200 3300 t (Call ENTER\(1\))1 780 1 1200 3540 t (Nu = 1)2 360 1 1200 3780 t (Lx = 0; Rx = +1)5 900 1 1200 4020 t (Ly = 0; Ry = +1)5 900 1 1200 4140 t (kx = 4; ky = 4)5 840 1 1200 4380 t (ndx = 3; ndy = 3)5 960 1 1200 4620 t (tstart = 0; tstop = 1; dt = 1)8 1740 1 1200 4860 t (errpar\(1\) = 1e-2; errpar\(2\) = 1e-4)5 2040 1 1200 5100 t ( Uniform grid.)2 840( #)1 300(ix = IUMB\(Lx,Rx,ndx,kx,nx\))2 1560 3 1200 5340 t ( Uniform grid.)2 840( #)1 300(iy = IUMB\(Ly,Ry,ndy,ky,ny\))2 1560 3 1200 5460 t ( Space for the solution.)4 1440( #)1 300(iU = ISTKGT\(Nu*\(nx-kx\)*\(ny-ky\),3\))2 1980 3 1200 5700 t (Call SETR\(Nu*\(nx-kx\)*\(ny-ky\),0e0,Ws\(iU\)\))1 2400 1 1200 5820 t (Call TTGR \(Ws\(iU\),Nu,kx,Ws\(ix\),nx, ky,Ws\(iy\),ny, tstart,tstop, dt,)5 3960 1 1200 6060 t (AF,BC,)1860 6180 w (errpar,)1860 6300 w (HANDLE\))1860 6420 w (Call LEAVE)1 600 1 1200 6660 t (Call WRAPUP)1 660 1 1200 6900 t (Stop)1200 7140 w cleartomark showpage saveobj restore end %%EndPage: 18 57 %%Page: 19 58 DpostDict begin /saveobj save def mark 58 pagesetup 10 R f (\261 4-19 \261)2 333 1 2713 480 t 10 CW f (End)1200 840 w 10 R f (The)720 1056 w 10 CW f (AF)900 1056 w 10 R f (subroutine uses one Port subroutine)4 1427 1 1045 1056 t 10 S f (\267)970 1212 w 10 CW f (SETERR)1076 1212 w 10 R f (is used to set an error state if)7 1145 1 1461 1212 t 10 I f (n n)1 50 1 2631 1212 t 7 I f (x x)1 31 1 2692 1232 t 10 I f (n n)1 50 1 2763 1212 t 7 I f (y y)1 31 1 2824 1232 t 10 R f (requires more space than reserved, in this case 100.)8 2050 1 2888 1212 t (The body of the subroutine)4 1082 1 970 1368 t 10 CW f (AF)2077 1368 w 10 R f (for \(A4.7\) is)2 496 1 2222 1368 t 10 CW f (Real xx\(100\),yy\(100\),D\(600\),x,y)1 1860 1 1200 1548 t (Integer p,q,i)1 780 1 1200 1668 t (External LR,BT)1 840 1 1200 1788 t (If \( nx*ny > 100 \) { Call SETERR\("AF - nx*ny .gt. 100",19,1,2\) })13 3840 1 1200 2028 t (Call BTMAP\(t,xi,yi,nx,ny, LR,BT, xx,yy,D\))3 2460 1 1200 2268 t ( Map into \(x,y\).)3 960( #)1 300(Call TTGRU\(nx,ny,D,Ux,Uy,Ut,Nu\))1 1860 3 1200 2508 t (Do i = 1, Nu)4 720 1 1200 2748 t ({)1320 2868 w (Do q = 1, ny)4 720 1 1320 2988 t ({)1440 3108 w (Do p = 1, nx)4 720 1 1440 3228 t ({)1560 3348 w (x = xx\(p+\(q-1\)*nx\); y = yy\(p+\(q-1\)*nx\))5 2280 1 1560 3468 t (a\(p,q,i,1\) = Ux\(p,q,i\) - .1*Uy\(p,q,i\))4 2220 1 1560 3708 t (a\(p,q,i,2\) = Uy\(p,q,i\) + .1*Ux\(p,q,i\))4 2220 1 1560 3828 t (AUx\(p,q,i,i,1\) = 1; AUy\(p,q,i,i,2\) = 1)5 2280 1 1560 3948 t (AUy\(p,q,i,i,1\) = -.1; AUx\(p,q,i,i,2\) = +.1)5 2520 1 1560 4068 t (F\(p,q,1\) = Ut\(p,q,1\) - x*y; FUt\(p,q,1,1\) = 1)7 2640 1 1560 4308 t (})1560 4428 w (})1440 4548 w (})1320 4668 w ( Map into \(xi,eta\).)3 1140( #)1 300(Call TTGRG\(nx,ny,D,Nu, A,AU,AUx,AUy, F,FU,FUx,FUy\))3 3000 3 1200 4908 t 10 R f (where the subroutines)2 876 1 720 5124 t 10 CW f (LR)1621 5124 w 10 R f (and)1766 5124 w 10 CW f (BT)1935 5124 w 10 R f ( body of the subroutine)4 927( The)1 205(will be described shortly.)3 1010 3 2080 5124 t 10 CW f (BC)4247 5124 w 10 R f (for \(A4.8\) is)2 496 1 4392 5124 t cleartomark showpage saveobj restore end %%EndPage: 19 58 %%Page: 20 59 DpostDict begin /saveobj save def mark 59 pagesetup 10 R f (\261 4-20 \261)2 333 1 2713 480 t 10 CW f (Real xx\(100\),yy\(100\),D\(600\),x,y)1 1860 1 1200 900 t (Integer i,j)1 660 1 1200 1020 t (External LR,BT)1 840 1 1200 1140 t (If \( nx*ny > 100 \) { Call SETERR\("BC - nx*ny .gt. 100",19,1,2\) })13 3840 1 1200 1380 t (Call BTMAP\(t,xi,yi,nx,ny, LR,BT, xx,yy,D\))3 2460 1 1200 1620 t ( Map into \(x,y\).)3 960( #)1 300(Call TTGRU\(nx,ny,D,Ux,Uy,Ut,Nu\))1 1860 3 1200 1860 t (Do j = 1, ny)4 720 1 1200 2100 t ({)1320 2220 w (Do i = 1, nx)4 720 1 1320 2340 t ({)1440 2460 w (x = xx\(i+\(j-1\)*nx\); y = yy\(i+\(j-1\)*nx\))5 2280 1 1440 2580 t ( Left or right.)3 900( #)1 300(If \( xi\(i\) == Lx | xi\(i\) == Rx \))9 1920 3 1440 2820 t ({)1560 2940 w (BU\(i,j,1,1\) = 1; B\(i,j,1\) = U\(i,j,1\)-t*x*y)5 2520 1 1560 3060 t (})1560 3180 w ( Bottom.)1 480( #)1 300(Else If \( yi\(j\) == Ly \))6 1380 3 1440 3300 t ({)1560 3420 w (B\(i,j,1\) = \(Ux\(i,j,1\)-t*y\) - \(Uy\(i,j,1\)-t*x\))4 2640 1 1560 3540 t ( Normal is \(1,-1\).)3 1080( #)1 300(BUx\(i,j,1,1\) = 1; BUy\(i,j,1,1\) = -1)5 2100 3 1560 3660 t (})1560 3780 w ( Top.)1 300(Else #)1 540 2 1440 3900 t ({)1560 4020 w (B\(i,j,1\) = -\(Ux\(i,j,1\)-t*y\) + \(Uy\(i,j,1\)-t*x\))4 2700 1 1560 4140 t ( Normal is \(-1,1\).)3 1080( #)1 300(BUx\(i,j,1,1\) = -1; BUy\(i,j,1,1\) = 1)5 2100 3 1560 4260 t (})1560 4380 w (})1440 4500 w (})1320 4620 w ( Map into \(xi,eta\).)3 1140( #)1 300(Call TTGRB\(nx,ny,D, Nu, BUx,BUy,BUt\))3 2160 3 1200 4860 t 10 R f (The body of the subroutines)4 1161 1 720 5076 t 10 CW f (HANDLE)1916 5076 w 10 R f (and)2311 5076 w 10 CW f (GERR)2490 5076 w 10 R f ( the error is same as before.)6 1165(for computing and printing)3 1110 2 2765 5076 t ( in the subroutine)3 706(The only change)2 673 2 720 5196 t 10 CW f (EWE)2127 5196 w 10 R f (for computing)1 572 1 2335 5196 t 10 I f (u u)1 50 1 2935 5196 t 10 R f (in the previous example is the code for computing)8 2027 1 3013 5196 t 10 I f (u u)1 50 1 720 5316 t 10 R f (.)770 5316 w cleartomark showpage saveobj restore end %%EndPage: 20 59 %%Page: 21 60 DpostDict begin /saveobj save def mark 60 pagesetup 10 R f (\261 4-21 \261)2 333 1 2713 480 t 10 CW f (Real t,xi\(nx\),yi\(ny\),U\(nx,ny,Nu\))1 1920 1 1200 1020 t (Integer nx,ny,Nu)1 960 1 1200 1140 t (Real xx\(1000\),yy\(1000\),D\(6000\),x,y)1 2040 1 1200 1380 t (Integer p,i,j)1 780 1 1200 1500 t (External LR,BT)1 840 1 1200 1620 t (If \( ny > 1000 \) { Call SETERR\("EWE - ny .gt. 1000",18,1,2\) })13 3660 1 1200 1860 t (Do p = 1, Nu)4 720 1 1200 2100 t ({)1320 2220 w (Do i = 1, nx)4 720 1 1320 2340 t ({)1440 2460 w (Call BTMAP\(t,xi\(i\),yi,1,ny, LR,BT, xx,yy,D\))3 2580 1 1500 2580 t (Do j = 1, ny)4 720 1 1440 2820 t ({)1560 2940 w (x = xx\(j\); y = yy\(j\))5 1200 1 1560 3060 t (U\(i,j,p\) = t*x*y)2 960 1 1560 3300 t (})1560 3420 w (})1440 3540 w (})1320 3660 w 10 R f (The subroutine)1 602 1 720 3876 t 10 CW f (LR)1347 3876 w 10 R f (is)1492 3876 w 10 CW f (Subroutine LR\(t,Lx,Rx,Lxt,Rxt\))1 1800 1 1200 4056 t (# To get the L and R end-points of the mapping in x.)12 3120 1 1080 4296 t (Real t,Lx,Rx,Lxt,Rxt)1 1200 1 1200 4536 t ( 1)1 120( =)1 180( 0; Rx)2 360(Lx =)1 300 4 1200 4776 t (Lxt = 0; Rxt = 0)5 960 1 1200 4896 t (Return)1200 5136 w (End)1200 5376 w 10 R f (and the subroutine)2 738 1 720 5556 t 10 CW f (BT)1483 5556 w 10 R f (is)1628 5556 w cleartomark showpage saveobj restore end %%EndPage: 21 60 %%Page: 22 61 DpostDict begin /saveobj save def mark 61 pagesetup 10 R f (\261 4-22 \261)2 333 1 2713 480 t 10 CW f (Subroutine BT\(t,x,f,g,fx,gx,ft,gt\))1 2040 1 1200 900 t (# To get the bottom and top of mapping in y.)10 2640 1 1080 1140 t (Real t,x,f,g,fx,gx,ft,gt)1 1440 1 1200 1380 t ( x)1 120( =)1 180( -1 + x; g)4 600(f =)1 240 4 1200 1620 t ( = 0)2 240( gt)1 480(ft = 0;)2 