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Lower left coordinates % are always set to 0. % /roundpagebbox { 7 dict begin /papersizes [8.5 inch 11 inch 14 inch 17 inch] def /mappapersize { /val exch def /slop .5 inch def /diff slop def /j 0 def 0 1 papersizes length 1 sub { /i exch def papersizes i get val sub abs dup diff le {/diff exch def /j i def} {pop} ifelse } for diff slop lt {papersizes j get} {val} ifelse } def pagebbox 0 0 put pagebbox 1 0 put pagebbox dup 2 get mappapersize 2 exch put pagebbox dup 3 get mappapersize 3 exch put end } bind def %%EndProlog %%BeginSetup mark /resolution 720 def setup 2 setdecoding %%EndSetup %%Page: 0 1 /saveobj save def mark 1 pagesetup 12 CW f (TTGU)2117 1230 w 12 B f (- A Package for Solving)4 1207 1 2435 1230 t (Time Varying Partial Differential Equations)4 2277 1 1741 1380 t (on a Union of Rectangles)4 1274 1 2243 1530 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 1770 t 10 R f (AT&T Bell Laboratories)2 993 1 2383 1950 t (Murray Hill, New Jersey 07974)4 1267 1 2246 2070 t 10 I f ( T)1 0( CT)1 56( RA AC)2 128( TR)1 61( ST)1 56(A AB BS)2 172 6 2643 2570 t 10 R f ( rectangles)1 430(A formulation is presented for partial differential equations on a union of)11 2920 2 1330 2830 t ( advantage of this for-)4 889( algorithm taking full)3 862( An)1 175(which facilitates their numerical solution.)4 1674 4 1080 2950 t (mulation is briefly outlined.)3 1117 1 1080 3070 t ( Fortran, called)2 637(An implementation of the algorithm in portable)6 1983 2 1330 3226 t 10 CW f (TTGU)3990 3226 w 10 R f (\(Transient)4270 3226 w ( equations on a Union of rectangles\), is described.)8 2036(Tensor Galerkin for partial differential)4 1564 2 1080 3346 t (It solves the same general type of partial differential equation as)10 2661 1 1080 3466 t 10 CW f (TTGR)3811 3466 w 10 R f ([15], but)1 354 1 4051 3466 t 10 CW f (TTGR)4440 3466 w 10 R f ( The)1 209( domain to a rectangle or domains can be easily mapped into rectangles.)12 2923(restricts the)1 468 3 1080 3586 t ( the spatial mesh and the accuracy desired in)8 1827(package is especially easy to use since only)7 1773 2 1080 3706 t ( time evolution is then)4 951( The)1 219( need to be specified.)4 901(the solution of the equations in time)6 1529 4 1080 3826 t ( user's guide to)3 664( A)1 139( accuracy.)1 420(automatically carried out to achieve the desired)6 1986 4 1080 3946 t 10 CW f (TTGU)4331 3946 w 10 R f (is)4613 3946 w (given along with many examples.)4 1346 1 1080 4066 t 10 B f (The examples are available through electronic mail.)6 2207 1 1901 4222 t 10 R f (There are 5 examples and the)5 1191 1 1330 4378 t 10 B f (fortran)2550 4378 w 10 R f ( double pre-)2 492(for each is available in single or)6 1299 2 2889 4378 t ( example, the command)3 954(cision. For)1 453 2 1080 4498 t (mail research!netlib)1 795 1 1540 4678 t (send only ttgux1 from port)4 1072 1 1540 4798 t (send only dttgux5 from port)4 1122 1 1540 4918 t (.)1540 5038 w ( mail the first example in single and the fifth example in)11 2322(will cause you to receive in the)6 1278 2 1080 5218 t (double precision.)1 688 1 1080 5338 t (October 29, 1990)2 696 1 720 5818 t cleartomark showpage saveobj restore %%EndPage: 0 1 %%Page: 1 2 /saveobj save def mark 2 pagesetup 12 CW f (TTGU)2117 1230 w 12 B f (- A Package for Solving)4 1207 1 2435 1230 t (Time Varying Partial Differential Equations)4 2277 1 1741 1380 t (on a Union of Rectangles)4 1274 1 2243 1530 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 1770 t 10 R f (AT&T Bell Laboratories)2 993 1 2383 1950 t (Murray Hill, New Jersey 07974)4 1267 1 2246 2070 t 10 B f (1. Introduction.)1 695 1 720 2430 t 10 R f ( \()1 72(Many physical problems require the solution of partial differential equations)9 3171 2 970 2586 t 10 B f (pde)4213 2586 w 10 R f ('s\) in two space)3 671 1 4369 2586 t ( sufficiently complex that their solution must be carried out numeri-)10 2714( these equations are)3 787(variables. Typically)1 819 3 720 2706 t (cally.)720 2826 w (This paper describes a formulation for solving systems of)8 2345 1 970 2982 t 10 B f (pde)3346 2982 w 10 R f ( on a union)3 468('s in two spatial variables,)4 1070 2 3502 2982 t ( formulation allows for terms of the form)7 1652( The)1 206( time.)1 229( and)1 195(of rectangles,)1 538 5 720 3102 t 10 B f (u)3566 3102 w 10 R f (,)3622 3102 w 10 B f (u)3673 3102 w 7 I f (x x)1 31 1 3740 3122 t 10 R f (,)3779 3102 w 10 B f (u)3830 3102 w 7 I f (y y)1 31 1 3897 3122 t 10 R f (,)3936 3102 w 10 B f (u)3987 3102 w 7 I f (t t)1 20 1 4054 3122 t 10 R f (,)4082 3102 w 10 B f (u)4132 3102 w 7 I f ( t)1 0(x xt)1 51 2 4199 3122 t 10 R f (,)4258 3102 w 10 B f (u)4308 3102 w 7 I f ( t)1 0(y yt)1 51 2 4375 3122 t 10 R f (,)4434 3102 w 10 B f (u)4484 3102 w 7 I f (x xx x)2 62 1 4551 3122 t 10 R f (,)4621 3102 w 10 B f (u)4671 3102 w 7 I f ( t)1 0(x xx xt)2 82 2 4738 3122 t 10 R f (,)4828 3102 w 10 B f (u)4878 3102 w 7 I f (x xy y)2 62 1 4945 3122 t 10 R f (,)5015 3102 w 10 B f (u)720 3222 w 7 I f ( t)1 0(x xy yt)2 82 2 787 3242 t 10 R f (,)877 3222 w 10 B f (u)928 3222 w 7 I f (y yy y)2 62 1 995 3242 t 10 R f (,)1065 3222 w 10 B f (u)1116 3222 w 7 I f ( t)1 0(y yy yt)2 82 2 1183 3242 t 10 R f (in the)1 226 1 1299 3222 t 10 B f (pde)1552 3222 w 10 R f ('s, and)1 268 1 1708 3222 t 10 B f (u)2003 3222 w 10 R f (,)2059 3222 w 10 B f (u)2111 3222 w 7 I f (x x)1 31 1 2178 3242 t 10 R f (,)2217 3222 w 10 B f (u)2269 3222 w 7 I f (y y)1 31 1 2336 3242 t 10 R f (,)2375 3222 w 10 B f (u)2427 3222 w 7 I f (t t)1 20 1 2494 3242 t 10 R f (,)2522 3222 w 10 B f (u)2574 3222 w 7 I f ( t)1 0(x xt)1 51 2 2641 3242 t 10 R f (,)2700 3222 w 10 B f (u)2752 3222 w 7 I f ( t)1 0(y yt)1 51 2 2819 3242 t 10 R f (in the boundary conditions \()4 1135 1 2905 3222 t 10 B f (bc)4067 3222 w 10 R f ('s \), where)2 427 1 4167 3222 t 10 B f (u)4621 3222 w 10 R f (is a vec-)2 336 1 4704 3222 t (tor of)1 240 1 720 3342 t 10 B f (pde)1006 3342 w 10 R f (variables, and)1 575 1 1208 3342 t 10 B f (u)1829 3342 w 7 I f (x x)1 31 1 1896 3362 t 10 R f (denotes)1981 3342 w 10 S f (\266)2332 3342 w 10 B f (u)2389 3342 w 10 I f (/ /)1 28 1 2477 3342 t 10 S f (\266)2513 3342 w 10 I f (x x)1 44 1 2570 3342 t 10 R f ( that the interfaces)3 794( the package demands)3 941( Currently,)1 479(, etc.)1 212 4 2614 3342 t ( mathematical formulation is)3 1197( The)1 220( rectangle.)1 430(between rectangles be either a point of a whole side of the)11 2473 4 720 3462 t (given in section 2.)3 733 1 720 3582 t ( users, who have a problem on one rectangle)8 1792(This package extends the functionality of TTGR[15] and)7 2278 2 970 3738 t (or have a problem that can be mapped onto one rectangle, should be using TTGR and reading [15].)17 3956 1 720 3858 t 10 B f (Getting Started if you have not read TTGR[15]:)7 2046 1 720 4098 t 10 R f ( familiar with TTGR[15], want to solve a sim-)8 1891(If you have not read this document before and are not)10 2179 2 970 4254 t (ple)720 4374 w 10 B f (pde)875 4374 w 10 R f ( fancy things having nothing to do with your immediate needs, just do the)13 3040(and have no interest in)4 936 2 1064 4374 t (following)720 4494 w 10 S f (\267)970 4650 w 10 R f (Read section 2, Statement of the Problem, p 3.)8 1857 1 1041 4650 t 10 S f (\267)970 4806 w 10 R f (Read section 4, Formulation, Example 1, p 7.)7 1813 1 1041 4806 t 10 S f (\267)970 4962 w 10 R f (Skim section 5, Software, pp 11-19.)5 1438 1 1041 4962 t 10 S f (\267)970 5118 w 10 R f (Read Appendix 1, Programs, Example 1, pp 1-9.)7 1945 1 1041 5118 t 10 S f (\267)970 5274 w 10 R f (Copy the example program \( either from a file or the paper \).)12 2430 1 1041 5274 t 10 S f (\267)970 5430 w 10 R f (Run it and make sure it works as advertised.)8 1767 1 1041 5430 t 10 S f (\267)970 5586 w 10 R f (Alter it to solve your problem; see the start of Appendix 1 for the steps involved here.)16 3426 1 1041 5586 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 5742 t (dozen lines of code in the example to get a problem solved.)11 2374 1 720 5862 t 10 B f (Getting started if you are familiar with TTGR[15]:)7 2167 1 720 6102 t 10 R f (If you are familiar with TTGR and want to solve a)10 2013 1 970 6258 t 10 B f (pde)3008 6258 w 10 R f (, you should do or know the following:)7 1561 1 3164 6258 t 10 S f (\267)970 6414 w 10 R f ( are three extra parameters, not)5 1299( There)1 295( TTGU.)1 329(Read pp. 10-11 indicating the calling sequence of)7 2064 4 1053 6414 t (needed in TTGR, which permit the user to specify the mesh.)10 2413 1 970 6534 t 10 S f (\267)970 6690 w 10 R f ( BC used for specifying the partial differential equations and the boundary)11 3003( and)1 171( AF)1 180(The subroutines)1 643 4 1043 6690 t (conditions have not been changed from TTGR.)6 1883 1 970 6810 t 10 S f (\267)970 6966 w 10 R f (If you have been using TSL2W to specify nonconstant boundary conditions or now have non con-)15 3994 1 1046 6966 t (stant boundary conditions, you will want to read about the subroutine ICON on p. 18.)14 3417 1 970 7086 t 10 S f (\267)970 7242 w 10 R f ( in the current implementation only banded direct solvers are)9 2486(The default options are the same, but)6 1508 2 1046 7242 t cleartomark showpage saveobj restore %%EndPage: 1 2 %%Page: 2 3 /saveobj save def mark 3 pagesetup 10 R f (- 2 -)2 166 1 2797 480 t ( the whole Jacobian must be computed. One)7 1855(available to solve the underlying matrix problem and)7 2215 2 970 840 t (cannot ask that only certain pieces of the Jacobian be computed as one could in TTGR.)15 3477 1 970 960 t 10 S f (\267)970 1116 w 10 R f (Read Appendix 1, Programs, Example 1, pp.1-9.)6 1945 1 1041 1116 t 10 B f (The Examples)1 609 1 720 1356 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 1512 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 1632 t 10 S f (\267)970 1788 w 10 R f (A simple)1 364 1 1041 1788 t 10 B f (pde)1430 1788 w 10 R f (, the heat equation, see Example 1, p-7.)7 1572 1 1586 1788 t 10 S f (\267)970 1944 w 10 R f (A coupled system of)3 824 1 1041 1944 t 10 B f (pde)1890 1944 w 10 R f (s, see Example 2, p 7.)5 871 1 2046 1944 t 10 S f (\267)970 2100 w 10 R f (A material interface, see Example 3, p 8)7 1604 1 1041 2100 t 10 S f (\267)970 2256 w 10 R f (A nonconstant initial conditions problem, see Example 4, p 9)9 2446 1 1041 2256 t 10 S f (\267)970 2412 w 10 R f (A static problem, see Example 5, pp 9-10)7 1656 1 1041 2412 t 10 B f (A Principle.)1 511 1 720 2652 t 10 R f (The guiding principle used during the design of)7 1901 1 720 2808 t 10 CW f (TTGU)2646 2808 w 10 R f (was)2911 2808 w (It is better a user complain the package runs slowly than)10 2246 1 1080 2988 t (complain the package cannot solve the problem at hand.)8 2243 1 1080 3108 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 3324 t (and problems \( that is, people whose time is more valuable than machine time \), should find)16 3834 1 720 3444 t 10 CW f (TTGU)4589 3444 w 10 R f (very)4863 3444 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 3564 t (find)720 3684 w 10 CW f (TTGU)908 3684 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 3684 t 10 CW f (TTGU)4354 3684 w 10 R f (up consid-)1 420 1 4620 3684 t (erably.)720 3804 w ( Galerkin's method in space, using B-splines, and a vari-)9 2338(The numerical solution technique employs)4 1732 2 970 3960 t ( section 3 for an out-)5 844(able order, variable time-step extrapolated backward difference procedure in time; see)10 3476 2 720 4080 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 4200 t 10 CW f (TTGU)4800 4200 w 10 R f ( 6)1 77( Section)1 352(for Transient Tensor product Galerkin for partial differential equations on a Union of rectangles.)13 3891 3 720 4320 t (describes ways of making)3 1073 1 720 4440 t 10 CW f (TTGU)1831 4440 w 10 R f ( alternative)1 460(run faster than the default settings allow and also describes)9 2471 2 2109 4440 t (ways of entering and using the package.)6 1599 1 720 4560 t (Appendix 1 presents the programs used to solve the examples discussed in section 4.)13 3385 