420 3 1200 1740 t ( = 1)2 240( gx)1 480(fx = 1;)2 420 3 1200 1860 t (Return)1200 2100 w (End)1200 2340 w 10 R f (The output of this program is)5 1169 1 720 2556 t 10 CW f ( 8.94e-08)1 600( =)1 120( 1.00e+00\))1 660(error in U\(.,)2 780 4 1140 2736 t ( of the stack allowed.)4 1320( 700000)1 540( /)1 120(used 3388)1 780 4 1140 2856 t 10 R f (The run-time for the above example was 28.1 seconds.)8 2184 1 720 3036 t 10 B f (Example 5 - A Static Problem.)5 1293 1 720 3276 t 10 R f (Consider the)1 508 1 970 3432 t 10 B f (pde)1503 3432 w 10 I f (f f)1 28 1 1220 3932 t 10 S f (= =)1 55 1 1313 3932 t 10 R f (0)1417 3932 w 10 I f (a a)1 50 1 1220 3772 t 7 R f (\( 2 \))2 91 1 1281 3732 t 10 S f (= =)1 55 1 1437 3772 t 10 B f (u)1565 3772 w 7 I f (y y)1 31 1 1632 3792 t 10 I f (a a)1 50 1 1220 3602 t 7 R f (\( 1 \))2 91 1 1281 3562 t 10 S f (= =)1 55 1 1437 3602 t 10 B f (u)1565 3602 w 7 I f (x x)1 31 1 1632 3622 t 10 R f (\(A4.9\))4777 3772 w ( 0 , 1 ])4 190(on the domain [)3 630 2 720 4128 t 10 S f (\264)1556 4128 w 10 R f ( with)1 203([ 0 , 1 ],)4 248 2 1619 4128 t 10 B f (bc)2095 4128 w 10 R f (s on the bottom)3 620 1 2195 4128 t 10 B f (b)1220 4308 w 10 S f (= =)1 55 1 1325 4308 t 10 I f (u u)1 50 1 1429 4308 t 7 I f (y y)1 31 1 1490 4328 t 10 R f (\(A4.10a\))4683 4308 w (and on the other sides)4 871 1 720 4488 t 10 B f (b)1220 4668 w 10 S f (= =)1 55 1 1325 4668 t 10 B f (u)1429 4668 w 10 S f (- -)1 55 1 1534 4668 t 10 I f (s s)1 39 1 1638 4668 t 10 R f (\()1685 4668 w 10 I f (t t)1 28 1 1726 4668 t 10 R f (,)1762 4668 w 10 I f (x x)1 44 1 1795 4668 t 10 R f (,)1847 4668 w 10 I f (y y)1 44 1 1880 4668 t 10 R f (\) \(A4.10b\))1 3108 1 1932 4668 t (where)720 4848 w 10 I f (s s)1 39 1 988 4848 t 10 R f (is chosen so that the solution is)6 1245 1 1052 4848 t 10 I f (u u)1 50 1 2322 4848 t 10 S f (= =)1 55 1 2421 4848 t 10 I f ( l)1 0( al)1 28( ea)1 50(R Re)1 105 4 2525 4848 t 10 R f (\()2716 4848 w 10 I f ( og g)2 50( lo)1 50( l)1 60(z z)1 39 4 2757 4848 t 10 R f (\()2964 4848 w 10 I f (z z)1 39 1 3005 4848 t 10 R f (\) \).)1 99 1 3052 4848 t (The following program solves the)4 1374 1 970 5004 t 10 B f (pde)2374 5004 w 10 R f (-)2530 5004 w 10 B f (bc)2563 5004 w 10 R f ( a)1 75(combination \(A4.9\)-\(A4.10\), using cubic B-splines over)5 2272 2 2693 5004 t ( carried out to)3 566( with the time-evolution)3 976( 0 , 1 \),)4 215(mesh consisting of 3 non-uniformly spaced, distinct points on \()9 2563 4 720 5124 t (10)720 5284 w 7 S f (- -)1 39 1 831 5244 t 7 R f (2)881 5244 w 10 R f ( non-uniform mesh used is)4 1086( The)1 210(absolute accuracy.)1 741 3 953 5284 t 10 I f (x x)1 44 1 3020 5284 t 7 I f (i i)1 20 1 3075 5304 t 10 S f (\272)3144 5284 w 10 R f (\()3240 5284 w 10 I f (n n)1 50 1 3306 5354 t 10 S f (- -)1 55 1 3380 5354 t 10 R f (1)3451 5354 w 10 I f (i i)1 28 1 3317 5224 t 10 S f (- -)1 55 1 3369 5224 t 10 R f (1)3440 5224 w 10 S1 f (_ ____)1 225 1 3291 5254 t 10 R f (\))3534 5284 w 7 I f (k k)1 31 1 3578 5244 t 10 R f (, for)1 171 1 3617 5284 t 10 I f (i i)1 28 1 3818 5284 t 10 S f (= =)1 55 1 3895 5284 t 10 R f (1 ,)1 83 1 3999 5284 t (. . .)2 125 1 4115 5259 t (,)4273 5284 w 10 I f (n n)1 50 1 4306 5284 t 10 R f ( the solu-)2 382(. Since)1 302 2 4356 5284 t ( \()1 41(tion has a log)3 566 2 720 5454 t 10 I f (z z)1 39 1 1335 5454 t 10 R f (\) singularity at)2 603 1 1382 5454 t 10 I f (z z)1 39 1 2020 5454 t 10 S f (= =)1 55 1 2108 5454 t 10 R f ( mesh is sufficient to give)5 1075(0, this grading of the)4 870 2 2212 5454 t 10 I f (O O)1 72 1 4191 5454 t 10 R f (\()4271 5454 w 10 I f (n n)1 50 1 4312 5454 t 7 S f (- -)1 39 1 4373 5414 t 7 I f (k k)1 31 1 4423 5414 t 10 R f (\) convergence)1 570 1 4470 5454 t (where)720 5574 w 10 I f (k k)1 44 1 990 5574 t 10 R f ( the grading, the convergence would only be)7 1788( Without)1 380(is the order of the spline used.)6 1213 3 1061 5574 t 10 I f (O O)1 72 1 4469 5574 t 10 R f (\()4549 5574 w 10 I f (n n)1 50 1 4590 5574 t 7 S f (- -)1 39 1 4651 5534 t 7 R f (1)4701 5534 w 10 R f ( it)1 84(\) and)1 204 2 4752 5574 t ( error at each time-step is printed out to)8 1615( The)1 210(would require many more points to get comparable accuracy.)8 2495 3 720 5694 t ( main program is)3 680( The)1 205(confirm the accuracy of the computed solution.)6 1888 3 720 5814 t cleartomark showpage saveobj restore end %%EndPage: 22 61 %%Page: 23 62 DpostDict begin /saveobj save def mark 62 pagesetup 10 R f (\261 4-23 \261)2 333 1 2713 480 t 10 CW f (# Main)1 360 1 1080 900 t (# To solve Laplaces equation with Real \( z*log\(z\) \) as solution.)11 3840 1 1080 1140 t (Real tstart,tstop,dt,Lx,Rx,Ly,Ry)1 1920 1 1200 1380 t (Real errpar\(2\))1 840 1 1200 1500 t (Integer Nu,kx,ix,nx, ky,iy,ny,ISTKGT,iU,ndx,ndy,i)2 2940 1 1200 1620 t (External AF,BC,HANDLE)1 1260 1 1200 1740 t (Common / CSTAK / Ds\(350000\); Double Precision Ds)7 2880 1 1200 2100 t ( The PORT Library stack and its aliases.)7 2400( #)1 300(Real Ws\(1000\))1 780 3 1200 2220 t (Real Rs\(1000\) ; Integer Is\(1000\) ; Complex Cs\(500\) ; Logical Ls\(1000\))10 4140 1 1200 2340 t (Equivalence \( Ds\(1\),Cs\(1\),Ws\(1\),Rs\(1\),Is\(1\),Ls\(1\) \))3 3060 1 1200 2460 t ( Initialize the PORT Library stack length.)6 2520( #)1 300(Call ISTKIN\(350000,4\))1 1260 3 1200 2700 t (Call ENTER\(1\))1 780 1 1200 2940 t (Nu = 1)2 360 1 1200 3180 t (Lx = 0; Rx = +1)5 900 1 1200 3420 t (Ly = 0; Ry = +1)5 900 1 1200 3540 t (kx = 4; ky = 4)5 840 1 1200 3780 t (ndx = 2; ndy = 2)5 960 1 1200 4020 t (tstart = 0; tstop = 1; dt = 1)8 1740 1 1200 4260 t (errpar\(1\) = 1e-2; errpar\(2\) = 1e-4)5 2040 1 1200 4500 t ( Space for x mesh.)4 1080( #)1 300(nx = ndx+2*\(kx-1\); ix = ISTKGT\(nx,3\))5 2160 3 1200 4740 t (Do i = 1, kx)4 720 1 1200 4860 t ( 0 and Rx mult = kx.)6 1200( #)1 360({ Ws\(ix+i-1\) = 0; Ws\(ix+nx-i\) = Rx })7 2160 3 1320 4980 t (Do i = 1, ndx-1 { Ws\(ix+kx-2+i\) = Rx*\(\(i-1\)/\(ndx-1e0\)\)**kx })9 3600 1 1200 5100 t ( Space for y mesh.)4 1080( #)1 300(ny = ndy+2*\(ky-1\); iy = ISTKGT\(ny,3\))5 2160 3 1200 5340 t (Do i = 1, ky)4 720 1 1200 5460 t ( 0 and Ry mult = ky.)6 1200( #)1 360({ Ws\(iy+i-1\) = 0; Ws\(iy+ny-i\) = Ry })7 2160 3 1320 5580 t (Do i = 1, ndy-1 { Ws\(iy+ky-2+i\) = Ry*\(\(i-1\)/\(ndy-1e0\)\)**ky })9 3600 1 1200 5700 t ( Space for the solution.)4 1440( #)1 300(iU = ISTKGT\(Nu*\(nx-kx\)*\(ny-ky\),3\))2 1980 3 1200 5940 t (Call SETR\(Nu*\(nx-kx\)*\(ny-ky\),0e0,Ws\(iU\)\))1 2400 1 1200 6060 t (Call TTGR \(Ws\(iU\),Nu,kx,Ws\(ix\),nx, ky,Ws\(iy\),ny, tstart,tstop, dt,)5 3960 1 1200 6300 t (AF,BC,)1860 6420 w (errpar,)1860 6540 w (HANDLE\))1860 6660 w (Call LEAVE)1 600 1 1200 6900 t (Call WRAPUP)1 660 1 1200 7140 t cleartomark showpage saveobj restore end %%EndPage: 23 62 %%Page: 24 63 DpostDict begin /saveobj save def mark 63 pagesetup 10 R f (\261 4-24 \261)2 333 1 2713 480 t 10 CW f (Stop)1200 840 w (End)1200 1080 w 10 R f (The body of the subroutine)4 1082 1 720 1296 t 10 CW f (AF)1827 1296 w 