1 970 4716 t ( in)1 118(Appendix 2 summarizes the basic procedures available)6 2279 2 970 4872 t 10 CW f (TTGU)3407 4872 w 10 R f (, along with their arguments, and)5 1393 1 3647 4872 t ( using)1 243(gives a list of error states and problems that may arise when)11 2415 2 720 4992 t 10 CW f (TTGU)3404 4992 w 10 R f (, along with the common causes of)6 1396 1 3644 4992 t (such difficulties.)1 666 1 720 5112 t ( respectively, and the)3 890(Appendix 1 and 2 of [15] give brief tutorials on B-splines and extrapolation)12 3180 2 970 5268 t (interested reader should peruse that document for background information.)8 2991 1 720 5388 t 10 B f (A Warning.)1 506 1 720 5628 t 10 R f ( behavior, please contact the)4 1161( the user finds bugs or unexpected)6 1400( If)1 122(This software is in an infant state.)6 1387 4 970 5784 t ( goal is to create a robust)6 1040( The)1 212( using the best tools at hand.)6 1178( code has been created modularly,)5 1397(author. The)1 493 5 720 5904 t (and widely applicable package for solving two-dimensional)6 2403 1 720 6024 t 10 B f (pde)3151 6024 w 10 R f ( modularity has hurt a bit.)5 1053( this)1 173(s. However,)1 507 3 3307 6024 t (For example, the)2 694 1 720 6144 t 10 B f (ode)1449 6144 w 10 R f (solver)1634 6144 w 10 CW f (IODE)1913 6144 w 10 R f ( discretized)1 466(from the Port Library has been used to solve the spatially)10 2386 2 2188 6144 t ( user can change the name of the)7 1388( the)1 158(equations. Thus,)1 694 3 720 6264 t 10 B f (pde)2997 6264 w 10 R f (defining subroutine, but not that for the)6 1646 1 3190 6264 t 10 B f (bc)4873 6264 w 10 R f (s:)4973 6264 w 10 CW f (IODE)720 6384 w 10 R f ( is a learning experience for that software as)8 1785( This)1 231(only has one subroutine name it can pass below it.)9 2036 3 988 6384 t ( linear algebra is robust, but not neces-)7 1554( The)1 205( spatial discretization scheme is very robust, but slow.)8 2159(well. The)1 402 4 720 6504 t ( than none)2 426( to quote a user: "Expensive solutions are better)8 1953( However,)1 446(sarily optimal in run-time and space.)5 1495 4 720 6624 t ( use it any way you see)6 944( Please)1 308( and mistrustful of this code.)5 1159( bottom line here is: be careful with)7 1437( The)1 207(at all.")1 265 6 720 6744 t ( any bugs or anomalous behavior to the author, and let the author know generally how things go)17 3936(fit, report)1 384 2 720 6864 t ( electronic mail address is)4 1037( My)1 189(with it.)1 284 3 720 6984 t (research!lck)2637 7164 w cleartomark showpage saveobj restore %%EndPage: 2 3 %%Page: 3 4 /saveobj save def mark 4 pagesetup 10 R f (- 3 -)2 166 1 2797 480 t 10 B f (The examples are available through electronic mail.)6 2207 1 1901 840 t 10 R f (There are 5 examples and the)5 1196 1 970 996 t 10 B f (fortran)2196 996 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 996 t (ple, the command)2 713 1 720 1116 t (mail research!netlib)1 795 1 1180 1296 t (send only ttgux1 from port)4 1072 1 1180 1416 t (send only dttgux5 from port)4 1122 1 1180 1536 t (.)1180 1656 w (will cause you to receive in the mail the first example in single and the fifth example in double precision.)19 4200 1 720 1836 t 10 B f ( of the Problem.)3 682(2. Statement)1 557 2 720 2076 t 10 R f ( can be handled by)4 808(The general form of equations that)5 1453 2 970 2232 t 10 CW f (TTGU)3271 2232 w 10 R f (is an essentially classical, text-book)4 1489 1 3551 2232 t (divergence-form)720 2352 w 10 B f (pde)1409 2352 w 10 R f (with a general set of boundary conditions \()7 1711 1 1590 2352 t 10 B f (bc)3326 2352 w 10 R f (s \).)1 122 1 3426 2352 t 10 B f (The pde-bc Formulation.)2 1070 1 720 2592 t 10 R f (On each rectangle, the general)4 1249 1 970 2748 t 10 B f (pde)2254 2748 w 10 R f (-)2410 2748 w 10 B f (bc)2443 2748 w 10 R f ( with the approach used in)5 1106(form that can be solved)4 977 2 2578 2748 t 10 CW f (TTGU)4697 2748 w 10 R f (is)4973 2748 w ( following equations, where)3 1118(given by the)2 498 2 720 2868 t 10 B f (u)2362 2868 w 10 R f (is a vector of)3 521 1 2444 2868 t 10 B f (pde)2991 2868 w 10 R f (variables of length)2 745 1 3173 2868 t 10 I f (n n)1 50 1 3944 2868 t 7 I f (u u)1 35 1 4005 2888 t 10 R f ( all physical laws)3 694(. Since)1 298 2 4048 2868 t (that are second order in space can be written in divergence form, the)12 2728 1 720 2988 t 10 B f (pde)3473 2988 w 10 R f ('s are assumed to be in semi-linear,)6 1411 1 3629 2988 t (divergence-form)720 3108 w 10 S f (\266)1245 3398 w 10 I f (x x)1 44 1 1302 3398 t 10 S f (\266)1271 3268 w 10 S1 f (_ __)1 131 1 1230 3298 t 10 B f (a)1403 3328 w 7 R f (\( 1 \))2 91 1 1458 3288 t 10 R f (\()1565 3328 w 10 I f (t t)1 28 1 1606 3328 t 10 R f (,)1642 3328 w 10 I f (x x)1 44 1 1708 3328 t 10 R f (,)1760 3328 w 10 I f (y y)1 44 1 1826 3328 t 10 R f (,)1878 3328 w 10 B f (u)1944 3328 w 10 R f (,)2008 3328 w 10 B f (u)2074 3328 w 7 I f (x x)1 31 1 2141 3348 t 10 R f (,)2188 3328 w 10 B f (u)2254 3328 w 7 I f (y y)1 31 1 2321 3348 t 10 R f (,)2368 3328 w 10 B f (u)2434 3328 w 7 I f (t t)1 20 1 2501 3348 t 10 R f (,)2537 3328 w 10 B f (u)2603 3328 w 7 I f ( t)1 0(x xt)1 51 2 2670 3348 t 10 R f (,)2737 3328 w 10 B f (u)2803 3328 w 7 I f ( t)1 0(y yt)1 51 2 2870 3348 t 10 R f (\))2937 3328 w 10 S f (+ +)1 55 1 3027 3328 t (\266)1245 3668 w 10 I f (y y)1 44 1 1302 3668 t 10 S f (\266)1271 3538 w 10 S1 f (_ __)1 131 1 1230 3568 t 10 B f (a)1403 3598 w 7 R f (\( 2 \))2 91 1 1458 3558 t 10 R f (\()1565 3598 w 10 I f (t t)1 28 1 1606 3598 t 10 R f (,)1642 3598 w 10 I f (x x)1 44 1 1708 3598 t 10 R f (,)1760 3598 w 10 I f (y y)1 44 1 1826 3598 t 10 R f (,)1878 3598 w 10 B f (u)1944 3598 w 10 R f (,)2008 3598 w 10 B f (u)2074 3598 w 7 I f (x x)1 31 1 2141 3618 t 10 R f (,)2188 3598 w 10 B f (u)2254 3598 w 7 I f (y y)1 31 1 2321 3618 t 10 R f (,)2368 3598 w 10 B f (u)2434 3598 w 7 I f (t t)1 20 1 2501 3618 t 10 R f (,)2537 3598 w 10 B f (u)2603 3598 w 7 I f ( t)1 0(x xt)1 51 2 2670 3618 t 10 R f (,)2737 3598 w 10 B f (u)2803 3598 w 7 I f ( t)1 0(y yt)1 51 2 2870 3618 t 10 R f (\))2937 3598 w 10 S f (= =)1 55 1 3027 3598 t 10 R f (\(2.1\))4849 3598 w 10 B f (f)1411 3868 w 10 R f (\()1452 3868 w 10 I f (t t)1 28 1 1493 3868 t 10 R f (,)1529 3868 w 10 I f (x x)1 44 1 1595 3868 t 10 R f (,)1647 3868 w 10 I f (y y)1 44 1 1713 3868 t 10 R f (,)1765 3868 w 10 B f (u)1831 3868 w 10 R f (,)1895 3868 w 10 B f (u)1961 3868 w 7 I f (x x)1 31 1 2028 3888 t 10 R f (,)2075 3868 w 10 B f (u)2141 3868 w 7 I f (y y)1 31 1 2208 3888 t 10 R f (,)2255 3868 w 10 B f (u)2321 3868 w 7 I f (t t)1 20 1 2388 3888 t 10 R f (,)2424 3868 w 10 B f (u)2490 3868 w 7 I f ( t)1 0(x xt)1 51 2 2557 3888 t 10 R f (,)2624 3868 w 10 B f (u)2690 3868 w 7 I f ( t)1 0(y yt)1 51 2 2757 3888 t 10 R f (\) ,)1 98 1 2824 3868 t (where)720 4098 w 10 B f (a)1002 4098 w 10 R f (and)1091 4098 w 10 B f (f)1274 4098 w 10 R f ( of their arguments, for)4 983(are vector-valued functions)2 1119 2 1346 4098 t 10 I f (L L)1 56 1 3488 4098 t 7 I f (x x)1 31 1 3555 4118 t 10 S f (\243)3635 4098 w 10 I f (x x)1 44 1 3731 4098 t 10 S f (\243)3816 4098 w 10 I f (R R)1 61 1 3912 4098 t 7 I f (x x)1 31 1 3984 4118 t 10 R f (and)4063 4098 w 10 I f (L L)1 56 1 4247 4098 t 7 I f (y y)1 31 1 4314 4118 t 10 S f (\243)4394 4098 w 10 I f (y y)1 44 1 4490 4098 t 10 S f (\243)4575 4098 w 10 I f (R R)1 61 1 4671 4098 t 7 I f (y y)1 31 1 4743 4118 t 10 R f ( is)1 107(. It)1 151 2 4782 4098 t (required that the length of)4 1041 1 720 4218 t 10 B f (a)1787 4218 w 10 R f (and)1863 4218 w 10 B f (f)2033 4218 w 10 R f (be equal to)2 440 1 2092 4218 t 10 I f (n n)1 50 1 2558 4218 t 7 I f (u u)1 35 1 2619 4238 t 10 R f (, the number of)3 613 1 2662 4218 t 10 B f (pde)3301 4218 w 10 R f (variables, that is, the length of the vec-)7 1557 1 3483 4218 t (tor)720 4338 w 10 B f (u)856 4338 w 10 R f ( boundary conditions are assumed to have the form)8 2041(. The)1 230 2 912 4338 t 10 B f (b)1220 4518 w 10 R f (\()1317 4518 w 10 I f (t t)1 28 1 1391 4518 t 10 R f (,)1427 4518 w 10 I f (x x)1 44 1 1493 4518 t 10 R f (,)1545 4518 w 10 I f (y y)1 44 1 1611 4518 t 10 R f (,)1663 4518 w 10 B f (u)1729 4518 w 10 R f (,)1793 4518 w 10 B f (u)1859 4518 w 7 I f (x x)1 31 1 1926 4538 t 10 R f (,)1973 4518 w 10 B f (u)2039 4518 w 7 I f (y y)1 31 1 2106 4538 t 10 R f (,)2153 4518 w 10 B f (u)2219 4518 w 7 I f (t t)1 20 1 2286 4538 t 10 R f (,)2322 4518 w 10 B f (u)2388 4518 w 7 I f ( t)1 0(x xt)1 51 2 2455 4538 t 10 R f (,)2522 4518 w 10 B f (u)2588 4518 w 7 I f ( t)1 0(y yt)1 51 2 2655 4538 t 10 R f (\))2755 4518 w 10 S f (= =)1 55 1 2845 4518 t 10 R f (0 \(2.2\))1 2091 1 2949 4518 t (where)720 4698 w 10 B f (b)991 4698 w 10 R f (is a vector-valued function, of length)5 1490 1 1075 4698 t 10 I f (n n)1 50 1 2593 4698 t 7 I f (u u)1 35 1 2654 4718 t 10 R f ( identically zero component of the)5 1387( Any)1 226( its arguments.)2 594(, of)1 136 4 2697 4698 t 10 B f (bc)720 4818 w 10 R f (vector)848 4818 w 10 B f (b)1125 4818 w 10 R f (is treated as an inactive)4 943 1 1209 4818 t 10 B f (bc)2180 4818 w 10 R f ( each of the)3 471(. If)1 144 2 2280 4818 t 10 B f (pde)2923 4818 w 10 R f ('s is second order in space, then each of the)9 1761 1 3079 4818 t 10 B f (bc)4868 4818 w 10 R f ('s)4968 4818 w ( any of the)3 436( If)1 120(will have to be active.)4 895 3 720 4938 t 10 B f (pde)2201 4938 w 10 R f ('s are of order less than 2 in space, some of the)11 1928 1 2357 4938 t 10 B f (bc)4315 4938 w 10 R f ('s must accord-)2 625 1 4415 4938 t ( conditions \()2 500( Initial)1 289(ingly be inactive.)2 691 3 720 5058 t 10 B f (ic)2200 5058 w 10 R f ('s\))2272 5058 w 10 B f (u)2402 5058 w 10 R f (\( 0 ,)2 124 1 2466 5058 t 10 I f (x x)1 44 1 2598 5058 t 10 R f (,)2650 5058 w 10 I f (y y)1 44 1 2683 5058 t 10 R f (\) must be supplied, but need not satisfy the)8 1713 1 2735 5058 t 10 B f (bc)4473 5058 w 10 R f ('s \(2.2\).)1 313 1 4573 5058 t (Note that the form of \(2.1\)-\(2.2\) encompasses parabolic \()8 2291 1 970 5214 t 10 I f (u u)1 50 1 3288 5214 t 7 I f (t t)1 20 1 3349 5234 t 10 S f (= =)1 55 1 3426 5214 t 10 I f (u u)1 50 1 3530 5214 t 7 I f (x xx x)2 62 1 3591 5234 t 10 S f (+ +)1 55 1 3710 5214 t 10 I f (u u)1 50 1 3814 5214 t 7 I f (y yy y)2 62 1 3875 5234 t 10 R f (\), elliptic \()2 423 1 3972 5214 t 10 I f (u u)1 50 1 4422 5214 t 7 I f (x xx x)2 62 1 4483 5234 t 10 S f (+ +)1 55 1 4602 5214 t 10 I f (u u)1 50 1 4706 5214 t 7 I f (y yy y)2 62 1 4767 5234 t 10 S f (= =)1 55 1 4886 5214 t 10 R f (0)4990 5214 w (\) and hyperbolic \()3 712 1 720 5334 t 10 I f (u u)1 50 1 1457 5334 t 7 I f (t t)1 20 1 1518 5354 t 10 S f (= =)1 55 1 1595 5334 t 10 I f (u u)1 50 1 1699 5334 t 7 I f (x x)1 31 1 1760 5354 t 10 S f (+ +)1 55 1 1848 5334 t 10 I f (u u)1 50 1 1952 5334 t 7 I f (y y)1 31 1 2013 5354 t 10 R f ( also encompasses)2 732( It)1 111(\) problems.)1 455 3 2077 5334 t 10 B f (pde)3400 5334 w 10 R f ('s that have no solution, such as)6 1274 1 3556 5334 t 10 I f (u u)1 50 1 1220 5514 t 7 I f (x x)1 31 1 1275 5533 t 7 R f (2)1275 5474 w 10 S f (+ +)1 55 1 1358 5514 t 10 I f (u u)1 50 1 1453 5514 t 7 I f (y y)1 31 1 1508 5533 t 7 R f (2)1508 5474 w 10 S f ( -)1 0( -)1 112(= =)1 55 3 1600 5514 t 10 R f (1)1783 5514 w (over the real field.)3 731 1 720 5694 t (The number of)2 593 1 970 5850 t 10 B f (pde)1588 5850 w 10 R f (variables,)1769 5850 w 10 I f (n n)1 50 1 2179 5850 t 7 I f (u u)1 35 1 2240 5870 t 10 R f (, is assumed to be the same for all rectangles.)9 1805 1 2283 5850 t 10 CW f (TTGU)970 6006 w 10 R f ( point or a whole side of the rectan-)8 1475(demands that the interface between rectangles be either a)8 2324 2 1241 6006 t ( the following situation)3 931(gle. Thus)1 397 2 720 6126 t cleartomark showpage saveobj restore %%EndPage: 3 4 %%Page: 4 5 /saveobj save def mark 5 pagesetup 10 R f (- 4 -)2 166 1 2797 480 t cleartomark saveobj restore %%BeginGlobal % % Version 3.3 drawing procedures for dpost. Automatically pulled in, but only % when needed. % /inpath false def /savematrix matrix def /Dl { inpath {pop pop neg lineto} {newpath neg moveto neg lineto stroke} ifelse } bind def /De { /y1 exch 2 div def /x1 exch 2 div def /savematrix savematrix currentmatrix def neg exch x1 add exch translate x1 y1 scale 0 0 1 0 360 inpath {1 0 moveto arc savematrix setmatrix} {newpath arc savematrix setmatrix stroke} ifelse } bind def /Da { /dy2 exch def /dx2 exch def /dy1 exch def /dx1 exch def dy1 add neg exch dx1 add exch dx1 dx1 mul dy1 dy1 mul add sqrt dy1 dx1 neg atan dy2 neg dx2 atan inpath {arc} {newpath arc stroke} ifelse } bind def /DA { /dy2 exch def /dx2 exch def /dy1 exch def /dx1 exch def dy1 add neg exch dx1 add exch dx1 dx1 mul dy1 dy1 mul add sqrt dy1 dx1 neg atan dy2 neg dx2 atan inpath {arcn} {newpath arcn stroke} ifelse } bind def /Ds { /y2 exch def /x2 exch def /y1 exch def /x1 exch def /y0 exch def /x0 exch def x0 5 x1 mul add 6 div y0 5 y1 mul add -6 div x2 5 x1 mul add 6 div y2 5 y1 mul add -6 div x1 x2 add 2 div y1 y2 add -2 div inpath {curveto} {newpath x0 x1 add 2 div y0 y1 add -2 div moveto curveto stroke} ifelse } bind def %%EndGlobal /saveobj save def mark 10 R f 1800 840 1800 1560 Dl 2520 840 1800 840 Dl 2520 1560 2520 840 Dl 1800 1560 2520 1560 Dl 2520 840 2520 1560 Dl 3240 840 2520 840 Dl 3240 1560 3240 840 Dl 2520 1560 3240 1560 Dl 3240 840 3240 1560 Dl 3960 840 3240 840 Dl 3960 1560 3960 840 Dl 3240 1560 3960 1560 Dl 2520 1560 2520 2280 Dl 3240 1560 2520 1560 Dl 3240 2280 3240 1560 Dl 2520 2280 3240 2280 Dl (Fig 2.1)1 284 1 2738 2460 t (is valid, but the following configuration)5 1594 1 720 2580 t 1800 2742 1800 3462 Dl 3960 2742 1800 2742 Dl 3960 3462 3960 2742 Dl 1800 3462 3960 3462 Dl (1)2855 3122 w 2520 3462 2520 4182 Dl 3240 3462 2520 3462 Dl 3240 4182 3240 3462 Dl 2520 4182 3240 4182 Dl (2)2855 3842 w (a)2498 3422 w (Fig 2.2)1 284 1 2738 4362 t ( when the user specifies a knot sequence)7 1671( main reason for this restriction is to insure that)9 1953( The)1 212(is not valid.)2 484 4 720 4602 t ( order)1 236(\(see the next section\), an error in the multiplicity of the knots is not made which would preclude high)18 4084 2 720 4722 t ( In)1 139( of this code if requested by users.)7 1406( restriction might be eliminated in future versions)7 2014(convergence. This)1 761 4 720 4842 t ( B-spline the knot at "a" in Fig. 2.2 should have multiplicity 3 to get)14 2890(rectangle 1 if one is using a cubic)7 1430 2 720 4962 t (fourth order convergence.)2 1032 1 720 5082 t 10 B f ( Method of Solution.)3 873(3. General)1 469 2 720 5322 t 10 R f (On rectangle)1 525 1 970 5478 t 10 I f (j j)1 28 1 1533 5478 t 10 R f (let the solution)2 621 1 1599 5478 t 10 B f (u)2258 5478 w 10 R f (\()2322 5478 w 10 I f (t t)1 28 1 2363 5478 t 10 R f (,)2399 5478 w 10 I f (x x)1 44 1 2432 5478 t 10 R f (,)2484 5478 w 10 I f (y y)1 44 1 2517 5478 t 10 R f ( tensor-)1 316(\), for a given instant in time, be approximated by a)10 2155 2 2569 5478 t (product of B-splines [21,1,2,3] of)4 1548 1 720 5598 t 10 B f (order)2345 5598 w 10 I f (k k)1 44 1 2660 5598 t 7 I f (x x)1 31 1 2715 5618 t 7 R f (,)2751 5618 w 7 I f (j j)1 20 1 2780 5618 t 10 R f (and)2884 5598 w 10 I f (k k)1 44 1 3104 5598 t 7 I f (y y)1 31 1 3159 5618 t 7 R f (,)3195 5618 w 7 I f (j j)1 20 1 3224 5618 t 10 R f (on)3328 5598 w 10 B f (meshes)3504 5598 w 10 I f (X X)1 61 1 3885 5598 t 10 R f (\( 1 \))2 132 1 3954 5598 t 10 S f (\243)4135 5598 w 10 R f (. . .)2 125 1 4223 5573 t 10 S f (\243)4381 5598 w 10 I f (X X)1 61 1 4477 5598 t 10 R f (\()4546 5598 w 10 I f ( X)1 0(N NX)1 128 2 4587 5598 t 7 I f (j j)1 20 1 4726 5618 t 10 R f (\), and)1 278 1 4762 5598 t 10 I f (Y Y)1 56 1 720 5718 t 10 R f (\( 1 \))2 132 1 784 5718 t 10 S f (\243)965 5718 w 10 R f (. . .)2 125 1 1053 5693 t 10 S f (\243)1211 5718 w 10 I f (Y Y)1 56 1 1307 5718 t 10 R f (\()1371 5718 w 10 I f ( Y)1 0(N NY)1 123 2 1412 5718 t 7 I f (j j)1 20 1 1546 5738 t 10 R f ( is, for any fixed)4 677(\). That)1 296 2 1582 5718 t 10 I f (y y)1 44 1 2586 5718 t 10 R f (, each component of the solution will be approximated in)9 2335 1 2630 5718 t 10 I f (x x)1 44 1 4996 5718 t 10 R f (by a piecewise polynomial function of degree less than)8 2308 1 720 5838 t 10 I f (k k)1 44 1 3067 5838 t 7 I f (x x)1 31 1 3122 5858 t 7 R f (,)3158 5858 w 7 I f (j j)1 20 1 3187 5858 t 10 R f (, with)1 242 1 3215 5838 t 10 I f (k k)1 44 1 3496 5838 t 7 I f (x x)1 31 1 3551 5858 t 7 R f (,)3587 5858 w 7 I f (j j)1 20 1 3616 5858 t 10 S f (- -)1 55 1 3660 5838 t 10 R f (2 continuous derivatives, where)3 1309 1 3731 5838 t 10 I f (k k)1 44 1 720 5958 t 7 I f (x x)1 31 1 775 5978 t 7 R f (,)811 5978 w 7 I f (j j)1 20 1 840 5978 t 10 S f (\263)909 5958 w 10 R f ( desires; a similar statement holds about the degree in)9 2202(2 is any integer the user)5 981 2 1005 5958 t 10 I f (y y)1 44 1 4220 5958 t 10 R f (. Let)1 215 1 4264 5958 t 10 I f (B B)1 61 1 4511 5958 t 7 I f (p p)1 35 1 4583 5978 t 10 R f (\()4634 5958 w 10 I f (x x)1 44 1 4675 5958 t 10 R f (\) be the)2 313 1 4727 5958 t (basis functions in)2 700 1 720 6078 t 10 I f (x x)1 44 1 1445 6078 t 10 R f (and)1514 6078 w 10 I f (C C)1 67 1 1683 6078 t 7 I f (q q)1 35 1 1761 6098 t 10 R f (\()1812 6078 w 10 I f (y y)1 44 1 1853 6078 t 10 R f (\) be those in)3 491 1 1905 6078 t 10 I f (y y)1 44 1 2421 6078 t 10 R f ( for each rectangle)3 738(. Then)1 280 2 2465 6078 t 10 I f (u u)1 50 1 1220 6258 t 7 I f (i i)1 20 1 1281 6278 t 10 R f (\()1317 6258 w 10 I f (t t)1 28 1 1358 6258 t 10 R f (,)1394 6258 w 10 I f (x x)1 44 1 1427 6258 t 10 R f (,)1479 6258 w 10 I f (y y)1 44 1 1512 6258 t 10 R f (\))1564 6258 w 10 S f (= =)1 55 1 1654 6258 t 7 I f (p p)1 35 1 1801 6358 t 15 S f (S)1774 6288 w 7 I f (q q)1 35 1 1971 6358 t 15 S f (S)1944 6288 w 10 I f (U U)1 72 1 2098 6258 t 7 I f (q q)1 35 1 2181 6278 t 7 R f (,)2221 6278 w 7 I f (p p)1 35 1 2244 6278 t 7 R f (,)2284 6278 w 7 I f (i i)1 20 1 2307 6278 t 10 R f (\()2343 6258 w 10 I f (t t)1 28 1 2384 6258 t 10 R f (\))2420 6258 w 10 I f (B B)1 61 1 2493 6258 t 7 I f (p p)1 35 1 2565 6278 t 10 R f (\()2616 6258 w 10 I f (x x)1 44 1 2657 6258 t 10 R f (\))2709 6258 w 10 I f (C C)1 67 1 2782 6258 t 7 I f (q q)1 35 1 2860 6278 t 10 R f (\()2911 6258 w 10 I f (y y)1 44 1 2952 6258 t 10 R f (\))3004 6258 w 10 I f (. .)1 25 1 3053 6258 t 10 R f (\(3.1\))4849 6258 w (If we set)2 649 1 720 6518 t 10 I f (h h)1 50 1 1547 6518 t 7 I f (x x)1 31 1 1608 6538 t 10 S f (= =)1 55 1 1696 6518 t 7 I f (i i)1 20 1 1876 6588 t 10 R f (max)1800 6518 w 10 S f (\357 \357)1 49 1 2005 6535 t 10 I f (X X)1 61 1 2086 6518 t 10 R f (\()2155 6518 w 10 I f (i i)1 28 1 2196 6518 t 10 S f (+ +)1 55 1 2248 6518 t 10 R f (1 \))1 91 1 2319 6518 t 10 S f (- -)1 55 1 2467 6518 t 10 I f (X X)1 61 1 2571 6518 t 10 R f (\()2640 6518 w 10 I f (i i)1 28 1 2681 6518 t 10 R f (\))2717 6518 w 10 S f (\357 \357)1 49 1 2791 6535 t 10 R f (, and)1 347 1 2807 6518 t 10 I f (h h)1 50 1 3332 6518 t 7 I f (y y)1 31 1 3393 6538 t 10 R f (similarly, then the error,)3 1430 1 3610 6518 t 10 S f ( \357)1 0( \357)1 24(\357 \357)1 49 3 720 6710 t 10 I f (u u)1 50 1 825 6693 t 7 I f (i i)1 20 1 886 6713 t 10 S f (- -)1 55 1 954 6693 t 10 I f (u u)1 50 1 1049 6693 t 11 R f (\303)1058 6688 w 7 I f (i i)1 20 1 1110 6713 t 10 S f ( \357)1 0( \357)1 24(\357 \357)1 49 3 1171 6710 t 7 S f (\245)1233 6713 w 10 S f (\272)1333 6693 w 7 I f (x x)1 31 1 1470 6763 t 7 R f (,)1506 6763 w 7 I f (y y)1 31 1 1529 6763 t 10 R f (max)1429 6693 w 10 S f (\357 \357)1 49 1 1634 6710 t 10 I f (u u)1 50 1 1715 6693 t 7 I f (i i)1 20 1 1776 6713 t 10 R f (\()1812 6693 w 10 I f (t t)1 28 1 1853 6693 t 10 R f (,)1889 6693 w 10 I f (x x)1 44 1 1922 6693 t 10 R f (,)1974 6693 w 10 I f (y y)1 44 1 2007 6693 t 10 R f (\))2059 6693 w 10 S f (- -)1 55 1 2149 6693 t 10 I f (u u)1 50 1 2253 6693 t 11 R f (\303)2262 6688 w 7 I f (i i)1 20 1 2314 6713 t 10 R f (\()2350 6693 w 10 I f (t t)1 28 1 2391 6693 t 10 R f (,)2427 6693 w 10 I f (x x)1 44 1 2460 6693 t 10 R f (,)2512 6693 w 10 I f (y y)1 44 1 2545 6693 t 10 R f (\))2597 6693 w 10 S f (\357 \357)1 49 1 2671 6710 t 10 R f (, is)1 123 1 2687 6693 t 10 I f (O O)1 72 1 2841 6693 t 10 R f (\()2921 6693 w 10 I f (h h)1 50 1 2962 6693 t 7 I f (x x)1 31 1 3017 6712 t (k k)1 31 1 3017 6639 t 4 I f (x x)1 18 1 3054 6653 t 10 S f (+ +)1 55 1 3125 6693 t 10 I f (h h)1 50 1 3220 6693 t 7 I f (y y)1 31 1 3275 6712 t (k k)1 31 1 3275 6639 t 4 I f (y y)1 18 1 3312 6653 t 10 R f ( B-spline)1 369(\), for some)2 447 2 3351 6693 t 10 I f (u u)1 50 1 4197 6693 t 11 R f (\303)4206 6688 w 7 I f (i i)1 20 1 4258 6713 t 10 R f ( Since)1 277(, see [3].)2 353 2 4286 6693 t 10 I f (k k)1 44 1 4946 6693 t 7 I f (x x)1 31 1 5001 6713 t 10 R f (and)720 6863 w 10 I f (k k)1 44 1 895 6863 t 7 I f (y y)1 31 1 950 6883 t 10 R f ( integer)1 309(may be taken to be any)5 953 2 1020 6863 t 10 S f (\263 \263)1 55 1 2314 6863 t 10 R f (2, this gives a powerful technique for approximating the solution)9 2655 1 2385 6863 t 10 B f (u)720 6983 w 10 R f (\()784 6983 w 10 I f (t t)1 28 1 825 6983 t 10 R f (,)861 6983 w 10 I f (x x)1 44 1 894 6983 t 10 R f (,)946 6983 w 10 I f (y y)1 44 1 979 6983 t 10 R f ( \(R-R-G\) method [25,19] to find essentially the)7 1929( can use the Rayleigh-Ritz-Galerkin)4 1464( We)1 195(\) in space.)2 421 4 1031 6983 t ( of the)2 265(projection of the solution)3 1020 2 720 7103 t 10 B f (pde)2035 7103 w 10 R f ( reduces the)2 486( This)1 233(onto the space of B-splines we have selected.)7 1842 3 2221 7103 t 10 B f (pde)4812 7103 w 10 R f ('s)4968 7103 w (in space and time to)4 799 1 720 7223 t 10 B f (ode)1544 7223 w 10 R f ('s in time [10,25] for the coefficients)6 1472 1 1694 7223 t 10 I f (U U)1 72 1 3191 7223 t 7 I f (q q)1 35 1 3274 7243 t 7 R f (,)3314 7243 w 7 I f (p p)1 35 1 3337 7243 t 7 R f (,)3377 7243 w 7 I f (i i)1 20 1 3400 7243 t 10 R f (\()3436 7223 w 10 I f (t t)1 28 1 3477 7223 t 10 R f (\) in the expansion \(3.1\).)4 954 1 3513 7223 t cleartomark showpage saveobj restore %%EndPage: 4 5 %%Page: 5 6 /saveobj save def mark 6 pagesetup 10 R f (- 5 -)2 166 1 2797 480 t (Thus, after the spatial discretization, only)5 1696 1 970 840 t 10 B f (ode)2699 840 w 10 R f ( these)1 239( Since)1 280('s in time remain to be solved.)6 1261 3 2849 840 t 10 B f (ode)4663 840 w 10 R f ('s are)1 227 1 4813 840 t ( vir-)1 172( This)1 232(known to be in general "stiff" [7,8], an implicit differencing scheme must be used to solve them.)16 3916 3 720 960 t (tually requires that the partial derivatives of the)7 1936 1 720 1080 t 10 B f (a)2687 1080 w 10 R f (and)2768 1080 w 10 B f (f)2943 1080 w 10 R f (in \(2.1\) and of the)4 742 1 3007 1080 t 10 B f (b)3780 1080 w 10 R f ( respect to their)3 639(in \(2.2\), with)2 534 2 3867 1080 t (arguments be known, either analytically or numerically.)6 2235 1 720 1200 t (The next step is the solution of these time-varying)8 2044 1 970 1356 t 10 B f (ode)3044 1356 w 10 R f ( we assume that some basic one-step)6 1500('s. Here)1 346 2 3194 1356 t 10 B f (ode)720 1476 w 10 R f ( [16], or an)3 548( example, a backwards-Euler or Crank-Nicholson scheme)6 2518( For)1 225(solver is available.)2 818 4 931 1476 t ( an explicit method such as Gragg's modified mid-point rule)9 2534(exponentially-fitted technique [13], or even)4 1786 2 720 1596 t ([12,13], could be used.)3 936 1 720 1716 t 10 CW f (TTGU)1783 1716 w 10 R f ( [20])1 197( See)1 200( the default time discretization scheme.)5 1594(uses backwards-Euler as)2 994 4 2055 1716 t (for a description of the method used to solve the nonlinear equations arising at each time-step.)15 3761 1 720 1836 t ( others, have the property that for a given time-step)9 2118(All the above techniques, and many)5 1466 2 970 1992 t 10 S f (d)4587 1992 w 10 R f (they pro-)1 371 1 4669 1992 t ( to)1 105(duce an approximate solution accurate)4 1547 2 720 2112 t 10 I f (O O)1 72 1 2399 2112 t 10 R f (\()2479 2112 w 10 S f (d)2520 2112 w 7 S f (g)2574 2072 w 10 R f (\), where typically)2 705 1 2619 2112 t 10 S f (g)3351 2112 w 10 R f ( if the equations are)4 795( Moreover,)1 470(is 1 or 2.)3 356 3 3419 2112 t (solved using time-steps of)3 1050 1 720 2232 t 10 S f (d)1796 2232 w 10 R f (and)1871 