10 R f (for \(A4.9\) is)2 496 1 1972 1296 t 10 CW f (Do i = 1, Nu)4 720 1 1200 1476 t ({)1320 1596 w (Do q = 1, ny)4 720 1 1320 1716 t ({)1440 1836 w (Do p = 1, nx)4 720 1 1440 1956 t ({)1560 2076 w (a\(p,q,i,1\) = Ux\(p,q,i\); a\(p,q,i,2\) = Uy\(p,q,i\))5 2760 1 1560 2196 t (AUx\(p,q,i,i,1\) = 1; AUy\(p,q,i,i,2\) = 1)5 2280 1 1560 2316 t (})1560 2436 w (})1440 2556 w (})1320 2676 w 10 R f (The body of the subroutine)4 1082 1 720 2892 t 10 CW f (BC)1827 2892 w 10 R f (for \(A4.10\) is)2 546 1 1972 2892 t 10 CW f (Do j = 1, ny)4 720 1 1200 3072 t ({)1320 3192 w (Do i = 1, nx)4 720 1 1320 3312 t ({)1440 3432 w ( Neumann data on bottom.)4 1440( #)1 300(If \( y\(j\) == Ly \))5 1020 3 1440 3552 t ({)1560 3672 w (b\(i,j,1\) = Uy\(i,j,1\))2 1200 1 1560 3792 t (BUy\(i,j,1,1\) = 1)2 960 1 1560 3912 t (})1560 4032 w ( Dirichlet data.)2 960(Else #)1 540 2 1440 4152 t ({)1560 4272 w (r = Sqrt\(x\(i\)**2+y\(j\)**2\))2 1500 1 1560 4392 t (If \( x\(i\) > 0 \) { theta = Atan\(y\(j\)/x\(i\)\) })10 2580 1 1560 4512 t ( theta = 2*Atan\(1e0\) })4 1320(Else {)1 1020 2 1560 4632 t (b\(i,j,1\) = U\(i,j,1\) - r*\(Cos\(theta\)*Log\(r\) - theta*Sin\(theta\)\))6 3720 1 1560 4872 t (BU\(i,j,1,1\) = 1)2 900 1 1560 4992 t (})1560 5112 w (})1440 5232 w (})1320 5352 w 10 R f (The body of the subroutines)4 1161 1 720 5568 t 10 CW f (HANDLE)1916 5568 w 10 R f (and)2311 5568 w 10 CW f (GERR)2490 5568 w 10 R f ( the error is same as before.)6 1165(for computing and printing)3 1110 2 2765 5568 t ( in the subroutine)3 706(The only change)2 673 2 720 5688 t 10 CW f (EWE)2127 5688 w 10 R f (for computing)1 572 1 2335 5688 t 10 I f (u u)1 50 1 2935 5688 t 10 R f (in the previous example is the code for computing)8 2027 1 3013 5688 t 10 I f (u u)1 50 1 720 5808 t 10 R f (.)770 5808 w cleartomark showpage saveobj restore end %%EndPage: 24 63 %%Page: 25 64 DpostDict begin /saveobj save def mark 64 pagesetup 10 R f (\261 4-25 \261)2 333 1 2713 480 t 10 CW f (Do p = 1, Nu)4 720 1 1200 900 t ({)1320 1020 w (Do i = 1, nx)4 720 1 1320 1140 t ({)1440 1260 w (Do j = 1, ny)4 720 1 1440 1380 t ({)1560 1500 w (r = Sqrt\(x\(i\)**2+y\(j\)**2\))2 1500 1 1560 1620 t (If \( x\(i\) > 0 \) { theta = Atan\(y\(j\)/x\(i\)\) })10 2580 1 1560 1740 t ( theta = 2*Atan\(1e0\) })4 1320(Else {)1 1020 2 1560 1860 t (If \( r > 0 \) { U\(i,j,p\) = r*\(Cos\(theta\)*Log\(r\) - theta*Sin\(theta\)\) })12 4080 1 1560 2100 t ( U\(i,j,p\) = 0 })4 900(Else {)1 840 2 1560 2220 t (})1560 2340 w (})1440 2460 w (})1320 2580 w 10 R f (The output of this program is)5 1169 1 720 2796 t 10 CW f ( 3.44e-02)1 600( =)1 120( 1.00e+00\))1 660(error in U\(.,)2 780 4 1140 2976 t ( of the stack allowed.)4 1320( 700000)1 540( /)1 120(used 2654)1 780 4 1140 3096 t 10 R f (The run-time for the above example was 8.7 seconds.)8 2134 1 720 3276 t 10 B f (Example 6 - Error Estimation.)4 1302 1 720 3516 t 10 R f ( 5 above, without using any information)6 1644(We consider estimating the error in the solution to example)9 2426 2 970 3672 t (about the exact solution to do it.)6 1286 1 720 3792 t ( solu-)1 231( The)1 211( scheme outlined in Example 6 of section 4.)8 1799(The error estimation is done according to the)7 1829 4 970 3948 t ( then the solution is obtained on a finer grid \(in this case, with twice)14 2745(tion is obtained as in example 5 above,)7 1575 2 720 4068 t ( the time evolution)3 768(the number of mesh points\), and finally the solution is obtained on the crude mesh with)15 3552 2 720 4188 t (carried out to one-tenth the accuracy.)5 1484 1 720 4308 t ( for)1 143(Since the problem is static, we expect that the error estimate in time should be very small and that)18 3927 2 970 4464 t (the spatial error will dominate.)4 1229 1 720 4584 t (The main program is)3 835 1 970 4740 t cleartomark showpage saveobj restore end %%EndPage: 25 64 %%Page: 26 65 DpostDict begin /saveobj save def mark 65 pagesetup 10 R f (\261 4-26 \261)2 333 1 2713 480 t 10 CW f (# Main)1 360 1 1080 900 t (# To get error estimates for Laplaces equation with Real \( z*log\(z\) \) as solution.)14 4920 1 1080 1140 t (Real tstart,tstop,dt,Lx,Rx,Ly,Ry, EERR,errE,errR)2 2880 1 1200 1380 t (Real errpar\(2\))1 840 1 1200 1500 t (Integer Nu,kx,ix,nx, ky,iy,ny,ISTKGT,iU,ndx,ndy,i,I1MACH,)2 3420 1 1200 1620 t (ixr,iyr,nxr,nyr,iUr,iUe)1680 1740 w (External AF,BC,HANDLE)1 1260 1 1200 1860 t (Common / CSTAK / Ds\(350000\); Double Precision Ds)7 2880 1 1200 2220 t ( The PORT Library stack and its aliases.)7 2400( #)1 300(Real Ws\(1000\))1 780 3 1200 2340 t (Real Rs\(1000\) ; Integer Is\(1000\) ; Complex Cs\(500\) ; Logical Ls\(1000\))10 4140 1 1200 2460 t (Equivalence \( Ds\(1\),Cs\(1\),Ws\(1\),Rs\(1\),Is\(1\),Ls\(1\) \))3 3060 1 1200 2580 t ( Initialize the PORT Library stack length.)6 2520( #)1 300(Call ISTKIN\(350000,4\))1 1260 3 1200 2820 t (Call ENTER\(1\))1 780 1 1200 3060 t (Nu = 1)2 360 1 1200 3300 t (Lx = 0; Rx = +1)5 900 1 1200 3540 t (Ly = 0; Ry = +1)5 900 1 1200 3660 t (kx = 4; ky = 4)5 840 1 1200 3900 t (ndx = 2; ndy = 2)5 960 1 1200 4140 t (tstart = 0; tstop = 1; dt = 1)8 1740 1 1200 4380 t (errpar\(1\) = 1e-2; errpar\(2\) = 1e-4)5 2040 1 1200 4620 t ( Space for x mesh.)4 1080( #)1 300(nx = ndx+2*\(kx-1\); ix = ISTKGT\(nx,3\))5 2160 3 1200 4860 t (Do i = 1, kx)4 720 1 1200 4980 t ( 0 and Rx mult = kx.)6 1200( #)1 360({ Ws\(ix+i-1\) = 0; Ws\(ix+nx-i\) = Rx })7 2160 3 1320 5100 t (Do i = 1, ndx-1 { Ws\(ix+kx-2+i\) = Rx*\(\(i-1\)/\(ndx-1e0\)\)**kx })9 3600 1 1200 5220 t ( Space for y mesh.)4 1080( #)1 300(ny = ndy+2*\(ky-1\); iy = ISTKGT\(ny,3\))5 2160 3 1200 5460 t (Do i = 1, ky)4 720 1 1200 5580 t ( 0 and Ry mult = ky.)6 1200( #)1 360({ Ws\(iy+i-1\) = 0; Ws\(iy+ny-i\) = Ry })7 2160 3 1320 5700 t (Do i = 1, ndy-1 { Ws\(iy+ky-2+i\) = Ry*\(\(i-1\)/\(ndy-1e0\)\)**ky })9 3600 1 1200 5820 t ( Space for the solution.)4 1440( #)1 300(iU = ISTKGT\(Nu*\(nx-kx\)*\(ny-ky\),3\))2 1980 3 1200 6060 t (Call SETR\(Nu*\(nx-kx\)*\(ny-ky\),0e0,Ws\(iU\)\))1 2400 1 1200 6180 t (iwunit = I1MACH\(2\))2 1080 1 1200 6420 t (Write\(iwunit,9000\))1200 6540 w (9000 Format\(" Solving on crude mesh."\))5 2280 1 1140 6660 t (Call TTGR \(Ws\(iU\),Nu,kx,Ws\(ix\),nx, ky,Ws\(iy\),ny, tstart,tstop, dt,)5 3960 1 1200 6900 t (AF,BC,)1860 7020 w (errpar,)1860 7140 w (HANDLE\))1860 7260 w cleartomark showpage saveobj restore end %%EndPage: 26 65 %%Page: 27 66 DpostDict begin /saveobj save def mark 66 pagesetup 10 R f (\261 4-27 \261)2 333 1 2713 480 t 10 CW f (dt = 1)2 360 1 1200 840 t ( Refine mesh.)2 780( #)1 300(ndx = 2*ndx-1; ndy = 2*ndy-1)5 1680 3 1200 1080 t ( Space for x mesh.)4 1080( #)1 300(nxr = ndx+2*\(kx-1\); ixr = ISTKGT\(nxr,3\))5 2340 3 1200 1320 t (Do i = 1, kx)4 720 1 1200 1440 t ( 0 and Rx mult = kx.)6 1200( #)1 360({ Ws\(ixr+i-1\) = 0; Ws\(ixr+nxr-i\) = Rx })7 2340 3 1320 1560 t (Do i = 1, ndx-1 { Ws\(ixr+kx-2+i\) = Rx*\(\(i-1\)/\(ndx-1e0\)\)**kx })9 3660 1 1200 1680 t ( Space for y mesh.)4 1080( #)1 300(nyr = ndy+2*\(ky-1\); iyr = ISTKGT\(nyr,3\))5 2340 3 1200 1920 t (Do i = 1, ky)4 720 1 1200 2040 t ( 0 and Ry mult = ky.)6 1200( #)1 360({ Ws\(iyr+i-1\) = 0; Ws\(iyr+nyr-i\) = Ry })7 2340 3 1320 2160 t (Do i = 1, ndy-1 { Ws\(iyr+ky-2+i\) = Ry*\(\(i-1\)/\(ndy-1e0\)\)**ky })9 3660 1 1200 2280 t ( Space for the solution.)4 1440( #)1 300(iUr = ISTKGT\(Nu*\(nxr-kx\)*\(nyr-ky\),3\))2 2160 3 1200 2520 t (Call SETR\(Nu*\(nxr-kx\)*\(nyr-ky\),0e0,Ws\(iUr\)\))1 2580 1 1200 2640 t (iwunit = I1MACH\(2\))2 1080 1 1200 2880 t (Write\(iwunit,9001\))1200 3000 w (9001 Format\(" Solving on Refined mesh."