2232 w 10 S f (d)2041 2232 w 10 I f (/ /)1 28 1 2098 2232 t 10 R f ( results of these two computations can be combined using extrapo-)10 2683(2, the)1 223 2 2134 2232 t ( to obtain a result accurate to)6 1159(lation [5,12])1 496 2 720 2352 t 10 I f (O O)1 72 1 2401 2352 t 10 R f (\()2481 2352 w 10 S f (d)2522 2352 w 7 R f (2)2576 2312 w 7 S f (g)2616 2312 w 10 R f ( process can be repeated indefinitely, so a basic pro-)9 2092(\). This)1 287 2 2661 2352 t (cess of accuracy)2 680 1 720 2472 t 10 S f (d)1439 2472 w 7 S f (g)1493 2432 w 10 R f (can be used to generate a sequence of processes of accuracy)10 2530 1 1569 2472 t 10 I f (O O)1 72 1 4139 2472 t 10 R f (\()4219 2472 w 10 S f (d)4260 2472 w 7 S f (g)4314 2432 w 10 R f (\),)4359 2472 w 10 I f (O O)1 72 1 4457 2472 t 10 R f (\()4537 2472 w 10 S f (d)4578 2472 w 7 R f (2)4632 2432 w 7 S f (g)4672 2432 w 10 R f (\) ,)1 74 1 4717 2472 t (. . .)2 125 1 4824 2447 t (,)4982 2472 w 10 I f (O O)1 72 1 720 2592 t 10 R f (\()800 2592 w 10 S f (d)841 2592 w 7 I f (P P)1 43 1 895 2552 t 7 S f (g)943 2552 w 10 R f (\) ,)1 74 1 988 2592 t (. . .)2 125 1 1095 2567 t (.)1270 2592 w ( available [17,18] for carrying out this extrapolation process and)9 2711(A step-size and order monitor is)5 1359 2 970 2748 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 2868 t 10 R f (deciding what time-step)2 984 1 1308 2868 t 10 S f (d)2329 2868 w 10 R f (and order)1 391 1 2415 2868 t 10 I f (P P)1 61 1 2843 2868 t 10 S f (g)2912 2868 w 10 R f (should be used, given the accuracy desired in the)8 2050 1 2990 2868 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 2988 t (integration then proceeds automatically, with no need for the user to worry about choosing)13 3622 1 720 3108 t 10 S f (d)4367 3108 w 10 R f (.)4416 3108 w (The algorithm implemented by)3 1241 1 720 3264 t 10 CW f (TTGU)1986 3264 w 10 R f (for solving such)2 644 1 2251 3264 t 10 B f (pde)2920 3264 w 10 R f ('s then consists of 3 steps:)5 1047 1 3076 3264 t ( the equations in space using R-R-G with B-splines.)8 2074(1\) Discretize)1 654 2 970 3456 t ( a one-step method for solving the resulting)7 1740(2\) Produce)1 577 2 970 3612 t 10 B f (ode)3312 3612 w 10 R f ('s.)3462 3612 w ( that one-step process to the extrapolation step-size and order monitor.)10 2809(3\) Feed)1 444 2 970 3768 t ( one must solve a linear system. For)7 1466(At the bottom level)3 784 2 970 3924 t 10 CW f (TTGU)3249 3924 w 10 R f (the user may specify that for the inte-)7 1522 1 3518 3924 t ( point iterative methods)3 971( this)1 179( At)1 159(rior of each rectangle a banded solver, with or without pivoting, be used.)12 3011 4 720 4044 t ( 6 gives a somewhat more detailed outline of the spa-)10 2180( Section)1 354(and sparse direct methods are not an option.)7 1786 3 720 4164 t (tial discretization process and the various parameters that describe the solution procedure.)11 3587 1 720 4284 t 10 B f (Mesh restrictions)1 740 1 720 4524 t 10 R f ( common side the order of the meshes and the mesh themselves must be)13 2916(When two rectangles share a)4 1154 2 970 4680 t ( if two rectangles meet at)5 1066(the same. For example,)3 968 2 720 4800 t 10 I f (x x)1 44 1 2791 4800 t 10 S f (= =)1 55 1 2859 4800 t 10 R f (0, then the)2 443 1 2930 4800 t 10 I f (y y)1 44 1 3410 4800 t 10 R f (meshes for these two rectangles must)5 1549 1 3491 4800 t (agree as well as)3 676 1 720 4920 t 10 I f (k k)1 44 1 1437 4920 t 7 I f (y y)1 31 1 1492 4940 t 10 R f ( 3.1 is)2 274(. Figure)1 352 2 1531 4920 t 10 B f (not permitted)1 602 1 2223 4920 t 10 R f ( if)1 103( Even)1 272(, while Figure 3.2 is permitted.)5 1318 3 2825 4920 t 10 I f (k k)1 44 1 4560 4920 t 7 I f (x x)1 31 1 4615 4940 t 7 R f (, 1)1 58 1 4651 4940 t 10 S f (= =)1 55 1 4733 4920 t 10 R f (2 and)1 236 1 4804 4920 t 10 I f (k k)1 44 1 720 5040 t 7 I f (x x)1 31 1 775 5060 t 7 R f (, 2)1 58 1 811 5060 t 10 S f (= =)1 55 1 893 5040 t 10 R f (4, Figure 3.2 would be fine.)5 1110 1 964 5040 t 2437 5922 2401 5922 Dl 2504 5922 2468 5922 Dl 2571 5922 2535 5922 Dl 2638 5922 2602 5922 Dl 2638 5886 2638 5922 Dl 2638 5809 2638 5845 Dl 2638 5734 2638 5770 Dl 2638 5657 2638 5693 Dl 2638 5582 2638 5618 Dl 2638 5505 2638 5541 Dl 2638 5430 2638 5466 Dl 2638 5353 2638 5389 Dl 2638 5278 2638 5314 Dl 2638 5202 2638 5238 Dl 2602 5202 2638 5202 Dl 2535 5202 2571 5202 Dl 2468 5202 2504 5202 Dl 2401 5202 2437 5202 Dl 2401 5238 2401 5202 Dl 2401 5314 2401 5278 Dl 2401 5389 2401 5353 Dl 2401 5466 2401 5430 Dl 2401 5541 2401 5505 Dl 2401 5618 2401 5582 Dl 2401 5693 2401 5657 Dl 2401 5770 2401 5734 Dl 2401 5845 2401 5809 Dl 2401 5922 2401 5886 Dl 2196 5680 2160 5680 Dl 2272 5680 2236 5680 Dl 2347 5680 2311 5680 Dl 2424 5680 2388 5680 Dl 2499 5680 2463 5680 Dl 2576 5680 2540 5680 Dl 2651 5680 2615 5680 Dl 2728 5680 2692 5680 Dl 2803 5680 2767 5680 Dl 2880 5680 2844 5680 Dl 2880 5644 2880 5680 Dl 2880 5577 2880 5613 Dl 2880 5510 2880 5546 Dl 2880 5443 2880 5479 Dl 2844 5443 2880 5443 Dl 2767 5443 2803 5443 Dl 2692 5443 2728 5443 Dl 2615 5443 2651 5443 Dl 2540 5443 2576 5443 Dl 2463 5443 2499 5443 Dl 2388 5443 2424 5443 Dl 2311 5443 2347 5443 Dl 2236 5443 2272 5443 Dl 2160 5443 2196 5443 Dl 2160 5479 2160 5443 Dl 2160 5546 2160 5510 Dl 2160 5613 2160 5577 Dl 2160 5680 2160 5644 Dl 2160 5202 2160 5922 Dl 2880 5202 2160 5202 Dl 2880 5922 2880 5202 Dl 2160 5922 2880 5922 Dl 2880 5202 2880 5922 Dl 3600 5202 2880 5202 Dl 3600 5922 3600 5202 Dl 2880 5922 3600 5922 Dl 3060 5922 3024 5922 Dl 3126 5922 3090 5922 Dl 3191 5922 3155 5922 Dl 3258 5922 3222 5922 Dl 3324 5922 3288 5922 Dl 3389 5922 3353 5922 Dl 3456 5922 3420 5922 Dl 3456 5886 3456 5922 Dl 3456 5809 3456 5845 Dl 3456 5734 3456 5770 Dl 3456 5657 3456 5693 Dl 3456 5582 3456 5618 Dl 3456 5505 3456 5541 Dl 3456 5430 3456 5466 Dl 3456 5353 3456 5389 Dl 3456 5278 3456 5314 Dl 3456 5202 3456 5238 Dl 3420 5202 3456 5202 Dl 3353 5202 3389 5202 Dl 3288 5202 3324 5202 Dl 3222 5202 3258 5202 Dl 3155 5202 3191 5202 Dl 3090 5202 3126 5202 Dl 3024 5202 3060 5202 Dl 3024 5238 3024 5202 Dl 3024 5314 3024 5278 Dl 3024 5389 3024 5353 Dl 3024 5466 3024 5430 Dl 3024 5541 3024 5505 Dl 3024 5618 3024 5582 Dl 3024 5693 3024 5657 Dl 3024 5770 3024 5734 Dl 3024 5845 3024 5809 Dl 3024 5922 3024 5886 Dl 2916 5778 2880 5778 Dl 2992 5778 2956 5778 Dl 3067 5778 3031 5778 Dl 3144 5778 3108 5778 Dl 3219 5778 3183 5778 Dl 3296 5778 3260 5778 Dl 3371 5778 3335 5778 Dl 3448 5778 3412 5778 Dl 3523 5778 3487 5778 Dl 3600 5778 3564 5778 Dl 3600 5742 3600 5778 Dl 3600 5675 3600 5711 Dl 3600 5610 3600 5646 Dl 3600 5544 3600 5580 Dl 3600 5477 3600 5513 Dl 3600 5412 3600 5448 Dl 3600 5346 3600 5382 Dl 3564 5346 3600 5346 Dl 3487 5346 3523 5346 Dl 3412 5346 3448 5346 Dl 3335 5346 3371 5346 Dl 3260 5346 3296 5346 Dl 3183 5346 3219 5346 Dl 3108 5346 3144 5346 Dl 3031 5346 3067 5346 Dl 2956 5346 2992 5346 Dl 2880 5346 2916 5346 Dl 2880 5382 2880 5346 Dl 2880 5448 2880 5412 Dl 2880 5513 2880 5477 Dl 2880 5580 2880 5544 Dl 2880 5646 2880 5610 Dl 2880 5711 2880 5675 Dl 2880 5778 2880 5742 Dl 2916 5634 2880 5634 Dl 2992 5634 2956 5634 Dl 3067 5634 3031 5634 Dl 3144 5634 3108 5634 Dl 3219 5634 3183 5634 Dl 3296 5634 3260 5634 Dl 3371 5634 3335 5634 Dl 3448 5634 3412 5634 Dl 3523 5634 3487 5634 Dl 3600 5634 3564 5634 Dl 3600 5598 3600 5634 Dl 3600 5544 3600 5580 Dl 3600 5490 3600 5526 Dl 3564 5490 3600 5490 Dl 3487 5490 3523 5490 Dl 3412 5490 3448 5490 Dl 3335 5490 3371 5490 Dl 3260 5490 3296 5490 Dl 3183 5490 3219 5490 Dl 3108 5490 3144 5490 Dl 3031 5490 3067 5490 Dl 2956 5490 2992 5490 Dl 2880 5490 2916 5490 Dl 2880 5526 2880 5490 Dl 2880 5580 2880 5544 Dl 2880 5634 2880 5598 Dl 3204 5922 3168 5922 Dl 3258 5922 3222 5922 Dl 3312 5922 3276 5922 Dl 3312 5886 3312 5922 Dl 3312 5809 3312 5845 Dl 3312 5734 3312 5770 Dl 3312 5657 3312 5693 Dl 3312 5582 3312 5618 Dl 3312 5505 3312 5541 Dl 3312 5430 3312 5466 Dl 3312 5353 3312 5389 Dl 3312 5278 3312 5314 Dl 3312 5202 3312 5238 Dl 3276 5202 3312 5202 Dl 3222 5202 3258 5202 Dl 3168 5202 3204 5202 Dl 3168 5238 3168 5202 Dl 3168 5314 3168 5278 Dl 3168 5389 3168 5353 Dl 3168 5466 3168 5430 Dl 3168 5541 3168 5505 Dl 3168 5618 3168 5582 Dl 3168 5693 3168 5657 Dl 3168 5770 3168 5734 Dl 3168 5845 3168 5809 Dl 3168 5922 3168 5886 Dl (1 2)1 770 1 2495 5582 t (Figure 3.1: This mesh is not permitted)6 1531 1 2114 6102 t 2437 6984 2401 6984 Dl 2504 6984 2468 6984 Dl 2571 6984 2535 6984 Dl 2638 6984 2602 6984 Dl 2638 6948 2638 6984 Dl 2638 6871 2638 6907 Dl 2638 6796 2638 6832 Dl 2638 6719 2638 6755 Dl 2638 6644 2638 6680 Dl 2638 6567 2638 6603 Dl 2638 6492 2638 6528 Dl 2638 6415 2638 6451 Dl 2638 6340 2638 6376 Dl 2638 6264 2638 6300 Dl 2602 6264 2638 6264 Dl 2535 6264 2571 6264 Dl 2468 6264 2504 6264 Dl 2401 6264 2437 6264 Dl 2401 6300 2401 6264 Dl 2401 6376 2401 6340 Dl 2401 6451 2401 6415 Dl 2401 6528 2401 6492 Dl 2401 6603 2401 6567 Dl 2401 6680 2401 6644 Dl 2401 6755 2401 6719 Dl 2401 6832 2401 6796 Dl 2401 6907 2401 6871 Dl 2401 6984 2401 6948 Dl 2196 6840 2160 6840 Dl 2272 6840 2236 6840 Dl 2347 6840 2311 6840 Dl 2424 6840 2388 6840 Dl 2499 6840 2463 6840 Dl 2576 6840 2540 6840 Dl 2651 6840 2615 6840 Dl 2728 6840 2692 6840 Dl 2803 6840 2767 6840 Dl 2880 6840 2844 6840 Dl 2880 6804 2880 6840 Dl 2880 6737 2880 6773 Dl 2880 6672 2880 6708 Dl 2880 6606 2880 6642 Dl 2880 6539 2880 6575 Dl 2880 6474 2880 6510 Dl 2880 6408 2880 6444 Dl 2844 6408 2880 6408 Dl 2767 6408 2803 6408 Dl 2692 6408 2728 6408 Dl 2615 6408 2651 6408 Dl 2540 6408 2576 6408 Dl 2463 6408 2499 6408 Dl 2388 6408 2424 6408 Dl 2311 6408 2347 6408 Dl 2236 6408 2272 6408 Dl 2160 6408 2196 6408 Dl 2160 6444 2160 6408 Dl 2160 6510 2160 6474 Dl 2160 6575 2160 6539 Dl 2160 6642 2160 6606 Dl 2160 6708 2160 6672 Dl 2160 6773 2160 6737 Dl 2160 6840 2160 6804 Dl 2196 6696 2160 6696 Dl 2272 6696 2236 6696 Dl 2347 6696 2311 6696 Dl 2424 6696 2388 6696 Dl 2499 6696 2463 6696 Dl 2576 6696 2540 6696 Dl 2651 6696 2615 6696 Dl 2728 6696 2692 6696 Dl 2803 6696 2767 6696 Dl 2880 6696 2844 6696 Dl 2880 6660 2880 6696 Dl 2880 6606 2880 6642 Dl 2880 6552 2880 6588 Dl 2844 6552 2880 6552 Dl 2767 6552 2803 6552 Dl 2692 6552 2728 6552 Dl 2615 6552 2651 6552 Dl 2540 6552 2576 6552 Dl 2463 6552 2499 6552 Dl 2388 6552 2424 6552 Dl 2311 6552 2347 6552 Dl 2236 6552 2272 6552 Dl 2160 6552 2196 6552 Dl 2160 6588 2160 6552 Dl 2160 6642 2160 6606 Dl 2160 6696 2160 6660 Dl 2160 6264 2160 6984 Dl 2880 6264 2160 6264 Dl 2880 6984 2880 6264 Dl 2160 6984 2880 6984 Dl 2880 6264 2880 6984 Dl 3600 6264 2880 6264 Dl 3600 6984 3600 6264 Dl 2880 6984 3600 6984 Dl 3060 6984 3024 6984 Dl 3126 6984 3090 6984 Dl 3191 6984 3155 6984 Dl 3258 6984 3222 6984 Dl 3324 6984 3288 6984 Dl 3389 6984 3353 6984 Dl 3456 6984 3420 6984 Dl 3456 6948 3456 6984 Dl 3456 6871 3456 6907 Dl 3456 6796 3456 6832 Dl 3456 6719 3456 6755 Dl 3456 6644 3456 6680 Dl 3456 6567 3456 6603 Dl 3456 6492 3456 6528 Dl 3456 6415 3456 6451 Dl 3456 6340 3456 6376 Dl 3456 6264 3456 6300 Dl 3420 6264 3456 6264 Dl 3353 6264 3389 6264 Dl 3288 6264 3324 6264 Dl 3222 6264 3258 6264 Dl 3155 6264 3191 6264 Dl 3090 6264 3126 6264 Dl 3024 6264 3060 6264 Dl 3024 6300 3024 6264 Dl 3024 6376 3024 6340 Dl 3024 6451 3024 6415 Dl 3024 6528 3024 6492 Dl 3024 6603 3024 6567 Dl 3024 6680 3024 6644 Dl 3024 6755 3024 6719 Dl 3024 6832 3024 6796 Dl 3024 6907 3024 6871 Dl 3024 6984 3024 6948 Dl 2916 6840 2880 6840 Dl 2992 6840 2956 6840 Dl 3067 6840 3031 6840 Dl 3144 6840 3108 6840 Dl 3219 6840 3183 6840 Dl 3296 6840 3260 6840 Dl 3371 6840 3335 6840 Dl 3448 6840 3412 6840 Dl 3523 6840 3487 6840 Dl 3600 6840 3564 6840 Dl 3600 6804 3600 6840 Dl 3600 6737 3600 6773 Dl 3600 6672 3600 6708 Dl 3600 6606 3600 6642 Dl 3600 6539 3600 6575 Dl 3600 6474 3600 6510 Dl 3600 6408 3600 6444 Dl 3564 6408 3600 6408 Dl 3487 6408 3523 6408 Dl 3412 6408 3448 6408 Dl 3335 6408 3371 6408 Dl 3260 6408 3296 6408 Dl 3183 6408 3219 6408 Dl 3108 6408 3144 6408 Dl 3031 6408 3067 6408 Dl 2956 6408 2992 6408 Dl 2880 6408 2916 6408 Dl 2880 6444 2880 6408 Dl 2880 6510 2880 6474 Dl 2880 6575 2880 6539 Dl 2880 6642 2880 6606 Dl 2880 6708 2880 6672 Dl 2880 6773 2880 6737 Dl 2880 6840 2880 6804 Dl 3204 6984 3168 6984 Dl 3258 6984 3222 6984 Dl 3312 6984 3276 6984 Dl 3312 6948 3312 6984 Dl 3312 6871 3312 6907 Dl 3312 6796 3312 6832 Dl 3312 6719 3312 6755 Dl 3312 6644 3312 6680 Dl 3312 6567 3312 6603 Dl 3312 6492 3312 6528 Dl 3312 6415 3312 6451 Dl 3312 6340 3312 6376 Dl 3312 6264 3312 6300 Dl 3276 6264 3312 6264 Dl 3222 6264 3258 6264 Dl 3168 6264 3204 6264 Dl 3168 6300 3168 6264 Dl 3168 6376 3168 6340 Dl 3168 6451 3168 6415 Dl 3168 6528 3168 6492 Dl 3168 6603 3168 6567 Dl 3168 6680 3168 6644 Dl 3168 6755 3168 6719 Dl 3168 6832 3168 6796 Dl 3168 6907 3168 6871 Dl 3168 6984 3168 6948 Dl 2916 6696 2880 6696 Dl 2992 6696 2956 6696 Dl 3067 6696 3031 6696 Dl 3144 6696 3108 6696 Dl 3219 6696 3183 6696 Dl 3296 6696 3260 6696 Dl 3371 6696 3335 6696 Dl 3448 6696 3412 6696 Dl 3523 6696 3487 6696 Dl 3600 6696 3564 6696 Dl 3600 6660 3600 6696 Dl 3600 6606 3600 6642 Dl 3600 6552 3600 6588 Dl 3564 6552 3600 6552 Dl 3487 6552 3523 6552 Dl 3412 6552 3448 6552 Dl 3335 6552 3371 6552 Dl 3260 6552 3296 6552 Dl 3183 6552 3219 6552 Dl 3108 6552 3144 6552 Dl 3031 6552 3067 6552 Dl 2956 6552 2992 6552 Dl 2880 6552 2916 6552 Dl 2880 6588 2880 6552 Dl 2880 6642 2880 6606 Dl 2880 6696 2880 6660 Dl (1 2)1 770 1 2495 6644 t (Figure 3.2: Permitted mesh with)4 1292 1 2021 7164 t 10 I f (k k)1 44 1 3338 7164 t 7 I f (y y)1 31 1 3393 7184 t 7 R f (, 1)1 58 1 3429 7184 t 10 S f (= =)1 55 1 3511 7164 t 10 I f (k k)1 44 1 3582 7164 t 7 I f (y y)1 31 1 3637 7184 t 7 R f (, 2)1 58 1 3673 7184 t cleartomark showpage saveobj restore %%EndPage: 5 6 %%Page: 6 7 /saveobj save def mark 7 pagesetup 10 R f (- 6 -)2 166 1 2797 480 t ( interfaces are the same independent of the rectangle used to compute the)12 3040(Currently the solution on)3 1030 2 970 840 t ( Figure 3.1 the value of the solution on the interfaces might)11 2458( With)1 258( coefficients.)1 523(solution from the B spline)4 1081 4 720 960 t ( Figure)1 315(depend on which B-spline coefficients were used, those from rectangle 1 or those from rectangle 2.)15 4005 2 720 1080 t ( of the)2 275(3.1 entails a level of approximation that the author did not wish to consider with the first version)17 4045 2 720 1200 t ( certain special problems was)4 1231( functionality was the primary objective and efficiency for)8 2432(code. Increased)1 657 3 720 1320 t ( in the next version of the program Figure 3.1 will be permitted.)12 2553( Perhaps)1 366(considered secondary.)1 886 3 720 1440 t 10 B f ( - Formulation.)2 647(4. Examples)1 542 2 720 1680 t 10 R f ( Appendix 1 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 1836 t (grams that solve these problems using)5 1524 1 720 1956 t 10 CW f (TTGU)2269 1956 w 10 R f (.)2509 1956 w 10 B f (Example 1 - A Simple Heat Equation.)6 1604 1 720 2196 t 10 R f (As a simple example of the use of)7 1356 1 970 2352 t 10 CW f (TTGU)2351 2352 w 10 R f (, consider solving the scalar heat equation)6 1672 1 2591 2352 t 10 I f (u u)1 50 1 1220 2532 t 7 I f (t t)1 20 1 1281 2552 t 10 S f (+ +)1 55 1 1349 2532 t 10 I f (u u)1 50 1 1444 2532 t 7 I f (x x)1 31 1 1505 2552 t 10 S f (+ +)1 55 1 1584 2532 t 10 I f (u u)1 50 1 1679 2532 t 7 I f (y y)1 31 1 1740 2552 t 10 S f (= =)1 55 1 1828 2532 t 10 B f (u)1932 2532 w 7 I f (x x)1 31 1 1999 2552 t 10 S f (+ +)1 55 1 2078 2532 t 10 B f (u)2173 2532 w 7 I f (x xx x)2 62 1 2240 2552 t 10 S f (+ +)1 55 1 2350 2532 t 10 I f (. .)1 25 1 2445 2532 t 10 R f (1)2478 2532 w 10 B f (u)2560 2532 w 7 I f (x xy y)2 62 1 2627 2552 t 10 S f (+ +)1 55 1 2737 2532 t 10 B f (u)2832 2532 w 7 I f (y y)1 31 1 2899 2552 t 10 S f (+ +)1 55 1 2978 2532 t 10 B f (u)3073 2532 w 7 I f (y yy y)2 62 1 3140 2552 t 10 S f (+ +)1 55 1 3250 2532 t 10 I f (. .)1 25 1 3345 2532 t 10 R f (1)3378 2532 w 10 B f (u)3460 2532 w 7 I f (x xy y)2 62 1 3527 2552 t 10 S f (+ +)1 55 1 3613 2532 t 10 I f (g g)1 50 1 3684 2532 t 10 R f (\()3742 2532 w 10 I f (t t)1 28 1 3783 2532 t 10 R f (,)3819 2532 w 10 I f (x x)1 44 1 3852 2532 t 10 R f (,)3904 2532 w 10 I f (y y)1 44 1 3937 2532 t 10 R f ( \(4.1\))1 977(\) ,)1 74 2 3989 2532 t (on the T-shaped region)3 923 1 720 2712 t 1800 2874 1800 3594 Dl 2520 2874 1800 2874 Dl 2520 3594 2520 2874 Dl 1800 3594 2520 3594 Dl (-1,1)1642 2894 w (-1,0)1642 3614 w (0,1)2458 2834 w 2520 2874 2520 3594 Dl 3240 2874 2520 2874 Dl 3240 3594 3240 2874 Dl 2520 3594 3240 3594 Dl (1,1)3178 2834 w 3240 2874 3240 3594 Dl 3960 2874 3240 2874 Dl 3960 3594 3960 2874 Dl 3240 3594 3960 3594 Dl (2,1)3960 2894 w (2,0)3960 3614 w 2520 3594 2520 4314 Dl 3240 3594 2520 3594 Dl 3240 4314 3240 3594 Dl 2520 4314 3240 4314 Dl (0,-1 1,-1)1 1036 1 2362 4334 t (0,0 1,0)1 1042 1 2359 3686 t (where the source term)3 916 1 720 4494 t 10 I f (g g)1 50 1 1672 4494 t 10 R f (\()1730 4494 w 10 I f (t t)1 28 1 1771 4494 t 10 R f (,)1807 4494 w 10 I f (x x)1 44 1 1840 4494 t 10 R f (,)1892 4494 w 10 I f (y y)1 44 1 1925 4494 t 10 R f (\) is chosen so that the solution is a known function,)10 2172 1 1977 4494 t 10 I f (u u)1 50 1 4186 4494 t 10 R f (\()4244 4494 w 10 I f (t t)1 28 1 4285 4494 t 10 R f (,)4321 4494 w 10 I f (x x)1 44 1 4354 4494 t 10 R f (\))4406 4494 w 10 S f (= =)1 55 1 4496 4494 t 10 I f (t t)1 28 1 4600 4494 t 10 R f (.)4636 4464 w 10 I f (x x)1 44 1 4669 4494 t 10 R f (.)4721 4464 w 10 I f (y y)1 44 1 4754 4494 t 10 R f (. The)1 242 1 4798 4494 t (boundary conditions are then taken to be)6 1625 1 720 4614 t 10 I f (u u)1 50 1 1220 4794 t 10 R f (\()1278 4794 w 10 I f (t t)1 28 1 1319 4794 t 10 R f (,)1355 4794 w 10 I f (x x)1 44 1 1388 4794 t 10 R f (,)1440 4794 w 10 I f (y y)1 44 1 1473 4794 t 10 R f (\))1525 4794 w 10 S f (= =)1 55 1 1574 4794 t 10 I f (t t)1 28 1 1645 4794 t 10 R f (.)1681 4764 w 10 I f (x x)1 44 1 1714 4794 t 10 R f (.)1766 4764 w 10 I f (y y)1 44 1 1799 4794 t 10 R f (\(4.2\))4849 4794 w (with initial conditions)2 879 1 720 5010 t 10 I f (u u)1 50 1 1220 5190 t 10 R f (\( 0 ,)2 124 1 1278 5190 t 10 I f (x x)1 44 1 1410 5190 t 10 R f (,)1462 5190 w 10 I f (y y)1 44 1 1495 5190 t 10 R f (\))1547 5190 w 10 S f (= =)1 55 1 1637 5190 t 10 R f (0)1741 5190 w 10 I f (. .)1 25 1 1823 5190 t 10 R f (\(4.3\))4849 5190 w (The)720 5406 w 10 B f (pde)900 5406 w 10 R f (\(4.1\) is equivalent to \(2.1\) with)5 1246 1 1081 5406 t 10 I f (a a)1 50 1 1220 5586 t 7 R f (\( 1 \))2 91 1 1281 5546 t 10 S f (= =)1 55 1 1437 5586 t 10 I f (u u)1 50 1 1541 5586 t 10 S f (+ +)1 55 1 1631 5586 t 10 I f (u u)1 50 1 1726 5586 t 7 I f (x x)1 31 1 1787 5606 t 10 S f (+ +)1 55 1 1866 5586 t 10 I f (. .)1 25 1 1961 5586 t 10 R f (1)1994 5586 w 10 I f (u u)1 50 1 2076 5586 t 7 I f (y y)1 31 1 2137 5606 t 10 R f (,)2184 5586 w 10 I f (a a)1 50 1 1228 5766 t 7 R f (\( 2 \))2 91 1 1289 5726 t 10 S f (= =)1 55 1 1445 5766 t 10 I f (u u)1 50 1 1549 5766 t 10 S f (+ +)1 55 1 1639 5766 t 10 I f (u u)1 50 1 1734 5766 t 7 I f (y y)1 31 1 1795 5786 t 10 S f (+ +)1 55 1 1874 5766 t 10 I f (. .)1 25 1 1969 5766 t 10 R f (1)2002 5766 w 10 I f (u u)1 50 1 2084 5766 t 7 I f (x x)1 31 1 2145 5786 t 10 R f (,)2192 5766 w 10 I f (f f)1 28 1 1236 5946 t 10 S f (= =)1 55 1 1329 5946 t 10 I f (u u)1 50 1 1433 5946 t 7 I f (t t)1 20 1 1494 5966 t 10 S f (+ +)1 55 1 1562 5946 t 10 I f (u u)1 50 1 1657 5946 t 7 I f (x x)1 31 1 1718 5966 t 10 S f (+ +)1 55 1 1797 5946 t 10 I f (u u)1 50 1 1892 5946 t 7 I f (y y)1 31 1 1953 5966 t 10 S f (- -)1 55 1 2032 5946 t 10 I f (g g)1 50 1 2127 5946 t 10 R f (\()2185 5946 w 10 I f (t t)1 28 1 2226 5946 t 10 R f (,)2262 5946 w 10 I f (x x)1 44 1 2295 5946 t 10 R f (,)2347 5946 w 10 I f (y y)1 44 1 2380 5946 t 10 R f (\))2432 5946 w (while the)1 369 1 720 6126 t 10 B f (bc)1114 6126 w 10 R f (s \(4.2\) are equivalent to \(2.2\) with)6 1364 1 1214 6126 t 10 B f (b)1220 6306 w 10 S f (= =)1 55 1 1325 6306 t 10 I f (u u)1 50 1 1429 6306 t 10 R f (\()1487 6306 w 10 I f (t t)1 28 1 1528 6306 t 10 R f (,)1564 6306 w 10 I f (x x)1 44 1 1597 6306 t 10 R f (,)1649 6306 w 10 I f (y y)1 44 1 1682 6306 t 10 R f (\))1734 6306 w 10 S f (- -)1 55 1 1815 6306 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 6306 t 10 R f ( only difference between this problem and)6 1697( The)1 207(See example 1 in Appendix 1 for code solving this problem.)10 2416 3 720 6486 t ( the domain. Both examples use the same pde, but for the)11 2344(example 1 in the documentation of TTGR[15] is)7 1976 2 720 6606 t (first example in TTGR[15] the domain is a rectangle.)8 2127 1 720 6726 t cleartomark showpage saveobj restore %%EndPage: 6 7 %%Page: 7 8 /saveobj save def mark 8 pagesetup 10 R f (- 7 -)2 166 1 2797 480 t 10 B f (Example 2 - Two Heat Equations.)5 1440 1 720 840 t 10 R f (Consider solving the coupled system of heat equations)7 2179 1 970 996 t 10 I f (u u)1 50 1 1220 1316 t 7 R f (2)1281 1336 w 7 I f (t t)1 20 1 1321 1336 t 10 S f (= =)1 55 1 1398 1316 t 10 I f (u u)1 50 1 1502 1316 t 7 R f (2)1563 1336 w 7 I f (x xx x)2 62 1 1603 1336 t 10 S f (+ +)1 55 1 1713 1316 t 10 I f (u u)1 50 1 1808 1316 t 7 R f (2)1869 1336 w 7 I f (y yy y)2 62 1 1909 1336 t 10 S f (- -)1 55 1 2019 1316 t 10 I f (u u)1 50 1 2114 1316 t 7 R f (1)2175 1336 w 10 I f (u u)1 50 1 2250 1316 t 7 R f (2)2311 1336 w 10 S f (+ +)1 55 1 2394 1316 t 10 I f (g g)1 50 1 2489 1316 t 7 R f (2)2550 1336 w 10 I f (u u)1 50 1 1220 1156 t 7 R f (1)1281 1176 w 7 I f (t t)1 20 1 1321 1176 t 10 S f (= =)1 55 1 1398 1156 t 10 I f (u u)1 50 1 1502 1156 t 7 R f (1)1563 1176 w 7 I f (x xx x)2 62 1 1603 1176 t 10 S f (+ +)1 55 1 1713 1156 t 10 I f (u u)1 50 1 1808 1156 t 7 R f (1)1869 1176 w 7 I f (y yy y)2 62 1 1909 1176 t 10 S f (- -)1 55 1 2019 1156 t 10 I f (u u)1 50 1 2090 1156 t 7 R f (1)2151 1176 w 10 I f (u u)1 50 1 2226 1156 t 7 R f (2)2287 1176 w 10 S f (+ +)1 55 1 2370 1156 t 10 I f (g g)1 50 1 2465 1156 t 7 R f (1)2526 1176 w 10 R f (\(4.4\))4849 1266 w (on the L shaped region)4 915 1 720 1496 t 2160 1658 2160 2378 Dl 2880 1658 2160 1658 Dl 2880 2378 2880 1658 Dl 2160 2378 2880 2378 Dl (0,1)2035 2398 w (0,2 1,2)1 970 1 2035 1678 t 2160 2378 2160 3098 Dl 2880 2378 2160 2378 Dl 2880 3098 2880 2378 Dl 2160 3098 2880 3098 Dl (0,0)2035 3118 w (1,0)2818 3178 w 2880 2378 2880 3098 Dl 3600 2378 2880 2378 Dl 3600 3098 3600 2378 Dl 2880 3098 3600 3098 Dl (0,2)3625 3118 w (1,2)3600 2398 w (1,1)2916 2362 w (where)720 3278 w 10 I f (g g)1 50 1 988 3278 t 7 R f (1)1049 3298 w 10 R f (and)1117 3278 w 10 I f (g g)1 50 1 1286 3278 t 7 R f (2)1347 3298 w 10 R f (are chosen so that the solution is given by)8 1671 1 1415 3278 t 10 I f (u u)1 50 1 1220 3458 t 7 R f (1)1281 3478 w 10 R f (\()1332 3458 w 10 I f (t t)1 28 1 1373 3458 t 10 R f (,)1409 3458 w 10 I f (x x)1 44 1 1442 3458 t 10 R f (,)1494 3458 w 10 I f (y y)1 44 1 1527 3458 t 10 R f (\))1579 3458 w 10 S f (= =)1 55 1 1669 3458 t 10 I f (e e)1 44 1 1773 3458 t 7 I f (t t)1 20 1 1828 3418 t 7 R f (\()1869 3418 w 7 I f (x x)1 31 1 1897 3418 t 7 S f (- -)1 39 1 1944 3418 t 7 I f (y y)1 31 1 1994 3418 t 7 R f (\))2030 3418 w 10 R f (and)2275 3458 w 10 I f (u u)1 50 1 2625 3458 t 7 R f (2)2686 3478 w 10 R f (\()2737 3458 w 10 I f (t t)1 28 1 2778 3458 t 10 R f (,)2814 3458 w 10 I f (x x)1 44 1 2847 3458 t 10 R f (,)2899 3458 w 10 I f (y y)1 44 1 2932 3458 t 10 R f (\))2984 3458 w 10 S f (= =)1 55 1 3074 3458 t 10 I f (e e)1 44 1 3178 3458 t 7 S f (- -)1 39 1 3233 3418 t 7 I f (t t)1 20 1 3283 3418 t 7 R f (\()3308 3418 w 7 I f (x x)1 31 1 3336 3418 t 7 S f (- -)1 39 1 3383 3418 t 7 I f (y y)1 31 1 3433 3418 t 7 R f (\))3469 3418 w 10 I f (. .)1 25 1 3516 3458 t 10 R f (The boundary conditions are then taken to be)7 1805 1 720 3638 t 10 I f (u u)1 50 1 1220 3818 t 7 R f (1)1281 3838 w 10 R f (\()1332 3818 w 10 I f (t t)1 28 1 1373 3818 t 10 R f (,)1409 3818 w 10 I f (x x)1 44 1 1442 3818 t 10 R f (,)1494 3818 w 10 I f (y y)1 44 1 1527 3818 t 10 R f (\))1579 3818 w 10 S f (= =)1 55 1 1628 3818 t 10 I f (e e)1 44 1 1699 3818 t 7 I f (t t)1 20 1 1754 3778 t 7 R f (\()1779 3778 w 7 I f (x x)1 31 1 1807 3778 t 7 S f (- -)1 39 1 1854 3778 t 7 I f (y y)1 31 1 1904 3778 t 7 R f (\))1940 3778 w 10 R f (and)2185 3818 w 10 I f (u u)1 50 1 2535 3818 t 7 R f (2)2596 