\))5 2400 1 1140 3120 t (Call TTGR \(Ws\(iUr\),Nu,kx,Ws\(ixr\),nxr, ky,Ws\(iyr\),nyr, tstart,tstop, dt,)5 4260 1 1200 3360 t (AF,BC,)1860 3480 w (errpar,)1860 3600 w (HANDLE\))1860 3720 w (dt = 1; errpar\(1\) = errpar\(1\)/10; errpar\(2\) = errpar\(2\)/10)8 3480 1 1200 3960 t ( Space for the solution.)4 1440( #)1 300(iUe = ISTKGT\(Nu*\(nx-kx\)*\(ny-ky\),3\))2 2040 3 1200 4200 t (Call SETR\(Nu*\(nx-kx\)*\(ny-ky\),0e0,Ws\(iUe\)\))1 2460 1 1200 4320 t (iwunit = I1MACH\(2\))2 1080 1 1200 4560 t (Write\(iwunit,9002\))1200 4680 w (9002 Format\(" Solving with errpar/10."\))4 2340 1 1140 4800 t (Call TTGR \(Ws\(iUe\),Nu,kx,Ws\(ix\),nx, ky,Ws\(iy\),ny, tstart,tstop, dt,)5 4020 1 1200 5040 t (AF,BC,)1860 5160 w (errpar,)1860 5280 w (HANDLE\))1860 5400 w (errR = EERR\(kx,ix,nx, ky,iy,ny, Ws\(iU\),Nu, ixr,nxr,iyr,nyr,Ws\(iUr\), tstop\))6 4440 1 1200 5640 t (errE = 0)2 480 1 1200 5880 t (Do i = 1, Nu*\(nx-kx\)*\(ny-ky\))4 1680 1 1200 6000 t ({)1320 6120 w (errE = Max\( errE, Abs\( Ws\(iU+i-1\)-Ws\(iUe+i-1\) \) \))7 2940 1 1320 6240 t (})1320 6360 w (iwunit = I1MACH\(2\))2 1080 1 1200 6600 t (Write\(iwunit,9003\) errE)1 1380 1 1200 6720 t (9003 Format\(" U error from U and Ue =",1p4e10.2\))8 2880 1 1140 6840 t (iwunit = I1MACH\(2\))2 1080 1 1200 7080 t (Write\(iwunit,9004\) errR)1 1380 1 1200 7200 t cleartomark showpage saveobj restore end %%EndPage: 27 66 %%Page: 28 67 DpostDict begin /saveobj save def mark 67 pagesetup 10 R f (\261 4-28 \261)2 333 1 2713 480 t 10 CW f (9004 Format\(" U error from U and Ur =",1p4e10.2\))8 2880 1 1140 840 t (Call LEAVE)1 600 1 1200 1080 t (Call WRAPUP)1 660 1 1200 1320 t (Stop)1200 1560 w (End)1200 1800 w 10 R f (and the subroutine for obtaining the error estimate is)8 2097 1 720 2016 t cleartomark showpage saveobj restore end %%EndPage: 28 67 %%Page: 29 68 DpostDict begin /saveobj save def mark 68 pagesetup 10 R f (\261 4-29 \261)2 333 1 2713 480 t 10 CW f (Real Function EERR\(kx,ix,nx, ky,iy,ny, U,Nu, ixr,nxr, iyr,nyr,Ur, t\))7 4080 1 1200 900 t (# To get and print the error estimate at each time-step.)10 3360 1 1080 1140 t ( U\(nx-kx,ny-ky,Nu\), Ur\(nxr-kx,nyr-ky,Nu\).)2 2460( #)1 300(Real U\(1\),Ur\(1\),t)1 1020 3 1200 1380 t (Integer kx,ix,nx, ky,iy,ny,Nu, ixr,nxr,iyr,nyr)3 2760 1 1200 1500 t (Common / CSTAK / Ds\(500\); Double Precision Ds)7 2700 1 1200 1740 t ( The PORT Library stack and its aliases.)7 2400( #)1 300(Real Ws\(1000\))1 780 3 1200 1860 t (Real Rs\(1000\) ; Integer Is\(1000\) ; Complex Cs\(500\) ; Logical Ls\(1000\))10 4140 1 1200 1980 t (Equivalence \( Ds\(1\),Cs\(1\),Ws\(1\),Rs\(1\),Is\(1\),Ls\(1\) \))3 3060 1 1200 2100 t (Real errU)1 540 1 1200 2340 t (Integer i,I1MACH,ISTKGT, iFA,iFAr,)2 2040 1 1200 2460 t (KA\(2\),ITA\(2\),NTA\(2\),IXA\(2\),NXA\(2\),MA\(2\), ILUMD,ixs,iys,nxs,nys)1 3720 1 1680 2580 t (Call ENTER\(1\))1 780 1 1200 2820 t (# Find the error in the solution at 2*kx * 2*ky points / fine mesh rectangle.)15 4620 1 1080 3060 t ( x search grid.)3 900( #)1 300(ixs = ILUMD\(Ws\(ixr\),nxr,2*kx,nxs\))2 1980 3 1200 3300 t ( y search grid.)3 900( #)1 300(iys = ILUMD\(Ws\(iyr\),nyr,2*ky,nys\))2 1980 3 1200 3420 t (KA\(1\) = kx; KA\(2\) = ky; ITA\(1\) = ix; ITA\(2\) = iy; NTA\(1\) = nx; NTA\(2\) = ny)17 4440 1 1200 3660 t (IXA\(1\) = ixs; IXA\(2\) = iys; NXA\(1\) = nxs; NXA\(2\) = nys)11 3240 1 1200 3780 t ( Get solution.)2 840( #)1 300(MA\(1\) = 0; MA\(2\) = 0)5 1200 3 1200 3900 t ( Approximate solution values.)3 1740( #)1 300(iFA = ISTKGT\(nxs*nys,3\))2 1380 3 1200 4140 t ( Evaluate them.)2 900( #)1 300(Call TSD1\(2,KA,Ws,ITA,NTA,U, Ws,IXA,NXA,MA, Ws\(iFA\)\))3 3120 3 1200 4380 t (KA\(1\) = kx; KA\(2\) = ky; ITA\(1\) = ixr; ITA\(2\) = iyr; NTA\(1\) = nxr; NTA\(2\) = nyr)17 4680 1 1200 4620 t (IXA\(1\) = ixs; IXA\(2\) = iys; NXA\(1\) = nxs; NXA\(2\) = nys)11 3240 1 1200 4740 t ( Get solution.)2 840( #)1 300(MA\(1\) = 0; MA\(2\) = 0)5 1200 3 1200 4860 t ( Approximate solution values.)3 1740( #)1 300(iFAr = ISTKGT\(nxs*nys,3\))2 1440 3 1200 5100 t ( Evaluate them.)2 900( #)1 300(Call TSD1\(2,KA,Ws,ITA,NTA,Ur, Ws,IXA,NXA,MA, Ws\(iFAr\)\))3 3240 3 1200 5340 t ( Error in solution values.)4 1560( #)1 300(errU = 0)2 480 3 1200 5580 t (Do i = 1, nxs*nys)4 1020 1 1200 5700 t ({)1320 5820 w (errU = Max\(errU,Abs\(Ws\(iFAr+i-1\)-Ws\(iFA+i-1\)\)\))2 2760 1 1320 5940 t (})1320 6060 w (Call LEAVE)1 600 1 1200 6300 t (EERR = errU; Return)3 1140 1 1200 6540 t (End)1200 6780 w 10 R f (The rest of the subroutines are the same as in example 5.)11 2262 1 720 6996 t (The output of this program is)5 1169 1 720 7152 t cleartomark showpage saveobj restore end %%EndPage: 29 68 %%Page: 30 69 DpostDict begin /saveobj save def mark 69 pagesetup 10 R f (\261 4-30 \261)2 333 1 2713 480 t 10 CW f (Solving on crude mesh.)3 1320 1 1140 900 t ( 3.44e-02)1 600( =)1 120( 1.00e+00\))1 660(error in U\(.,)2 780 4 1140 1020 t (Solving on Refined mesh.)3 1440 1 1140 1140 t ( 1.50e-02)1 600( =)1 120( 1.00e+00\))1 660(error in U\(.,)2 780 4 1140 1260 t (Solving with errpar/10.)2 1380 1 1140 1380 t ( 3.44e-02)1 600( =)1 120( 1.00e+00\))1 660(error in U\(.,)2 780 4 1140 1500 t ( 0.00e+00)1 600(U error from U and Ue =)6 1380 2 1140 1620 t ( 4.27e-02)1 600(U error from U and Ur =)6 1380 2 1140 1740 t ( of the stack allowed.)4 1320( 700000)1 540( /)1 120(used 3426)1 780 4 1140 1860 t 10 R f (The run-time for the above example was 37.7 seconds.)8 2184 1 720 2040 t ( the time error estimate is small, and the spatial error)10 2245(The output shows that, just as we expected,)7 1825 2 970 2196 t ( the error estimates are quite good, and over-estimates.)8 2184(dominates. Also,)1 700 2 720 2316 t ( had been used in the above run, the errors would have been)12 2631(If a uniform grid)3 725 2 970 2472 t 10 CW f (3.44e-02)4371 2472 w 10 R f (and)4896 2472 w 10 CW f (2.68e-02)720 2592 w 10 R f ( convergence rate of)3 816( A)1 125(, or substantially worse than achieved above.)6 1809 3 1200 2592 t 10 I f (O O)1 72 1 3977 2592 t 10 R f (\()4057 2592 w 10 I f (n n)1 50 1 4098 2592 t 7 S f (- -)1 39 1 4159 2552 t 7 I f (k k)1 31 1 4209 2552 t 10 R f (\) is a)2 198 1 4256 2592 t 10 I f ( t)1 0( ot)1 28(l lo)1 78 3 4481 2592 t 10 R f (better than)1 426 1 4614 2592 t 10 I f (O O)1 72 1 720 2712 t 10 R f (\()800 2712 w 10 I f (n n)1 50 1 841 2712 t 7 S f (- -)1 39 1 902 2672 t 7 R f (1)952 2672 w 10 R f (\), even