3838 w 10 R f (\()2647 3818 w 10 I f (t t)1 28 1 2688 3818 t 10 R f (,)2724 3818 w 10 I f (x x)1 44 1 2757 3818 t 10 R f (,)2809 3818 w 10 I f (y y)1 44 1 2842 3818 t 10 R f (\))2894 3818 w 10 S f (= =)1 55 1 2943 3818 t 10 I f (e e)1 44 1 3014 3818 t 7 S f (- -)1 39 1 3069 3778 t 7 I f (t t)1 20 1 3119 3778 t 7 R f (\()3144 3778 w 7 I f (x x)1 31 1 3172 3778 t 7 S f (- -)1 39 1 3219 3778 t 7 I f (y y)1 31 1 3269 3778 t 7 R f (\))3305 3778 w 10 R f (\(4.5\))4849 3818 w (with initial conditions)2 879 1 720 4034 t 10 I f (u u)1 50 1 1220 4214 t 7 R f (1)1281 4234 w 10 R f (\( 0 ,)2 124 1 1332 4214 t 10 I f (x x)1 44 1 1464 4214 t 10 R f (,)1516 4214 w 10 I f (y y)1 44 1 1549 4214 t 10 R f (\))1601 4214 w 10 S f (= =)1 55 1 1691 4214 t 10 R f (1)1795 4214 w 10 S f (= =)1 55 1 1894 4214 t 10 I f (u u)1 50 1 1998 4214 t 7 R f (2)2059 4234 w 10 R f (\( 0 ,)2 124 1 2110 4214 t 10 I f (x x)1 44 1 2242 4214 t 10 R f (,)2294 4214 w 10 I f (y y)1 44 1 2327 4214 t 10 R f (\). \(4.6\))1 2661 1 2379 4214 t (The)720 4430 w 10 B f (pde)900 4430 w 10 R f (\(4.4\) is equivalent to \(2.1\) with)5 1246 1 1081 4430 t 10 I f (a a)1 50 1 1220 4610 t 7 R f (1)1275 4629 w (\( 1 \))2 91 1 1275 4570 t 10 S f (= =)1 55 1 1431 4610 t 10 I f (u u)1 50 1 1535 4610 t 7 R f (1)1596 4630 w 7 I f (x x)1 31 1 1636 4630 t 10 R f (,)1683 4610 w 10 I f (a a)1 50 1 1815 4610 t 7 R f (2)1870 4629 w (\( 1 \))2 91 1 1870 4570 t 10 S f (= =)1 55 1 2026 4610 t 10 I f (u u)1 50 1 2130 4610 t 7 R f (2)2191 4630 w 7 I f (x x)1 31 1 2231 4630 t 10 R f (,)2278 4610 w 10 I f (a a)1 50 1 1228 4790 t 7 R f (1)1283 4809 w (\( 2 \))2 91 1 1283 4750 t 10 S f (= =)1 55 1 1439 4790 t 10 I f (u u)1 50 1 1543 4790 t 7 R f (1)1604 4810 w 7 I f (y y)1 31 1 1644 4810 t 10 R f (,)1691 4790 w 10 I f (a a)1 50 1 1823 4790 t 7 R f (2)1878 4809 w (\( 2 \))2 91 1 1878 4750 t 10 S f (= =)1 55 1 2034 4790 t 10 I f (u u)1 50 1 2138 4790 t 7 R f (2)2199 4810 w 7 I f (y y)1 31 1 2239 4810 t 10 R f (,)2286 4790 w 10 I f (f f)1 28 1 1236 4970 t 7 R f (1)1275 4990 w 10 S f (= =)1 55 1 1367 4970 t 10 I f (u u)1 50 1 1471 4970 t 7 R f (1)1532 4990 w 7 I f (t t)1 20 1 1572 4990 t 10 S f (+ +)1 55 1 1640 4970 t 10 I f (u u)1 50 1 1735 4970 t 7 R f (1)1796 4990 w 10 I f (u u)1 50 1 1871 4970 t 7 R f (2)1932 4990 w 10 S f (- -)1 55 1 2015 4970 t 10 I f (g g)1 50 1 2110 4970 t 7 R f (1)2171 4990 w 10 R f (,)2222 4970 w 10 I f (f f)1 28 1 2362 4970 t 7 R f (2)2401 4990 w 10 S f (= =)1 55 1 2493 4970 t 10 I f (u u)1 50 1 2597 4970 t 7 R f (2)2658 4990 w 7 I f (t t)1 20 1 2698 4990 t 10 S f (+ +)1 55 1 2766 4970 t 10 I f (u u)1 50 1 2861 4970 t 7 R f (1)2922 4990 w 10 I f (u u)1 50 1 2997 4970 t 7 R f (2)3058 4990 w 10 S f (- -)1 55 1 3141 4970 t 10 I f (g g)1 50 1 3236 4970 t 7 R f (2)3297 4990 w 10 R f (while the)1 369 1 720 5150 t 10 B f (bc)1114 5150 w 10 R f (s \(4.5\) are equivalent to \(2.2\) with)6 1364 1 1214 5150 t 10 I f (b b)1 50 1 1220 5330 t 7 R f (1)1281 5350 w 10 S f (= =)1 55 1 1373 5330 t 10 I f (u u)1 50 1 1477 5330 t 7 R f (1)1538 5350 w 10 R f (\()1589 5330 w 10 I f (t t)1 28 1 1630 5330 t 10 R f (,)1666 5330 w 10 I f (x x)1 44 1 1699 5330 t 10 R f (,)1751 5330 w 10 I f (y y)1 44 1 1784 5330 t 10 R f (\))1836 5330 w 10 S f (- -)1 55 1 1917 5330 t 10 I f (e e)1 44 1 2012 5330 t 7 I f (t t)1 20 1 2067 5290 t 7 R f (\()2092 5290 w 7 I f (x x)1 31 1 2120 5290 t 7 S f (- -)1 39 1 2167 5290 t 7 I f (y y)1 31 1 2217 5290 t 7 R f (\))2253 5290 w 10 R f (and)2498 5330 w 10 I f (b b)1 50 1 2848 5330 t 7 R f (2)2909 5350 w 10 S f (= =)1 55 1 3001 5330 t 10 I f (u u)1 50 1 3105 5330 t 7 R f (2)3166 5350 w 10 R f (\()3217 5330 w 10 I f (t t)1 28 1 3258 5330 t 10 R f (,)3294 5330 w 10 I f (x x)1 44 1 3327 5330 t 10 R f (,)3379 5330 w 10 I f (y y)1 44 1 3412 5330 t 10 R f (\))3464 5330 w 10 S f (- -)1 55 1 3545 5330 t 10 I f (e e)1 44 1 3640 5330 t 7 S f (- -)1 39 1 3695 5290 t 7 I f (t t)1 20 1 3745 5290 t 7 R f (\()3770 5290 w 7 I f (x x)1 31 1 3798 5290 t 7 S f (- -)1 39 1 3845 5290 t 7 I f (y y)1 31 1 3895 5290 t 7 R f (\))3931 5290 w 10 I f (. .)1 25 1 3978 5330 t 10 R f ( example and)2 536( only difference between this)4 1162( The)1 206(See example 2 in Appendix 1 for code solving this problem.)10 2416 4 720 5510 t (example 2 in the documentation'of TTGR[15] is the domain. In [15] the domain is a simple rectangle.)16 4078 1 720 5630 t 10 B f (More Meaty Examples)2 970 1 720 5870 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 6026 t 10 B f (ode)720 6146 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 6146 t ( method has the property that the term)7 1537( That)1 235( Galerkin's method [19,25].)3 1118(ences is)1 314 4 720 6266 t 10 B f (a \()1 91 1 3951 6266 t 10 R f (.)4050 6236 w (\) in \(2.1\) is forced to be)6 957 1 4083 6266 t ( Since)1 274(continuous everywhere, see [19].)3 1327 2 720 6386 t 10 B f (a \()1 91 1 2348 6386 t 10 R f (.)2447 6356 w ( contin-)1 309(\) is a flux, physically, and physicists expect fluxes to be)10 2251 2 2480 6386 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 6506 t ( not automatically make)3 990(methods such as finite-differences, least-squares and collocation do)7 2764 2 720 6626 t 10 B f (a \()1 91 1 4508 6626 t 10 R f (.)4607 6596 w (\) continu-)1 400 1 4640 6626 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 6746 t (use of Galerkin's method in)4 1115 1 720 6866 t 10 CW f (TTGU)1860 6866 w 10 R f (is important in their formulation.)4 1314 1 2125 6866 t cleartomark showpage saveobj restore %%EndPage: 7 8 %%Page: 8 9 /saveobj save def mark 9 pagesetup 10 R f (- 8 -)2 166 1 2797 480 t 10 B f (Example 3 - Interfaces)3 962 1 720 840 t 10 R f ( shows how to deal with material interfaces.)7 1779( It)1 113(This is example 3 in the documentation of TTGR[15].)8 2178 3 970 996 t ( with three rectangles piled one on another, each with its own material constant,)13 3225(Assume a layered structure)3 1095 2 720 1116 t 10 S f (k)720 1236 w 10 R f (\()807 1236 w 10 I f (x x)1 44 1 848 1236 t 10 R f (,)900 1236 w 10 I f (y y)1 44 1 933 1236 t 10 R f ( a heat flow problem, the modeling equations might look like)10 2446(\). For)1 247 2 985 1236 t 10 B f (u)1220 1416 w 7 I f (t t)1 20 1 1287 1436 t 10 S f ( \321)1 120(= =)1 55 2 1364 1416 t 10 R f (.)1571 1386 w (\()1628 1416 w 10 S f (k)1693 1416 w 10 R f (\()1780 1416 w 10 I f (x x)1 44 1 1821 1416 t 10 R f (,)1873 1416 w 10 I f (y y)1 44 1 1906 1416 t 10 R f (\))1958 1416 w 10 S f (\321)2031 1416 w 10 B f (u)2134 1416 w 10 R f (\))2222 1416 w 10 S f (+ +)1 55 1 2303 1416 t 10 I f (g g)1 50 1 2398 1416 t 10 R f (\()2456 1416 w 10 I f (t t)1 28 1 2497 1416 t 10 R f (,)2533 1416 w 10 I f (x x)1 44 1 2566 1416 t 10 R f (,)2618 1416 w 10 I f (y y)1 44 1 2651 1416 t 10 R f (\) \(4.7\))1 2337 1 2703 1416 t ( 0 , 1 ])4 190(on the domain [)3 630 2 720 1596 t 10 S f (\264)1556 1596 w 10 R f ( where)1 268([ 0 , 3 ],)4 248 2 1619 1596 t 10 S f (k)2160 1596 w 10 R f (is piecewise constant on the different rectangles, say,)7 2120 1 2240 1596 t 10 S f (k \272)1 151 1 1220 1961 t (\354)1420 1774 w (\357)1420 1874 w (\355)1420 1974 w (\357)1420 2074 w (\356)1420 2174 w 10 R f (1)1535 2111 w 10 I f (/ /)1 28 1 1593 2111 t 10 R f (3 2)1 273 1 1629 2111 t 10 S f (< <)1 55 1 1918 2111 t 10 I f (y y)1 44 1 1989 2111 t 10 S f (\243)2041 2111 w 10 R f (3)2104 2111 w (1)1535 1971 w 10 I f (/ /)1 28 1 1593 1971 t 10 R f (2 1)1 273 1 1629 1971 t 10 S f (< <)1 55 1 1918 1971 t 10 I f (y y)1 44 1 1989 1971 t 10 S f (\243)2041 1971 w 10 R f (2)2104 1971 w (1 0)1 273 1 1535 1831 t 10 S f (\243)1816 1831 w 10 I f (y y)1 44 1 1879 1831 t 10 S f (\243)1931 1831 w 10 R f (1)1994 1831 w (and)720 2346 w 10 I f (g g)1 50 1 889 2346 t 10 R f (is chosen so that)3 658 1 964 2346 t 10 I f (u u)1 50 1 1220 2711 t 10 S f (\272)1311 2711 w (\354)1415 2524 w (\357)1415 2624 w (\355)1415 2724 w (\357)1415 2824 w (\356)1415 2924 w 10 R f (3)1530 2861 w 10 I f (t t)1 28 1 1588 2861 t 10 R f (\()1648 2861 w 10 I f (y y)1 44 1 1689 2861 t 10 S f (- -)1 55 1 1757 2861 t 10 R f ( 2)1 231(1 \))1 91 2 1828 2861 t 10 S f (< <)1 55 1 2166 2861 t 10 I f (y y)1 44 1 2237 2861 t 10 S f (\243)2289 2861 w 10 R f (3)2352 2861 w 10 I f (. .)1 25 1 2410 2861 t (t t)1 28 1 1530 2721 t 10 R f (\( 2)1 91 1 1590 2721 t 10 I f (y y)1 44 1 1689 2721 t 10 S f (- -)1 55 1 1757 2721 t 10 R f ( 1)1 231(1 \))1 91 2 1828 2721 t 10 S f (< <)1 55 1 2166 2721 t 10 I f (y y)1 44 1 2237 2721 t 10 S f (\243)2289 2721 w 10 R f (2)2352 2721 w 10 I f ( y)1 0(t ty)1 72 2 1530 2581 t 10 R f (0)1940 2581 w 10 S f (\243)1998 2581 w 10 I f (y y)1 44 1 2061 2581 t 10 S f (\243)2113 2581 w 10 R f (1)2176 2581 w (The)720 3096 w 10 B f (bc)900 3096 w 10 R f (s on the bottom and top are given by, say, insulation)10 2088 1 1000 3096 t 10 B f (u)1220 3276 w 7 I f (N N)1 47 1 1287 3296 t 10 S f (= =)1 55 1 1391 3276 t 10 R f (0 \(4.8a\))1 3545 1 1495 3276 t (where)720 3456 w 10 B f (u)988 3456 w 7 I f (N N)1 47 1 1055 3476 t 10 R f (is the normal derivative, and on the sides Dirichlet data)9 2208 1 1135 3456 t 10 B f (u)1220 3636 w 10 S f (= =)1 55 1 1325 3636 t 10 I f (s s)1 39 1 1429 3636 t 10 R f (\()1476 3636 w 10 I f (t t)1 28 1 1517 3636 t 10 R f (,)1553 3636 w 10 I f (x x)1 44 1 1586 3636 t 10 R f (,)1638 3636 w 10 I f (y y)1 44 1 1671 3636 t 10 R f (\) \(4.8b\))1 3317 1 1723 3636 t (is used, for some known)4 974 1 720 3816 t 10 I f (s s)1 39 1 1719 3816 t 10 R f (.)1758 3816 w (The)970 3972 w 10 B f (pde)1150 3972 w 10 R f (is equivalent to \(2.1\) with)4 1030 1 1331 3972 t 10 I f (f f)1 28 1 1220 4472 t 10 S f (= =)1 55 1 1313 4472 t 10 B f (u)1417 4472 w 7 I f (t t)1 20 1 1484 4492 t 10 S f (- -)1 55 1 1552 4472 t 10 I f ( .)1 0( .)1 33(g g)1 50 3 1647 4472 t (a a)1 50 1 1220 4312 t 7 R f (\( 2 \))2 91 1 1281 4272 t 10 S f ( k)1 104(= =)1 55 2 1437 4312 t 10 R f (\()1628 4312 w 10 I f (x x)1 44 1 1669 4312 t 10 R f (,)1721 4312 w 10 I f (y y)1 44 1 1754 4312 t 10 R f (\))1806 4312 w 10 B f (u)1879 4312 w 7 I f (y y)1 31 1 1946 4332 t 10 I f (a a)1 50 1 1220 4142 t 7 R f (\( 1 \))2 91 1 1281 4102 t 10 S f ( k)1 104(= =)1 55 2 1437 4142 t 10 R f (\()1628 4142 w 10 I f (x x)1 44 1 1669 4142 t 10 R f (,)1721 4142 w 10 I f (y y)1 44 1 1754 4142 t 10 R f (\))1806 4142 w 10 B f (u)1879 4142 w 7 I f (x x)1 31 1 1946 4162 t 10 R f (The bottom and top)3 786 1 720 4652 t 10 B f (bc)1531 4652 w 10 R f (s are equivalent to \(2.2\) with)5 1148 1 1631 4652 t 10 B f (b)1220 4832 w 10 S f (= =)1 55 1 1325 4832 t 10 I f (u u)1 50 1 1429 4832 t 7 I f (y y)1 31 1 1490 4852 t 10 R f (and those on the side are equivalent to)7 1528 1 720 5012 t 10 B f (b)1220 5192 w 10 S f (= =)1 55 1 1325 5192 t 10 B f (u)1429 5192 w 10 S f (- -)1 55 1 1534 5192 t 10 I f (s s)1 39 1 1638 5192 t 10 R f (\()1685 5192 w 10 I f (t t)1 28 1 1726 5192 t 10 R f (,)1762 5192 w 10 I f (x x)1 44 1 1795 5192 t 10 R f (,)1847 5192 w 10 I f (y y)1 44 1 1880 5192 t 10 R f (\))1932 5192 w 10 I f (. .)1 25 1 1981 5192 t 10 R f ( of this problem is that the normal component of)9 1971(The sporting aspect)2 788 2 720 5372 t 10 S f (k \321)1 158 1 3508 5372 t 10 B f (u)3698 5372 w 10 R f (, that is)2 300 1 3754 5372 t 10 S f (k)4083 5372 w 10 I f (u u)1 50 1 4170 5372 t 7 I f (y y)1 31 1 4231 5392 t 10 R f (, across each mate-)3 770 1 4270 5372 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 5492 t ( the documentation of TTGR,)4 1205( In)1 138( this problem.)2 563( example 3 of Appendix 1 for code solving)8 1746( See)1 198(be the case.)2 470 6 720 5612 t ( treated as one rectangle and to take into consideration the material interfaces, multiple)13 3608(this example was)2 712 2 720 5732 t ( is treated as being defined on 3 unit squares)9 1817( TTGU the problem)3 808( For)1 193( inserted at the interfaces.)4 1038(knots were)1 464 5 720 5852 t (stacked on top of each other.)5 1147 1 720 5972 t 10 B f (Example 4 - Non constant boundary conditions)6 2008 1 720 6212 t 10 R f ( in Example 2)3 561( However)1 416(This example has the same spatial domain and the same pde as in Example 2.)14 3093 3 970 6368 t ( 0, so that the solution can be represented initally by the constant 1, and in this example the)18 3641(the initial time is)3 679 2 720 6488 t ( initally)1 312(intial time is set to 1, so that)7 1143 2 720 6608 t 10 I f (u u)1 50 1 2203 6608 t 10 R f (is not a constant and the B-spline coefficients have to be determined)11 2759 1 2281 6608 t (before the PDE is solved.)4 1018 1 720 6728 t cleartomark showpage saveobj restore %%EndPage: 8 9 %%Page: 9 10 /saveobj save def mark 10 pagesetup 10 R f (- 9 -)2 166 1 2797 480 t 10 B f (Example 5 - A Static Problem)5 1268 1 720 840 t 10 R f (This example shows how to solve a static)7 1699 1 970 996 t 10 B f (pde)2700 996 w 10 R f (and also shows that using a non-uniform grid can be)9 2152 1 2888 996 t ( the)1 147( Let)1 183(very useful and effective.)3 1013 3 720 1116 t 10 B f (pde)2088 1116 w 10 R f (be Laplace's equation)2 875 1 2269 1116 t 10 I f (u u)1 50 1 1220 1296 t 7 I f (x xx x)2 62 1 1281 1316 t 10 S f (+ +)1 55 1 1391 1296 t 10 I f (u u)1 50 1 1486 1296 t 7 I f (y yy y)2 62 1 1547 1316 t 10 S f (= =)1 55 1 1657 1296 t 10 R f (0 \(4.11\))1 3288 1 1752 1296 t (on the domain)2 572 1 720 1476 t 1800 1638 1800 2358 Dl 2520 1638 1800 1638 Dl 2520 2358 2520 1638 Dl 1800 2358 2520 2358 Dl (2,0)1675 1658 w (1,0)1675 2378 w (2,1)2520 1658 w 1800 2358 1800 3078 Dl 2520 2358 1800 2358 Dl 2520 3078 2520 2358 Dl 1800 3078 2520 3078 Dl (0,0)1675 3098 w (0,1)2458 3158 w 2520 2358 2520 3078 Dl 3240 2358 2520 2358 Dl 3240 3078 3240 2358 Dl 2520 3078 3240 3078 Dl (0,2)3178 3158 w 3240 2358 3240 3078 Dl 3960 2358 3240 2358 Dl 3960 3078 3960 2358 Dl 3240 3078 3960 3078 Dl (0,3)3960 3098 w (1,3)3960 2378 w 3240 1638 3240 2358 Dl 3960 1638 3240 1638 Dl 3960 2358 3960 1638 Dl 3240 2358 3960 2358 Dl (2,3 2,2)1 -720 1 3960 1658 t (1,1 1,2)1 648 1 2556 2342 t ( of any analytic function)4 992( the Real part)3 544( Since)1 275(This domain was suggested by a problem posed by G.C.Scott.)9 2509 4 720 3258 t (solves Laplace's equation, a good choice for)6 1812 1 720 3378 t 10 I f (u u)1 50 1 2562 3378 t 10 R f (is the Real part of)4 730 1 2642 3378 t 10 I f (z z)1 39 1 3402 3378 t 10 R f (log \()1 193 1 3473 3378 t 10 I f (z z)1 39 1 3698 3378 t 10 R f (\), where)1 331 1 3769 3378 t 10 I f (z z)1 39 1 4130 3378 t 10 S f (= =)1 55 1 4218 3378 t 10 I f (x x)1 44 1 4322 3378 t 10 S f (+ +)1 55 1 4406 3378 t 10 I f ( y)1 0( y)1 76(i i)1 28 3 4501 3378 t 10 R f (. Dirichlet)1 435 1 4605 3378 t 10 B f (bc)720 3498 w 10 R f ( and top of the domain, consistent with that choice for)10 2176(s on the left, right)4 712 2 820 3498 t 10 I f (u u)1 50 1 3735 3498 t 10 R f ( the bot-)2 337( For)1 191(can be stipulated.)2 700 3 3812 3498 t (tom Neumann data)2 760 1 720 3618 t 10 I f (u u)1 50 1 1505 3618 t 7 I f (y y)1 31 1 1566 3638 t 10 S f (= =)1 55 1 1654 3618 t 10 R f ( gives the boundary conditions)4 1227( This)1 228(0 can be chosen.)3 659 3 1758 3618 t 10 I f (u u)1 50 1 1220 4278 t 10 R f (\( 0 ,)2 124 1 1278 4278 t 10 I f (y y)1 44 1 1410 4278 t 10 R f (\))1462 4278 w 10 S f (= =)1 55 1 1552 4278 t 10 R f (2)1765 4348 w 10 S f ( p)1 71(- -)1 55 2 1689 4218 t 10 I f (y y)1 44 1 1847 4218 t 10 S1 f (_ ____)1 232 1 1674 4248 t 10 I f (. .)1 25 1 1924 4278 t (u u)1 50 1 1220 4078 t 10 R f (\()1278 4078 w 10 I f (x x)1 44 1 1319 4078 t 10 R f (, 1 \))2 124 1 1371 4078 t 10 S f (= =)1 55 1 1552 4078 t 10 I f ( l)1 0( al)1 28( ea)1 50(R Re)1 105 4 1656 4078 t 10 R f (\()1847 4078 w 10 I f ( og g)2 50( lo)1 50( l)1 60(z z)1 39 4 1888 4078 t 10 R f (\()2095 4078 w 10 I f (z z)1 39 1 2136 4078 t 10 R f (\) \))1 74 1 2183 4078 t 10 I f (u u)1 50 1 1220 3938 t 10 R f (\( 1 ,)2 124 1 1278 3938 t 10 I f (y y)1 44 1 1410 3938 t 10 R f (\))1462 3938 w 10 S f (= =)1 55 1 1552 3938 t 10 I f ( l)1 0( al)1 28( ea)1 50(R Re)1 105 4 1656 3938 t 10 R f (\()1847 3938 w 10 I f ( og g)2 50( lo)1 50( l)1 60(z z)1 39 4 1888 3938 t 10 R f (\()2095 3938 w 10 I f (z z)1 39 1 2136 3938 t 10 R f (\) \))1 74 1 2183 3938 t 10 I f (u u)1 50 1 1220 3778 t 7 I f (y y)1 31 1 1281 3798 t 10 R f (\()1328 3778 w 10 I f (x x)1 44 1 1369 3778 t 10 R f (, 0 \))2 124 1 1421 3778 t 10 S f (= =)1 55 1 1602 3778 t 10 R f (0)1706 3778 w (\(4.12\))4799 4028 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 4508 t (has singular first partials at)4 1082 1 720 4628 t 10 I f (z z)1 39 1 1827 4628 t 10 S f (= =)1 55 1 1890 4628 t 10 R f (0.)1961 4628 w (The)970 4784 w 10 B f (pde)1150 4784 w 10 R f (is equivalent to \(2.1\) with)4 1030 1 1331 4784 t 10 I f (f f)1 28 1 1220 5284 t 10 S f (= =)1 55 1 1313 5284 t 10 R f (0)1417 5284 w 10 I f (a a)1 50 1 1220 5124 t 7 R f (\( 2 \))2 91 1 1281 5084 t 10 S f (= =)1 55 1 1437 5124 t 10 B f (u)1541 5124 w 7 I f (y y)1 31 1 1608 5144 t 10 I f (a a)1 50 1 1220 4954 t 7 R f (\( 1 \))2 91 1 1281 4914 t 10 S f (= =)1 55 1 1437 4954 t 10 B f (u)1541 4954 w 7 I f (x x)1 31 1 1608 4974 t 10 R f (The boundary conditions are equivalent to \(2.2\) with)7 2108 1 720 5444 t 10 B f (b)1220 5909 w 10 S f (= =)1 55 1 1325 5909 t (\354)1437 5622 w (\357)1437 5722 w (\357)1437 5822 w (\355)1437 5922 w (\357)1437 6022 w (\357)1437 6122 w (\356)1437 6222 w 10 I f (u u)1 50 1 1486 6159 t 10 R f (\( 0 ,)2 124 1 1544 6159 t 10 I f (y y)1 44 1 1676 6159 t 10 R f (\))1728 6159 w 10 S f (+ +)1 55 1 1818 6159 t 10 R f (2)1987 6229 w 10 S f (p)1947 6099 w 10 I f (y y)1 44 1 2034 6099 t 10 S1 f (_ ___)1 161 1 1932 6129 t 10 I f (u u)1 50 1 1486 5959 t 10 R f (\()1544 5959 w 10 I f (x x)1 44 1 1585 5959 t 10 R f (, 1 \))2 124 1 1637 5959 t 10 S f (- -)1 55 1 1818 5959 t 10 I f ( l)1 0( al)1 28( ea)1 50(R Re)1 105 4 1922 5959 t 10 R f (\()2113 5959 w 10 I f ( og g)2 50( lo)1 50( l)1 60(z z)1 39 4 2154 5959 t 10 R f (\()2361 5959 w 10 I f (z z)1 39 1 2402 5959 t 10 R f (\) \))1 74 1 2449 5959 t 10 I f (u u)1 50 1 1486 5819 t 10 R f (\( 1 ,)2 124 1 1544 5819 t 10 I f (y y)1 44 1 1676 5819 t 10 R f (\))1728 5819 w 10 S f (- -)1 55 1 1818 5819 t 10 I f ( l)1 0( al)1 28( ea)1 50(R Re)1 105 4 1922 5819 t 10 R f (\()2113 5819 w 10 I f ( og g)2 50( lo)1 50( l)1 60(z z)1 39 4 2154 5819 t 10 R f (\()2361 5819 w 10 I f (z z)1 39 1 2402 5819 t 10 R f (\) \))1 74 1 2449 5819 t 10 I f (u u)1 50 1 1486 5659 t 7 I f (y y)1 31 1 1547 5679 t 10 R f (\()1594 5659 w 10 I f (x x)1 44 1 1635 5659 t 10 R f (, 0 \))2 124 1 1687 5659 t (The fact that there is no)5 945 1 720 6430 t 10 B f (u)1690 6430 w 7 I f (t t)1 20 1 1757 6450 t 10 R f (term in)1 286 1 1810 6430 t 10 I f (f f)1 28 1 2121 6430 t 10 R f (is ok, since)2 447 1 2174 6430 t 10 CW f (TTGU)2646 6430 w 10 R f (does not require it.)3 749 1 2911 6430 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 6586 t ( this problem)2 542( the only difference between)4 1167( Again)1 302(example 5 in Appendix 1 for code solving this problem.)9 2309 4 720 6706 t (and example 5 in the documentation of TTGR[15] is the domain. In [15] the domain is a simple rectangle.)18 4239 1 720 6826 t cleartomark showpage saveobj restore %%EndPage: 9 10 %%Page: 11 11 /saveobj save def mark 11 pagesetup 10 R f (- 11 -)2 216 1 2772 480 t 10 B f ( for the pde-bc Problem.)4 1040(5. Software)1 507 2 720 840 t 10 R f ( called)1 266(This section is a brief user's manual for a software package)10 2391 2 970 996 t 10 CW f (TTGU)3655 996 w 10 R f (, \(Transient Tensor Galerkin)3 1145 1 3895 996 t (method for)1 441 1 720 1116 t 10 B f (pde)1186 1116 w 10 R f (s on a Union of rectangles\), implementing the algorithm outlined in section 3.)12 3109 1 1342 1116 t (Before invoking)1 652 1 970 1272 t 10 CW f (TTGU)1647 1272 w 10 R f (the user must)2 533 1 1912 1272 t 10 S f (\267)970 1428 w 10 R f (Make B-spline meshes for)3 1051 1 1041 1428 t 10 I f (x x)1 44 1 2117 1428 t 10 R f (and)2186 1428 w 10 I f (y y)1 44 1 2355 1428 t 10 R f (on each rectangle.)2 722 1 2424 1428 t 10 S f (\267)970 1584 w 10 R f (Make initial conditions for the B-spline coefficients)6 2070 1 1041 1584 t 10 B f (U)3136 1584 w 10 R f (in \(3.1\).)1 319 1 3233 1584 t 10 S f (\267)970 1740 w 10 R f (Write subroutines)1 713 1 1041 1740 t 10 S f (\267)1220 1896 w 10 CW f (AF)1326 1896 w 10 R f (- to evaluate)2 493 1 1471 1896 t 10 B f (a)1989 1896 w 10 R f (and)2064 1896 w 10 B f (f)2233 1896 w 10 R f (in \(2.1\).)1 319 1 2291 1896 t 10 S f (\267)1220 2052 w 10 CW f (BC)1326 2052 w 10 R f (- to evaluate)2 493 1 1471 2052 t 10 B f (b)1989 2052 w 10 R f (in \(2.2\).)1 319 1 2070 2052 t 10 S f (\267)1220 2208 w 10 CW f (HANDLU)1326 2208 w 10 R f (- to output \(print\) the solution results.)6 1503 1 1711 2208 t (Each of these preparatory steps are illustrated in Appendix 1 and will not be described here.)15 3664 1 720 2364 t ( coefficients are stored in such a way that a user with one rectangle can use either)16 3534(The B-spline)1 536 2 970 2520 t ( the structure defined below it is easy)7 1489(TTGR[15] or TTGU without changing data structures. Moreover, with)8 2831 2 720 2640 t ( e.g. TSD1 which evaluates the solution from one rectangle. Because each)11 3048(to use other available software,)4 1272 2 720 2760 t ( using a different mesh, one cannot simply add an index stipulating the rectangle. The fol-)15 3592(rectangle is stored)2 728 2 720 2880 t (lowing storage map for the B-spline coefficients may look cumbersome, but it works.)12 3418 1 720 3000 t (Assume that there are)3 867 1 970 3156 t 10 I f (n n)1 50 1 1862 3156 t 7 I f (r r)1 27 1 1923 3176 t 10 R f (rectangles and let)2 698 1 1983 3156 t 10 I f (s s)1 39 1 1784 3386 t 7 I f (j j)1 20 1 1834 3406 t 10 S f (= =)1 55 1 1878 3386 t 10 R f (1)1949 3386 w 10 S f (+ +)1 55 1 2015 3386 t 7 I f (m m)1 50 1 2121 3486 t 15 S f (S)2102 3416 w 7 I f (j j)1 20 1 2089 3286 t 7 S f (- -)1 39 1 2120 3286 t 7 R f (1)2170 3286 w 10 R f (\()2215 3386 w 10 I f ( X)1 0(N NX)1 128 2 2256 3386 t 7 I f (i i)1 20 1 2395 3406 t 10 S f (- -)1 55 1 2439 3386 t 10 I f (k k)1 44 1 2510 3386 t 7 I f (x x)1 31 1 2565 3406 t 7 R f (,)2601 3406 w 7 I f (m m)1 50 1 2624 3406 t 10 R f (\))2690 3386 w 10 S f (\264)2739 3386 w 10 R f (\()2802 3386 w 10 I f ( Y)1 0(N NY)1 123 2 2843 3386 t 7 I f (i i)1 20 1 2977 3406 t 10 S f (- -)1 55 1 3021 3386 t 10 I f (k k)1 44 1 3092 3386 t 7 I f (y y)1 31 1 3147 3406 t 7 R f (,)3183 3406 w 7 I f (m m)1 50 1 3206 3406 t 10 R f ( 0)1 91( for)1 223(\) ,)1 74 3 3272 3386 t 10 S f (\243 \243)1 55 1 3676 3386 t 10 I f (j j)1 28 1 3747 3386 t 10 S f (\243)3783 3386 w 10 I f (n n)1 50 1 3846 3386 t 7 I f (r r)1 27 1 3907 3406 t 10 I f (. .)1 25 1 3950 3386 t 10 R f (Then the B-spline coefficients)3 1302 1 720 3646 t 10 B f (b)2079 3646 w 7 I f (p p)1 35 1 2146 3666 t 7 R f (,)2186 3666 w 7 I f (q q)1 35 1 2209 3666 t 7 R f (,)2249 3666 w 7 I f (i i)1 20 1 2272 3666 t 10 R f (for rectangle)1 538 1 2357 3646 t 10 I f (j j)1 28 1 2952 3646 t 10 R f ( dimensioned)1 569(in \(3.1\) is stored in a matrix U)7 1434 2 3037 3646 t (\()720 3766 w 10 I f (s s)1 39 1 761 3766 t 7 I f (r r)1 27 1 811 3786 t 7 S f (+ +)1 39 1 854 3786 t 7 R f (1)904 3786 w 10 S f (- -)1 55 1 963 3766 t 10 R f (1 ,)1 83 1 1034 3766 t 10 I f (n n)1 50 1 1125 3766 t 7 I f (u u)1 35 1 1186 3786 t 10 R f (\) at position U\()3 644 1 1237 3766 t 10 I f (s s)1 39 1 1889 3766 t 7 I f (j j)1 20 1 1939 3786 t 10 S f (- -)1 55 1 1983 3766 t 10 R f (1)2054 3766 w 10 S f (+ +)1 55 1 2120 3766 t 10 I f (p p)1 50 1 2191 3766 t 10 S f (+ +)1 55 1 2265 3766 t 10 R f (\()2336 3766 w 10 I f (q q)1 50 1 2377 3766 t 10 S f (- -)1 55 1 2451 3766 t 10 R f (1 \))1 91 1 2522 3766 t 10 S f (\264)2629 3766 w 10 R f (\()2692 3766 w 10 I f ( X)1 0(N NX)1 128 2 2733 3766 t 7 I f (j j)1 20 1 2872 3786 t 10 S f (- -)1 55 1 2916 3766 t 10 I f (k k)1 44 1 2987 3766 t 7 I f (x x)1 31 1 3042 3786 t 7 R f (,)3078 3786 w 7 I f (j j)1 20 1 3107 3786 t 10 R f (\) ,)1 74 1 3143 3766 t 10 I f (i i)1 28 1 3225 3766 t 10 R f (\) The subroutines AF and BC will only be)8 1779 1 3261 3766 t ( The)1 214(given the U coefficients for one rectangle at a time so the above notation will be irrelevant to them.)18 4106 2 720 3886 t ( is)1 99( It)1 119(use of the above data structure is shown in HANDLU and with a B-spline evaluator