when)2 512 1 1003 2712 t 10 I f (n n)1 50 1 1540 2712 t 10 R f (is only 2 or 3.)4 553 1 1615 2712 t ( is typical and indi-)4 796( This)1 234( example 6.)2 475(Notice that the run-time for example 5 is 4 or 5 times less than)13 2565 4 970 2868 t ( is a)2 171( It)1 116(cates that obtaining error estimates is expensive.)6 1963 3 720 2988 t 10 I f ( y)1 0( ll ly)2 72( al)1 28( ea)1 50(r re)1 83 5 3000 2988 t 10 R f (good idea to apply the above scheme when-)7 1777 1 3263 2988 t ( reasonable.)1 488(ever working on a new problem and concern about the accuracy of the computed solution is)15 3832 2 720 3108 t ( and)1 174(Once the mesh)2 605 2 720 3228 t 10 CW f (errpar)1529 3228 w 10 R f (have been chosen appropriately and confirmed, production runs can dispense)9 3121 1 1919 3228 t ( a)1 77( alternative is)2 552( The)1 212( some error estimation is a good idea.)7 1547( But)1 202(with the error estimates and their expense.)6 1730 6 720 3348 t ( piles of number cost more to obtain.)7 1471( Good)1 272(pile of cheap numbers that may be garbage.)7 1740 3 720 3468 t (We have from the above estimates that)6 1552 1 970 3624 t 10 S f ( \357)1 0( \357)1 24(\357 \357)1 49 3 1220 3821 t 10 B f (u)1316 3804 w 7 I f ( e)1 0( ue)1 31( ru)1 35(t tr)1 47 4 1383 3824 t 10 S f (- -)1 55 1 1544 3804 t 10 B f (u)1639 3804 w 10 S f ( \357)1 0( \357)1 24(\357 \357)1 49 3 1719 3821 t 7 S f (\245)1781 3824 w 10 S f (\243)1881 3804 w 10 R f (4. 27)1 183 1 1977 3804 t 10 S f (\264)2168 3804 w 10 R f (10)2231 3804 w 7 S f (- -)1 39 1 2342 3764 t 7 R f (2)2392 3764 w 10 R f (\(A4.11\))4727 3804 w (and from that, and the assumption that the spatial error is)10 2278 1 720 3984 t 10 I f (O O)1 72 1 3023 3984 t 10 R f (\()3103 3984 w 10 I f (n n)1 50 1 3144 3984 t 7 S f (- -)1 39 1 3205 3944 t 7 I f (k k)1 31 1 3255 3944 t 10 R f (\), it's tempting to say that)5 1028 1 3302 3984 t 10 S f ( \357)1 0( \357)1 24(\357 \357)1 49 3 1220 4181 t 10 B f (u)1316 4164 w 7 I f ( e)1 0( ue)1 31( ru)1 35(t tr)1 47 4 1383 4184 t 10 S f (- -)1 55 1 1544 4164 t 10 B f (u)1639 4164 w 7 I f (r r)1 27 1 1706 4184 t 10 S f ( \357)1 0( \357)1 24(\357 \357)1 49 3 1765 4181 t 7 S f (\245)1827 4184 w 10 S f (\243)1927 4164 w ( \357)1 0( \357)1 24(\357 \357)1 49 3 2015 4181 t 10 B f (u)2111 4164 w 7 I f ( e)1 0( ue)1 31( ru)1 35(t tr)1 47 4 2178 4184 t 10 S f (- -)1 55 1 2339 4164 t 10 B f (u)2434 4164 w 10 S f ( \357)1 0( \357)1 24(\357 \357)1 49 3 2514 4181 t 7 S f (\245)2576 4184 w 10 S f (\264)2667 4164 w 10 R f (2)2754 4164 w 7 S f (- -)1 39 1 2815 4124 t 7 I f (k k)1 31 1 2865 4124 t 10 S f (~)2945 4144 w (~)2945 4169 w 10 R f (1. 6)1 133 1 3041 4164 t 10 S f (\264)3182 4164 w 10 R f (10)3245 4164 w 7 S f (- -)1 39 1 3356 4124 t 7 R f (3)3406 4124 w 10 R f (\(A4.12\))4727 4164 w ( is typical of the estimates \(A4.11\))6 1439( This)1 237(which underestimates the actual error by an order of magnitude!)9 2644 3 720 4344 t ( and is quite reliable because it)6 1291( estimates the error in the crude mesh solution)8 1930( \(A4.11\))1 374(and \(A4.12\).)1 518 4 720 4464 t 10 I f ( y)1 0( ly)1 44(o on nl)2 128 3 4868 4464 t 10 R f (assumes that)1 515 1 720 4584 t 10 B f (u)1267 4584 w 7 I f (r r)1 27 1 1334 4604 t 10 R f (is more accurate than)3 871 1 1401 4584 t 10 B f (u)2305 4584 w 10 R f ( estimates the error in the refined solution and is not so)11 2283(. \(A4.12\))1 396 2 2361 4584 t (reliable, because it also assumes that the spatial error is)9 2261 1 720 4704 t 10 I f (O O)1 72 1 3012 4704 t 10 R f (\()3092 4704 w 10 I f (n n)1 50 1 3133 4704 t 7 S f (- -)1 39 1 3194 4664 t 7 I f (k k)1 31 1 3244 4664 t 10 R f ( is valid for large)4 702( assumption)1 487(\). That)1 297 3 3291 4704 t 10 I f (n n)1 50 1 4807 4704 t 10 R f (, but)1 183 1 4857 4704 t (not necessarily for small)3 994 1 720 4824 t 10 I f (n n)1 50 1 1744 4824 t 10 R f ( but not the refined)4 784( we can reliably estimate the error in the crude solution,)10 2276(. So)1 186 3 1794 4824 t (one.)720 4944 w ( be applied to the solution at any point in time, not just)12 2415(The above error estimation scheme can)5 1655 2 970 5100 t 10 CW f (tstop)720 5220 w 10 R f (as was done above.)3 770 1 1045 5220 t 10 B f (Example Summary)1 825 1 720 5460 t 10 R f ( time usage of each of the examples on two widely disparate)11 2695(We summarize the memory and)4 1375 2 970 5616 t ( in double precision, under Unix and)6 1474(machines, the Cray-1 in single precision under COS and a Vax 11/750)11 2846 2 720 5736 t ( two tables provide a rough guide to the resource utilization)10 2600( These)1 310(using a floating-point accelerator.)3 1410 3 720 5856 t (expected when running)2 931 1 720 5976 t 10 CW f (TTGR)1676 5976 w 10 R f (on various machines.)2 846 1 1941 5976 t 10 S f (_ ______________________________________)1 1902 1 1929 6056 t 10 R f (Single Precision on a Cray-1)4 1149 1 2305 6176 t 10 S f (_ ______________________________________)1 1902 1 1929 6196 t 10 R f ( \(secs\))1 257( Time)1 361( \(words\))1 335(Example Memory)1 849 4 1979 6316 t 10 S f (_ ______________________________________)1 1902 1 1929 6336 t 10 R f ( 0.1025)1 761(1 825)1 792 2 2131 6456 t 10 S f (_ ______________________________________)1 1902 1 1929 6476 t 10 R f ( 4.1503)1 761(2 9398)1 792 2 2131 6596 t 10 S f (_ ______________________________________)1 1902 1 1929 6616 t 10 R f ( 0.2031)1 761(3 1260)1 792 2 2131 6736 t 10 S f (_ ______________________________________)1 1902 1 1929 6756 t 10 R f ( 0.3645)1 761(4 3388)1 792 2 2131 6876 t 10 S f ( \347)1 -1902(_ ______________________________________)1 1902 2 1929 6980 t (\347)1929 6956 w (\347)1929 6856 w (\347)1929 6756 w (\347)1929 6656 w (\347)1929 6556 w (\347)1929 6456 w (\347)1929 6356 w (\347)1929 6256 w (\347)1929 6156 w (\347)2409 6980 w (\347)2409 6896 w (\347)2409 6796 w (\347)2409 6696 w (\347)2409 6596 w (\347)2409 6496 w (\347)2409 6396 w (\347)2409 6296 w (\347)3238 6980 w (\347)3238 6896 w (\347)3238 6796 w (\347)3238 6696 w (\347)3238 6596 w (\347)3238 6496 w (\347)3238 6396 w (\347)3238 6296 w (\347)3831 6980 w (\347)3831 6956 w (\347)3831 6856 w (\347)3831 6756 w (\347)3831 6656 w (\347)3831 6556 w (\347)3831 6456 w (\347)3831 6356 w (\347)3831 6256 w (\347)3831 6156 w cleartomark showpage saveobj restore end %%EndPage: 30 69 %%Page: 31 70 DpostDict begin /saveobj save def mark 70 pagesetup 10 R f (\261 4-31 \261)2 333 1 2713 480 t 10 S f (_ ______________________________________)1 1902 1 1929 740 t 10 R f ( 0.1207)1 761(5 2654)1 792 2 2131 860 t 10 S f (_ ______________________________________)1 1902 1 1929 880 t 10 R f ( 0.5042)1 761(6 3426)1 792 2 2131 1000 t 10 S f ( \347)1 -1902(_ ______________________________________)1 1902 2 1929 1020 t (\347)1929 920 w (\347)1929 820 w (\347)2409 1020 w (\347)2409 940 w (\347)2409 840 w (\347)3238 1020 w (\347)3238 940 w (\347)3238 840 w (\347)3831 1020 w (\347)3831 920 w (\347)3831 820 w (_ ______________________________________)1 1902 1 1929 1196 t 10 R f (Double Precision on a Vax 11/750)5 1379 1 2190 1316 t 10 S f (_ ______________________________________)1 1902 1 1929 1336 t 10 R f ( \(secs\))1 257( Time)1 361( \(words\))1 335(Example Memory)1 849 4 1979 1456 t 10 S f (_ ______________________________________)1 1902 1 1929 1476 t 10 R f ( 5.5)1 711(1 1356)1 817 2 2131 1596 t 10 S f (_ ______________________________________)1 1902 1 1929 1616 t 10 R f ( 465.8)1 711(2 18324)1 817 2 2131 1736 t 10 S f (_ ______________________________________)1 1902 1 1929 1756 t 10 R f ( 11.4)1 711(3 2139)1 817 2 2131 1876 t 10 S f (_ ______________________________________)1 1902 1 1929 1896 t 10 R f ( 30.4)1 711(4 6460)1 817 2 2131 2016 t 10 S f (_ ______________________________________)1 1902 1 1929 2036 t 10 R f ( 9.5)1 711(5 5020)1 817 2 2131 2156 t 10 S f (_ ______________________________________)1 1902 1 1929 2176 t 10 R f ( 41.7)1 711(6 6530)1 817 2 2131 2296 t 10 S f ( \347)1 -1902(_ ______________________________________)1 1902 2 1929 2316 t (\347)1929 2296 w (\347)1929 2196 w (\347)1929 2096 w (\347)1929 1996 w (\347)1929 1896 w (\347)1929 1796 w (\347)1929 1696 w (\347)1929 1596 w (\347)1929 1496 w (\347)1929 1396 w (\347)1929 1296 w (\347)2409 2316 w (\347)2409 2236 w (\347)2409 2136 w (\347)2409 2036 w (\347)2409 1936 w (\347)2409 1836 w (\347)2409 1736 w (\347)2409 1636 w (\347)2409 1536 w (\347)2409 1436 w (\347)3238 2316 w (\347)3238 2236 w (\347)3238 2136 w (\347)3238 2036 w (\347)3238 1936 w (\347)3238 1836 w (\347)3238 1736 w (\347)3238 1636 w (\347)3238 1536 w (\347)3238 1436 w (\347)3831 2316 w (\347)3831 2296 w (\347)3831 2196 w (\347)3831 2096 w (\347)3831 1996 w (\347)3831 1896 w (\347)3831 1796 w (\347)3831 1696 w (\347)3831 1596 w (\347)3831 1496 w (\347)3831 1396 w (\347)3831 1296 w cleartomark showpage saveobj restore end %%EndPage: 31 70 %%Page: 1 71 DpostDict begin /saveobj save def mark 71 pagesetup 10 B f (Appendix 5)1 493 1 2633 840 t (Routine, Common and Error State Summary)5 1932 1 1914 1200 t 10 R f (This appendix summarizes the calling sequences for the routines of)9 2759 1 970 1596 t 10 CW f (TTGR)3763 1596 w 10 R f (, the contents of the rela-)5 1037 1 4003 1596 t ( meant to serve as a reference guide, not)8 1643( is terse and)3 492( It)1 117(tively public Common regions and the error states.)7 2068 4 720 1716 t ( top level of)3 480( The)1 205(as a tutorial.)2 491 3 720 1836 t 10 CW f (TTGR)1921 1836 w 10 R f (is)2186 1836 w 10 CW f (Call TTGR\(U,Nu,kx,x,nx, ky,y,ny,)2 1920 1 1200 2016 t (tstart,tstop,dt,)1800 2136 w (AF,BC,)1800 2256 w (errpar,)1800 2376 w (HANDLE\))1800 2496 w 10 R f (The)720 2712 w 10 CW f (AF)900 2712 w 10 R f (procedure is of the form)4 964 1 1045 2712 t 10 CW f (Subroutine AF\(t,x,nx,y,ny,U,Ux,Uy,Ut,Utx,Uty,Nu,)1 2880 1 1200 2892 t (A,AU,AUx,AUy,AUt,AUtx,AUty,)2040 3012 w (f,fU,fUx,fUy,fUt,fUtx,fUty\))2040 3132 w 10 R f (the)720 3312 w 10 CW f (BC)867 3312 w 10 R f (procedure is of the form)4 964 1 1012 3312 t 10 CW f (Subroutine BC\(t,Lx,Rx,Ly,Ry,U,Ux,Uy,Ut,Utx,Uty,Nu,)1 3000 1 1200 3492 t (B,BU,BUx,BUy,BUt,BUtx,BUty\))2040 3612 w 10 R f (and the)1 291 1 720 3792 t 10 CW f (HANDLE)1036 3792 w 10 R f (procedure has the form)3 922 1 1421 3792 t 10 CW f (Subroutine HANDLE\(t0,U0,V0,t,U,V,Nu,nxmk,nymk,kx,x,nx,ky,y,ny,dt,tstop\))1 4260 1 1200 3972 t 10 R f (The "Return-End")1 728 1 720 4188 t 10 CW f (HANDLE)1473 4188 w 10 R f (procedure is)1 490 1 1858 4188 t 10 CW f (TTGRH)2373 4188 w 10 R f (.)2673 4188 w (The "Return-End")1 728 1 720 4344 t 10 CW f (BC)1473 4344 w 10 R f (procedure is)1 490 1 1618 4344 t 10 CW f (TTGRP)2133 4344 w 10 R f (, for when)2 407 1 2433 4344 t 10 I f (n n)1 50 1 2865 4344 t 7 I f (u u)1 35 1 2926 4364 t 10 S f (= =)1 55 1 3018 4344 t 10 R f (0.)3122 4344 w (The statistics printing procedure is)4 1382 1 720 4500 t 10 CW f (TTGRX)2127 4500 w 10 R f (.)2427 4500 w (The basic knob twiddling routine for)5 1468 1 970 4656 t 10 CW f (TTGR)2463 4656 w 10 R f (is)2728 4656 w 10 CW f (Call TTGRV\(j,f,r,i,l\))1 1260 1 1200 4836 t 10 R f (The following table summarizes the values that can be set by)10 2435 1 720 5052 t 10 CW f (TTGRV)3180 5052 w 10 S f (_ __________________________________)1 1725 1 2017 5132 t 10 R f ( to)1 103( Set)1 279( Default)1 577(Name j)1 605 4 2128 5252 t 10 S f (_ __________________________________)1 1725 1 2017 5272 t (_ __________________________________)1 1725 1 2017 5292 t 10 CW f (theta)2097 5472 w 10 R f (1 1)1 492 1 2694 5472 t 10 CW f (f)3546 5472 w (beta)2127 5592 w 10 R f (2 1)1 492 1 2694 5592 t 10 CW f (f)3546 5592 w (gamma)2097 5712 w 10 R f (3 1)1 492 1 2694 5712 t 10 CW f (f)3546 5712 w (delta)2097 5832 w 10 R f (4 0)1 492 1 2694 5832 t 10 CW f (f)3546 5832 w 10 S f (_ __________________________________)1 1725 1 2017 5852 t 10 CW f (hfract)2067 5972 w 10 R f (1001 1)1 567 1 2619 5972 t 10 CW f (r)3546 5972 w (egive)2097 6092 w 10 R f (1002 100)1 617 1 2619 6092 t 10 CW f (r)3546 6092 w 10 S f (_ __________________________________)1 1725 1 2017 6112 t 10 CW f (kj)2187 6232 w 10 R f (2001 0)1 567 1 2619 6232 t 10 CW f (i)3546 6232 w (minit)2097 6352 w 10 R f (2002 10)1 592 1 2619 6352 t 10 CW f (i)3546 6352 w (maxit)2097 6472 w 10 R f (2003 50)1 592 1 2619 6472 t 10 CW f (i)3546 6472 w (kmax)2127 6592 w 10 R f (2004 10)1 592 1 2619 6592 t 10 CW f (i)3546 6592 w (kinit)2097 6712 w 10 R f (2005 2)1 567 1 2619 6712 t 10 CW f (i)3546 6712 w (mmax)2127 6832 w 10 R f (2006 15)1 592 1 2619 6832 t 10 CW f (i)3546 6832 w (mxq)2157 6952 w 10 R f (2008)2619 6952 w 10 CW f (0 i)1 475 1 3131 6952 t 10 S f ( \347)1 -1725(_ __________________________________)1 1725 2 2017 6980 t (\347)2017 6932 w (\347)2017 6832 w (\347)2017 6732 w (\347)2017 6632 w (\347)2017 6532 w (\347)2017 6432 w (\347)2017 6332 w (\347)2017 6232 w (\347)2017 6132 w (\347)2017 6032 w (\347)2017 5932 w (\347)2017 5832 w (\347)2017 5732 w (\347)2017 5632 w (\347)2017 5532 w (\347)2017 5432 w (\347)2017 5332 w (\347)2017 5232 w (\347)2502 6980 w (\347)2502 6932 w (\347)2502 6832 w (\347)2502 6732 w (\347)2502 6632 w (\347)2502 6532 w (\347)2502 6432 w (\347)2502 6332 w (\347)2502 6232 w (\347)2502 6132 w (\347)2502 6032 w (\347)2502 5932 w (\347)2502 5832 w (\347)2502 5732 w (\347)2502 5632 w (\347)2502 5532 w (\347)2502 5432 w (\347)2502 5332 w (\347)2502 5232 w (\347)2936 6980 w (\347)2936 6932 w (\347)2936 6832 w (\347)2936 6732 w (\347)2936 6632 w (\347)2936 6532 w (\347)2936 6432 w (\347)2936 6332 w (\347)2936 6232 w (\347)2936 6132 w (\347)2936 6032 w (\347)2936 5932 w (\347)2936 5832 w (\347)2936 5732 w (\347)2936 5632 w (\347)2936 5532 w (\347)2936 5432 w (\347)2936 5332 w (\347)2936 5232 w (\347)3386 6980 w (\347)3386 6932 w (\347)3386 6832 w (\347)3386 6732 w (\347)3386 6632 w (\347)3386 6532 w (\347)3386 6432 w (\347)3386 6332 w (\347)3386 6232 w (\347)3386 6132 w (\347)3386 6032 w (\347)3386 5932 w (\347)3386 5832 w (\347)3386 5732 w (\347)3386 5632 w (\347)3386 5532 w (\347)3386 5432 w (\347)3386 5332 w (\347)3386 5232 w (\347)3742 6980 w (\347)3742 6932 w (\347)3742 6832 w (\347)3742 6732 w (\347)3742 6632 w (\347)3742 6532 w (\347)3742 6432 w (\347)3742 6332 w (\347)3742 6232 w (\347)3742 6132 w (\347)3742 6032 w (\347)3742 5932 w (\347)3742 5832 w (\347)3742 5732 w (\347)3742 5632 w (\347)3742 5532 w (\347)3742 5432 w (\347)3742 5332 w (\347)3742 5232 w cleartomark showpage saveobj restore end %%EndPage: 1 71 %%Page: 2 72 DpostDict begin /saveobj save def mark 72 pagesetup 10 R f (\261 5-2 \261)2 283 1 2738 480 t 10 CW f (myq)2157 840 w 10 R f (2009)2619 840 w 10 CW f (0 i)1 475 1 3131 840 t (la)2187 960 w 10 R f (2010 1)1 567 1 2619 960 t 10 CW f (i)3546 960 w (pieces)2067 1080 w 10 R f (2011 +2)1 595 1 2619 1080 t 10 CW f (i)3546 1080 w (pc)2187 1200 w 10 R f (2012 0)1 567 1 2619 1200 t 10 CW f (i)3546 1200 w (accel)2097 1320 w 10 R f (2013 0)1 567 1 2619 1320 t 10 CW f (i)3546 1320 w 10 S f (_ __________________________________)1 1725 1 2017 1340 t 10 CW f (xpoly)2097 1460 w 10 R f (3001)2619 1460 w 10 CW f (False l)1 595 1 3011 1460 t (erputs)2067 1580 w 10 R f (3002)2619 1580 w 10 CW f (False l)1 595 1 3011 1580 t 10 S f (_ __________________________________)1 1725 1 2017 1600 t 10 CW f (N)2170 1720 w 10 R f (\(i\) 4000+i)1 631 1 2230 1720 t 10 S f (- -)1 55 1 3133 1720 t 10 CW f (i)3546 1720 w 10 S f ( \347)1 -1725(_ __________________________________)1 1725 2 2017 1740 t (\347)2017 1720 w (\347)2017 1620 w (\347)2017 1520 w (\347)2017 1420 w (\347)2017 1320 w (\347)2017 1220 w (\347)2017 1120 w (\347)2017 1020 w (\347)2017 920 w (\347)2017 820 w (\347)2502 1740 w (\347)2502 1720 w (\347)2502 1620 w (\347)2502 1520 w (\347)2502 1420 w (\347)2502 1320 w (\347)2502 1220 w (\347)2502 1120 w (\347)2502 1020 w (\347)2502 920 w (\347)2502 820 w (\347)2936 1740 w (\347)2936 1720 w (\347)2936 1620 w (\347)2936 1520 w (\347)2936 1420 w (\347)2936 1320 w (\347)2936 1220 w (\347)2936 1120 w (\347)2936 1020 w (\347)2936 920 w (\347)2936 820 w (\347)3386 1740 w (\347)3386 1720 w (\347)3386 1620 w (\347)3386 1520 w (\347)3386 1420 w (\347)3386 1320 w (\347)3386 1220 w (\347)3386 1120 w (\347)3386 1020 w (\347)3386 920 w (\347)3386 820 w (\347)3742 1740 w (\347)3742 1720 w (\347)3742 1620 w (\347)3742 1520 w (\347)3742 1420 w (\347)3742 1320 w (\347)3742 1220 w (\347)3742 1120 w (\347)3742 1020 w (\347)3742 920 w (\347)3742 820 w 10 R f (where)720 1956 w 10 CW f (mxq)988 1956 w 10 S f (= =)1 55 1 1193 1956 t 10 R f (0 means that)2 505 1 1297 1956 t 10 CW f (kx)1827 1956 w 10 R f (quadrature points will be used in)5 1307 1 1972 1956 t 10 I f (x x)1 44 1 3304 1956 t 10 R f (, similarly for)2 547 1 3348 1956 t 10 CW f (myq)3920 1956 w 10 R f (.)4100 1956 w (The procedural knob twiddler is)4 1281 1 970 2112 t 10 CW f (Call TTGRR\(U,Nu,kx,x,nx, ky,y,ny,)2 1980 1 1140 2292 t (tstart,tstop,dt,)1800 2412 w (AF,BC,)1800 2532 w (ERROR,errpar,)1800 2652 w (HANDLE\))1800 2772 w 10 R f (where the)1 390 1 720 2952 t 10 CW f (ERROR)1135 2952 w 10 R f (procedure has the form)3 922 1 1460 2952 t 10 CW f (Logical Function ERROR\(U,Nu,nxmk,kx,x,nx, ky,y,ny,t,dt,)3 3300 1 1200 3132 t (errpar,)2580 3252 w (erputs,)2580 3372 w (eU,eV\))2580 3492 w 10 R f (The)720 3672 w 10 CW f (TTGR)900 3672 w 10 R f (coordinate mapping routines for)3 1284 1 1165 3672 t 10 CW f (AF)2474 3672 w 10 R f (are)2619 3672 w 10 CW f (Call TTGRU\(nx,ny,D, Ux,Uy,Ut, Nu\))3 1980 1 1200 3852 t 10 R f (to map from internal to user coordinates, and)7 1797 1 720 4032 t 10 CW f (Call TTGRG\( nx,ny,D, Nu,)3 1440 1 1200 4212 t (A,AU,AUx,AUy,)1920 4332 w (F,FU,FUx,FUy\))1920 4452 w 10 R f (to map from user to internal coordinates.)6 1628 1 720 4632 t (For mapping)1 524 1 720 4788 t 10 CW f (BC)1279 4788 w 10 R f (from internal to user coordinates there is)6 1679 1 1434 4788 t 10 CW f (TTGRU)3183 4788 w 10 R f (as used above in)3 687 1 3518 4788 t 10 CW f (AF)4240 4788 w 10 R f (, and to map)3 524 1 4360 4788 t 10 CW f (BC)4920 4788 w 10 R f (from user back into internal coordinates)5 1594 1 720 4908 t 10 CW f (Call TTGRB\( nx,ny,D, Nu, BUx,BUy,BUt \))5 2280 1 1200 5088 t 10 B f (Common Regions)1 758 1 720 5388 t 10 R f (When the user cannot evaluate any of)6 1501 1 970 5544 t 10 CW f (AF)2496 5544 w 10 R f (or)2641 5544 w 10 CW f (BC)2749 5544 w 10 R f (that fact can be signaled to)5 1067 1 2894 5544 t 10 CW f (TTGR)3986 5544 w 10 R f (via)4251 5544 w 10 CW f (Common / TTGRF / Failed; Logical Failed)6 2340 1 1200 5724 t 10 B f (Naming Conventions)1 898 1 720 6024 t 10 R f (The naming convention for)3 1105 1 970 6180 t 10 CW f (TTGR)2105 6180 w 10 R f ( for the Port Library: all hidden \( not user call-)10 1918(is the same as that)4 747 2 2375 6180 t ( names, there)2 533( users avoid such)3 694( If)1 119(able \) subroutines have names beginning with a letter followed by a digit.)12 2974 4 720 6300 t (will be no name conflicts.)4 1035 1 720 6420 t cleartomark showpage saveobj restore end %%EndPage: 2 72 %%Page: 3 73 DpostDict begin /saveobj save def mark 73 pagesetup 10 R f (\261 5-3 \261)2 283 1 2738 480 t 10 B f (Error States.)1 554 1 720 840 t 10 R f ( provides a list of the error states [14] that may be encountered when using)14 3011(This section)1 487 2 970 996 t 10 CW f (TTGR)4495 996 w 10 R f (. Some)1 305 1 4735 996 t ( aid the user in finding bugs \(if they exist\) in the user-)12 2297(interpretation of these error messages is made to)7 2023 2 720 1116 t (supplied code)1 556 1 720 1236 t 10 CW f (AF)1305 1236 w 10 R f (,)1425 1236 w 10 CW f (BC)1479 1236 w 10 R f (or)1628 1236 w 10 CW f (HANDLE)1740 1236 w 10 R f ( \(entry to\))2 409( each level of)3 546(. For)1 218 3 2100 1236 t 10 CW f (TTGR)3303 1236 w 10 R f (, the error message and number for a)7 1497 1 3543 1236 t (given error state is the same, always reflecting the error from the bottom layer of)14 3321 1 720 1356 t 10 CW f (TTGR)4072 1356 w 10 R f ( list of error)3 492(. The)1 236 2 4312 1356 t ( the complete set of error states for the)8 1559(states below, along with interpretation, is)5 1656 2 720 1476 t 10 CW f (TTGR)3963 1476 w 10 R f (package as obtained)2 809 1 4231 1476 t (from the lowest level of)4 954 1 720 1596 t 10 CW f (TTGR)1699 1596 w 10 R f (.)1939 1596 w (There are many internal variables of)5 1478 1 970 1752 t 10 CW f (TTGR)2479 1752 w 10 R f (that may be controlled by the subroutine)6 1651 1 2750 1752 t 10 CW f (TTGRV)4433 1752 w 10 R f (. Thus,)1 307 1 4733 1752 t (there are many more ways to call)6 1342 1 720 1872 t 10 CW f (TTGR)2090 1872 w 10 R f ( the obvious ones involving data in the call-)8 1765(with bad data than just)4 917 2 2358 1872 t (ing sequence for)2 675 1 720 1992 t 10 CW f (TTGR)1428 1992 w 10 R f ( is complete for)3 651( list of error states given below)6 1285(. The)1 238 3 1668 1992 t 10 CW f (TTGR)3876 1992 w 10 R f (/)4116 1992 w 10 CW f (TTGRV)4144 1992 w 10 R f (, and therefore)2 596 1 4444 1992 t (involves variables not mentioned in the calling sequence for)8 2450 1 720 2112 t 10 CW f (TTGR)3236 2112 w 10 R f ( you are only using)4 790(. If)1 147 2 3476 2112 t 10 CW f (TTGR)4443 2112 w 10 R f (, and not)2 357 1 4683 2112 t 10 CW f (TTGRV)720 2232 w 10 R f (as well, then the error states mentioning variables not in the)10 2473 1 1054 2232 t 10 CW f (TTGR)3561 2232 w 10 R f ( if you)2 281( So)1 166(call can not occur.)3 758 3 3835 2232 t ( can-)1 199(bump into an error state below that mentions an undescribed or unknown variable, simply ignore it, it)16 4121 2 720 2352 t (not happen to you.)3 744 1 720 2472 t ( .lt. 1.)2 231( Nu)1 239(1 -)1 283 3 770 2664 t ( .lt. 2.)2 231( kx)1 217(2 -)1 283 3 770 2820 t ( .lt. 2*kx.)2 381( nx)1 217(3 -)1 283 3 770 2976 t ( .lt. 2.)2 231( ky)1 217(4 -)1 283 3 770 3132 t ( .lt. 2*ky.)2 381( ny)1 217(5 -)1 283 3 770 3288 t ( user-chosen value for the time-step dt is too small, that is, tstart+dt)12 2693( The)1 205( on input.)2 381( dt=0)1 301(6 -)1 283 5 770 3444 t 10 S f (\272)4658 3444 w 10 R f (tstart.)4738 3444 w ( and tstop-tstart must have the same sign.)7 1649( dt)1 128( has wrong sign on input.)5 1011( dt)1 195(7 -)1 283 5 770 3600 t ( .lt. kx-1.)2 364( mxq)1 295(8 -)1 283 3 770 3756 t ( .lt. ky-1.)2 364( myq)1 295(9 -)1 283 3 770 3912 t ( must be 1 or 2.)5 622( Abs\(LA\))1 477(10 -)1 283 3 770 4068 t ( must be 0 or 2.)5 622( Pieces)1 372(11 -)1 283 3 770 4224 t ( must be 0, 1 or 2.)6 722( PC)1 240(12 -)1 283 3 770 4380 t ( must be 0, 1 or 2.)6 722( Accel)1 349(13 -)1 283 3 770 4536 t ( PC .gt. 0 need Abs\(LA\) = 2.)7 1155( For)1 256(14 -)1 283 3 770 4692 t ( PC .gt. 0 need Accel = 1 or 2.)9 1210( For)1 256(15 -)1 283 3 770 4848 t ( from)1 219( dt)1 195(1000 -)1 283 3 770 5004 t 10 CW f (HANDLE)1492 5004 w 10 R f (has wrong sign. Recoverable.)3 1178 1 1877 5004 t ( raise dt in)3 419( Cannot)1 406(1001 -)1 283 3 770 5160 t 10 CW f (HANDLE)1903 5160 w 10 R f (when Failure is set. Recoverable.)4 1325 1 2288 5160 t ( .le. 0 returned by)4 736( E\(i\))1 272(1002 -)1 283 3 770 5316 t 10 CW f (ERROR)2094 5316 w 10 R f ( a relative)2 411( Having)1 353( error request is too small.)5 1088( The)1 213(. Recoverable.)1 581 5 2394 5316 t (error request on a variable going to 0 can cause this.)10 2081 1 1170 5436 t ( have mgq=k when one of the)6 1187( Must)1 256( and Order=0. Recoverable.)3 1105( mxq=kx-1)1 534(1003 -)1 283 5 770 5592 t 10 CW f (pde)4160 5592 w 10 R f (s is of zero order.)4 695 1 4340 5592 t (1004 -)1 283 1 770 5748 t 10 CW f (pde)1170 5748 w 10 R f ( is no)2 217( There)1 282(\(i\) is vacuous. Recoverable.)3 1111 3 1350 5748 t 10 I f (i i)1 28 1 2985 5748 t 7 I f ( h)1 0(t th)1 55 2 3024 5708 t 10 CW f (pde)3112 5748 w 10 R f (.)3292 5748 w ( The)1 205( BCs. Recoverable.)2 771( Improper)1 488(1005 -)1 283 4 770 5904 t 10 B f (bc)2542 5904 w 10 R f (s and)1 208 1 2642 5904 t 10 CW f (pde)2875 5904 w 10 R f (s do not match properly.)4 974 1 3055 5904 t (1006 -)1 283 1 770 6060 t 10 CW f (pde)1170 6060 w 10 R f ( The)1 214(system not in minimal order form. Recoverable.)6 1974 2 1384 6060 t 10 CW f (pde)3607 6060 w 10 R f (can have derivatives removed)3 1218 1 3822 6060 t (from it.)1 300 1 1170 6180 t ( few boundary conditions. Recoverable.)4 1591( Too)1 278(1007 -)1 283 3 770 6336 t ( many boundary conditions. Recoverable.)4 1664( Too)1 278(1008 -)1 283 3 770 6492 t ( are too many mixed)4 821( There)1 282( boundary conditions are overdetermined. Recoverable.)5 2214( Mixed)1 378(1009 -)1 283 5 770 6648 t 10 B f (bc)4773 6648 w 10 R f (s.)4873 6648 w ( mixed)1 275( The)1 205( Mixed BCs. Recoverable.)3 1057( Singular)1 456(1010 -)1 283 5 770 6804 t 10 B f (bc)3071 6804 w 10 R f (s were singular so frequently that dt went to 0.)9 1853 1 3171 6804 t ( too many Dirichlet)3 810( are)1 155( There)1 291( boundary conditions are overdetermined. Recoverable.)5 2259( Dirichlet)1 472(1011 -)1 283 6 770 6960 t 10 B f (bc)1170 7080 w 10 R f (s.)1270 7080 w cleartomark showpage saveobj restore end %%EndPage: 3 73 %%Page: 5 74 DpostDict begin /saveobj save def mark 74 pagesetup 10 R f (\261 5-5 \261)2 283 1 2738 480 t ( Dirichlet)1 381( The)1 206( Dirichlet BCs. Recoverable.)3 1154( Singular)1 456(1012 -)1 283 5 770 840 t 10 B f (bc)3276 840 w 10 R f (s were singular so frequently that dt went)7 1664 1 3376 840 t (to 0.)1 178 1 1170 960 t ( Jacobian for the)3 698( The)1 217( Jacobian. Recoverable.)2 971( Singular)1 456(1013 -)1 283 5 770 1116 t 10 CW f (pde)3432 1116 w 10 R f ( that dt)2 304(was singular so frequently)3 1087 2 3649 1116 t (went to 0.)2 397 1 1170 1236 t ( of stack space for LU decomposition. Recoverable.)7 2070( Out)1 267(1014 -)1 283 3 770 1392 t ( many iterations needed in splitting. Recoverable.)6 1981( Too)1 278(1015 -)1 283 3 770 1548 t ( problem may be very badly)5 1228( The)1 226( has become too small.)4 997( time-step)1 417( The)1 225( Recoverable.)1 568( dt=0.)1 326(1016 -)1 283 8 770 1704 t ( Another)1 389(scaled, that is units like light-years and micro-grams are being used simultaneously.)11 3481 2 1170 1824 t ( accuracy requirement, like errpar\(2\)=0 when the solution is exceedingly)9 3003(cause is too small an)4 867 2 1170 1944 t (small.)1170 2064 w ( returned from)2 576( dt=0)1 301(1017 -)1 283 3 770 2220 t 10 CW f (HANDLE)1955 2220 w 10 R f ( lowered dt and it became too small.)7 1448( Handle)1 338(. Recoverable.)1 573 3 2315 2220 t (1018 -)1 283 1 770 2376 t 10 CW f (AF)1170 2376 w 10 R f (,)1290 2376 w 10 CW f (BC)1340 2376 w 10 R f ( = True occurred in)4 770( Failed)1 300(failure. Recoverable.)1 833 3 1485 2376 t 10 CW f (AF)3413 2376 w 10 R f (or)3558 2376 w 10 CW f (BC)3666 2376 w 10 R f (so often that dt went to 0.)6 1019 1 3811 2376 t ( number of Newton iterations was pre-)6 1547( The)1 206( predicted. Recoverable.)2 971( many Newton iterations)3 985( Too)1 278(1019 -)1 283 6 770 2532 t ( Jacobian, see next error)4 970( cause is a bad)4 584( Probable)1 407(dicted to be too large so often that dt went to 0.)11 1909 4 1170 2652 t (state.)1170 2772 w ( many Newton iterations were needed so)6 1630( Too)1 212( needed. Recoverable.)2 882( many Newton iterations)3 985( Too)1 278(1020 -)1 283 6 770 2928 t ( possible)1 360( Another)1 384( cause is an incorrectly computed Jacobian.)6 1774( Probable)1 412(often that dt went to 0.)5 940 5 1170 3048 t ( another possible cause is a very badly condi-)8 1821( Yet)1 195(cause is that Minit and/or Maxit are too small.)8 1854 3 1170 3168 t ( stringent an error request or a mesh that is too)10 1988( possible causes: too)3 861( Further)1 358(tioned Jacobian.)1 663 4 1170 3288 t (non-uniform.)1170 3408 w cleartomark showpage saveobj restore end %%EndPage: 5 74 %%Trailer DpostDict begin done end %%Pages: 74 %%DocumentFonts: Courier Times-Bold Times-Italic Times-Roman Times-Roman Symbol