in Appendix 1.)17 4102 3 720 4006 t (not as strange as it looks on the surface.)8 1589 1 720 4126 t (The outer layer of the)4 864 1 970 4282 t 10 CW f (TTGU)1859 4282 w 10 R f (package is called)2 681 1 2124 4282 t 10 CW f (TTGU)2830 4282 w 10 R f (and is invoked by)3 708 1 3095 4282 t 10 CW f (Call TTGU \(U,nu,nr,kx,X,nx,ixb,ky,Y,ny,iyb,)2 2580 1 1140 4642 t (tstart,tstop,dt,)1800 4762 w (AF,BC,)1800 4882 w (errpar,)1800 5002 w (HANDLU\))1800 5122 w (The input to TTGU is)4 1200 1 720 5362 t (U)1020 5554 w 10 R f ( \(3.1\) for the initial values of the)7 1382( B-spline coefficients)2 876(- The)1 305 3 1520 5554 t 10 B f (pde)4120 5554 w 10 R f (variables)4313 5554 w 10 B f (u)4710 5554 w 10 R f (on all)1 237 1 4803 5554 t ( below for ways to get)5 925( See)1 201( description for the storage of U.)6 1342( above)1 271(rectangles. See)1 631 5 1670 5674 t (the initial conditions,)2 848 1 1670 5794 t 10 CW f (ICON)2543 5794 w 10 R f (.)2783 5794 w 10 CW f (nu)1020 5950 w 10 R f ( number)1 330(- The)1 305 2 1520 5950 t 10 I f (n n)1 50 1 2180 5950 t 7 I f (u u)1 35 1 2241 5970 t 10 R f (of)2309 5950 w 10 B f (pde)2417 5950 w 10 R f (variables)2598 5950 w 10 B f (u)2983 5950 w 10 R f (.)3039 5950 w 10 CW f (nr)1020 6106 w 10 R f ( number)1 330(- The)1 305 2 1520 6106 t 10 I f (n n)1 50 1 2180 6106 t 7 I f (r r)1 27 1 2241 6126 t 10 R f (of rectangles.)1 537 1 2301 6106 t 10 CW f (kx)1020 6262 w 10 R f ( array of length)3 648(- An)1 272 2 1520 6262 t 10 I f (n n)1 50 1 2477 6262 t 7 I f (r r)1 27 1 2538 6282 t 10 R f (where)2610 6262 w 10 CW f (kx\(j\))2890 6262 w 10 R f ( be used in)3 469(gives the B-spline order to)4 1108 2 3227 6262 t 10 I f (x x)1 44 1 4842 6262 t 10 R f (for)4924 6262 w (rectangle)1670 6382 w 10 I f (j j)1 28 1 2060 6382 t 10 R f (.)2088 6382 w 10 CW f (kx)2233 6382 w 10 S f (\263)2378 6382 w 10 R f (2 is necessary.)2 579 1 2458 6382 t 10 CW f (X)1020 6538 w 10 R f ( B-spline mesh to be used in)6 1547(- The)1 305 2 1520 6538 t 10 I f (x x)1 44 1 3466 6538 t 10 R f (. Elements)1 491 1 3510 6538 t 10 CW f (X\(IXB\(J\)\))4095 6538 w 10 R f (through)4729 6538 w 10 CW f (X\(IXB\(J\)+NX\(J\)-1\))1670 6658 w 10 R f (correspond to the mesh in)4 1084 1 2728 6658 t 10 I f (x x)1 44 1 3850 6658 t 10 R f (for rectangle)1 519 1 3932 6658 t 10 I f (j j)1 28 1 4488 6658 t 10 R f ( multi-)1 282(. The)1 242 2 4516 6658 t (plicity of)1 372 1 1670 6778 t 10 CW f (X\(IXB\(J\)\))2075 6778 w 10 R f (and)2648 6778 w 10 CW f (X\(IXB\(J\)+NX\(J\)-1\))2825 6778 w 10 R f (must be at least)3 643 1 3878 6778 t 10 CW f (kx\(j\))4590 6778 w 10 R f (for)4924 6778 w 10 I f (J J)1 44 1 1670 6898 t 10 S f (= =)1 55 1 1738 6898 t 10 R f (1 ,)1 83 1 1809 6898 t 10 I f (. .)1 25 1 1900 6898 t 10 R f (...)1925 6898 w 10 CW f (nr)2050 6898 w 10 R f ( uniform meshes,)2 739( Port Library routines for making)5 1454(. The)1 255 3 2170 6898 t 10 CW f (UMB)4667 6898 w 10 R f (and)4896 6898 w 10 CW f (LUMB)1670 7018 w 10 R f (, guarantee the first and last mesh points have multiplicity)9 2315 1 1910 7018 t 10 CW f (kx)4250 7018 w 10 R f (.)4370 7018 w cleartomark showpage saveobj restore %%EndPage: 11 11 %%Page: 12 12 /saveobj save def mark 12 pagesetup 10 R f (- 12 -)2 216 1 2772 480 t 10 CW f (nx)1020 840 w 10 R f ( array of length)3 621(- An)1 272 2 1520 840 t 10 I f (n n)1 50 1 2441 840 t 7 I f (r r)1 27 1 2502 860 t 10 R f (where)2565 840 w 10 CW f (nx\(j\))2861 840 w 10 R f (gives the length of the mesh array)6 1371 1 3189 840 t 10 CW f (X)4624 840 w 10 R f (for rect-)1 327 1 4713 840 t (angle)1670 960 w 10 I f (j j)1 28 1 1911 960 t 10 R f (beginning at)1 497 1 1964 960 t 10 CW f (ixb\(j\))2486 960 w 10 R f (.)2846 960 w 10 CW f (ixb)1020 1116 w 10 R f ( array of length)3 612(- An)1 272 2 1520 1116 t 10 I f (n n)1 50 1 2429 1116 t 7 I f (r r)1 27 1 2490 1136 t 10 R f (where)2550 1116 w 10 CW f (ixb\(j\))2818 1116 w 10 R f (points to the beginning of the)5 1180 1 3203 1116 t 10 I f (x x)1 44 1 4409 1116 t 10 R f (mesh for rect-)2 561 1 4479 1116 t (angle)1670 1236 w 10 I f (j j)1 28 1 1911 1236 t 10 R f (in the array)2 454 1 1964 1236 t 10 CW f (X)2443 1236 w 10 R f (.)2503 1236 w 10 CW f (ky)1020 1392 w 10 R f ( array of length)3 648(- An)1 272 2 1520 1392 t 10 I f (n n)1 50 1 2477 1392 t 7 I f (r r)1 27 1 2538 1412 t 10 R f (where)2610 1392 w 10 CW f (ky\(j\))2890 1392 w 10 R f ( be used in)3 469(gives the B-spline order to)4 1108 2 3227 1392 t 10 I f (y y)1 44 1 4842 1392 t 10 R f (for)4924 1392 w (rectangle)1670 1512 w 10 I f (j j)1 28 1 2060 1512 t 10 R f (.)2088 1512 w 10 CW f (ky)2233 1512 w 10 S f (\263)2378 1512 w 10 R f (2 is necessary.)2 579 1 2458 1512 t 10 CW f (Y)1020 1668 w 10 R f ( B-spline mesh to be used in)6 1547(- The)1 305 2 1520 1668 t 10 I f (y y)1 44 1 3466 1668 t 10 R f (. Elements)1 491 1 3510 1668 t 10 CW f (Y\(IYB\(J\)\))4095 1668 w 10 R f (through)4729 1668 w 10 CW f (Y\(IYB\(J\)+NY\(J\)-1\))1670 1788 w 10 R f (correspond to the mesh in)4 1084 1 2728 1788 t 10 I f (y y)1 44 1 3850 1788 t 10 R f (for rectangle)1 519 1 3932 1788 t 10 I f (j j)1 28 1 4488 1788 t 10 R f ( multi-)1 282(. The)1 242 2 4516 1788 t (plicity of)1 372 1 1670 1908 t 10 CW f (Y\(IYB\(J\)\))2075 1908 w 10 R f (and)2648 1908 w 10 CW f (Y\(IYB\(J\)+NY\(J\)-1\))2825 1908 w 10 R f (must be at least)3 643 1 3878 1908 t 10 CW f (ky\(j\))4590 1908 w 10 R f (for)4924 1908 w 10 I f (J J)1 44 1 1670 2028 t 10 S f (= =)1 55 1 1738 2028 t 10 R f (1 ,)1 83 1 1809 2028 t (. . .)2 125 1 1925 2003 t 10 CW f (nr)2075 2028 w 10 R f ( Port Library routines for making uniform meshes,)7 2174(. The)1 253 2 2195 2028 t 10 CW f (UMB)4669 2028 w 10 R f (and)4896 2028 w 10 CW f (LUMB)1670 2148 w 10 R f (, guarantee the first and last mesh points have multiplicity)9 2315 1 1910 2148 t 10 CW f (ky)4250 2148 w 10 R f (.)4370 2148 w 10 CW f (ny)1020 2304 w 10 R f ( array of length)3 621(- An)1 272 2 1520 2304 t 10 I f (n n)1 50 1 2441 2304 t 7 I f (r r)1 27 1 2502 2324 t 10 R f (where)2565 2304 w 10 CW f (ny\(j\))2861 2304 w 10 R f (gives the length of the mesh array)6 1371 1 3189 2304 t 10 CW f (Y)4624 2304 w 10 R f (for rect-)1 327 1 4713 2304 t (angle)1670 2424 w 10 I f (j j)1 28 1 1911 2424 t 10 R f (beginning at)1 497 1 1964 2424 t 10 CW f (iyb\(j\))2486 2424 w 10 R f (.)2846 2424 w 10 CW f (iyb)1020 2580 w 10 R f ( array of length)3 612(- An)1 272 2 1520 2580 t 10 I f (n n)1 50 1 2429 2580 t 7 I f (r r)1 27 1 2490 2600 t 10 R f (where)2550 2580 w 10 CW f (iyb\(j\))2818 2580 w 10 R f (points to the beginning of the)5 1180 1 3203 2580 t 10 I f (y y)1 44 1 4409 2580 t 10 R f (mesh for rect-)2 561 1 4479 2580 t (angle)1670 2700 w 10 I f (j j)1 28 1 1911 2700 t 10 R f (in the array)2 454 1 1964 2700 t 10 CW f (Y)2443 2700 w 10 R f (.)2503 2700 w 10 CW f (tstart)1020 2856 w 10 R f ( integration at time)3 758(- Start)1 339 2 1520 2856 t 10 CW f (tstart)2642 2856 w 10 R f (.)3002 2856 w 10 CW f (tstop)1020 3012 w 10 R f ( integration at time)3 779(- Stop)1 334 2 1520 3012 t 10 CW f (tstop)2665 3012 w 10 R f (.)2965 3012 w 10 CW f (tstop)3117 3012 w 10 R f (should be a variable,)3 850 1 3450 3012 t 10 I f ( t)1 0(n no ot)2 128 2 4333 3012 t 10 R f (a constant, in)2 546 1 4494 3012 t (the program calling)2 782 1 1670 3132 t 10 CW f (TTGU)2477 3132 w 10 R f (; see output description below.)4 1224 1 2717 3132 t 10 CW f (dt)1020 3288 w 10 R f ( performance of)2 636( The)1 205( initial choice for the time-step.)5 1254(- The)1 305 4 1520 3288 t 10 CW f (TTGU)3945 3288 w 10 R f (is substantially inde-)2 830 1 4210 3288 t (pendent of the initial value of)5 1209 1 1670 3408 t 10 CW f (dt)2910 3408 w 10 R f ( is sufficient that)3 687(chosen. It)1 419 2 3061 3408 t 10 CW f (dt)4198 3408 w 10 R f (be within several)2 692 1 4348 3408 t ( value of)2 389( The)1 225(orders of magnitude of being "correct.")5 1662 3 1670 3528 t 10 CW f (dt)3991 3528 w 10 R f (will automatically be)2 884 1 4156 3528 t (adjusted by)1 464 1 1670 3648 t 10 CW f (TTGU)2200 3648 w 10 R f ( least possi-)2 482(to obtain the solution to the desired accuracy at the)9 2087 2 2471 3648 t ( Thus,)1 300(ble cost.)1 358 2 1670 3768 t 10 CW f (dt)2378 3768 w 10 R f (should be a variable,)3 901 1 2548 3768 t 10 I f ( t)1 0(n no ot)2 128 2 3499 3768 t 10 R f ( calling)1 323(a constant, in the user's)4 1040 2 3677 3768 t (program.)1670 3888 w 10 CW f (AF)1020 4044 w 10 R f ( subroutine for specifying the)4 1260(- A)1 222 2 1520 4044 t 10 B f (a)3048 4044 w 10 R f (and)3144 4044 w 10 B f (f)3334 4044 w 10 R f (terms in the)2 516 1 3413 4044 t 10 B f (pde)3976 4044 w 10 R f (\(2.1\).)4179 4044 w 10 CW f (AF)4537 4044 w 10 R f (must be)1 336 1 4704 4044 t ( subprogram)1 521( user-supplied)1 583( This)1 248(declared External in the user's calling program.)6 2018 4 1670 4164 t (will be described later.)3 909 1 1670 4284 t 10 CW f (BC)1020 4440 w 10 R f ( specifying the boundary conditions)4 1524( subroutine for)2 632(- A)1 222 3 1520 4440 t 10 B f (b)3946 4440 w 10 R f (in \(2.2\).)1 342 1 4050 4440 t 10 CW f (BC)4535 4440 w 10 R f (must be)1 337 1 4703 4440 t ( subprogram)1 521( user-supplied)1 583( This)1 248(declared External in the user's calling program.)6 2018 4 1670 4560 t ( there are no)3 495( If)1 116(will be described later.)3 909 3 1670 4680 t 10 B f (bc)3216 4680 w 10 R f (s then the dummy subroutine)4 1165 1 3316 4680 t 10 CW f (TTGUP)4542 4680 w 10 R f (may)4868 4680 w (be used in place of)4 748 1 1670 4800 t 10 CW f (BC)2443 4800 w 10 R f (.)2563 4800 w 10 CW f (errpar)1020 4956 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 4956 t ( two components govern, roughly, the relative)6 1842( The)1 206(solution of the equations in time.)5 1322 3 1670 5076 t ( the)1 148( For)1 190(and absolute error in the computed solution.)6 1768 3 1670 5196 t 10 I f (i i)1 28 1 3802 5196 t 7 I f ( h)1 0(t th)1 55 2 3841 5156 t 10 R f (component of the)2 701 1 3930 5196 t 10 B f (pde)4657 5196 w 10 R f (solu-)4840 5196 w (tion)1670 5316 w 10 B f (u)1851 5316 w 10 R f (, the error at each time-step in the time integration will be at most)13 2619 1 1907 5316 t 10 CW f (errpar)1920 5472 w 10 R f (\( 1 \))2 132 1 2288 5472 t 10 I f (* *)1 50 1 2436 5472 t 10 S f ( \357)1 0( \357)1 24(\357 \357)1 49 3 2486 5489 t 10 I f (u u)1 50 1 2558 5472 t 7 I f (i i)1 20 1 2619 5492 t 10 S f ( \357)1 0( \357)1 24(\357 \357)1 49 3 2647 5489 t 7 S f (\245)2709 5492 w 10 S f (+ +)1 55 1 2808 5472 t 10 CW f (errpar)2903 5472 w 10 R f (\( 2 \))2 132 1 3271 5472 t (Thus,)1670 5628 w 10 CW f (errpar)1921 5628 w 10 R f ( solution accurate to an absolute error of)7 1624(\(1\)=0 gives the)2 607 2 2281 5628 t 10 CW f (errpar)4539 5628 w 10 R f (\(2\),)4899 5628 w (and)1670 5748 w 10 CW f (errpar)1888 5748 w 10 R f ( a relative error of)4 771(\(2\)=0 gives the solution accurate to)5 1482 2 2248 5748 t 10 CW f (errpar)4539 5748 w 10 R f (\(1\).)4899 5748 w (The choice of)2 560 1 1670 5868 t 10 CW f (errpar)2261 5868 w 10 R f (\(1\) and)1 292 1 2621 5868 t 10 CW f (errpar)2945 5868 w 10 R f ( sound)1 271( A)1 129(\(2\) is highly problem dependent.)4 1335 3 3305 5868 t ( the scale of the problem is such that)8 1499(technique is the following: If)4 1184 2 1670 5988 t 10 I f (S S)1 50 1 4383 5988 t 10 R f (is the smallest)2 577 1 4463 5988 t (value for which a prescribed relative error tolerance is desired, then the choice)12 3126 1 1670 6108 t 10 CW f (errpar)2280 6288 w 10 R f (\(1\))2640 6288 w 10 S f (= =)1 55 1 2781 6288 t 10 R f (10)2885 6288 w 7 S f (- -)1 39 1 2996 6248 t 7 R f (2)3046 6248 w 10 R f (;)3139 6288 w 10 CW f (errpar)3392 6288 w 10 R f (\(2\))3752 6288 w 10 S f (= =)1 55 1 3893 6288 t 10 R f (10)3997 6288 w 7 S f (- -)1 39 1 4108 6248 t 7 R f (2)4158 6248 w 10 S f (\264)4209 6288 w 10 R f (S)4289 6288 w ( values down to around S in size.)7 1405(will essentially give 1% relative accuracy in all)7 1965 2 1670 6468 t ( will have absolute error smaller than 10)7 1702(Values below S)2 655 2 1670 6588 t 7 S f (- -)1 39 1 4038 6548 t 7 R f (2)4088 6548 w 10 S f (\264)4139 6588 w 10 R f ( should be)2 437(S. Users)1 371 2 4232 6588 t (very careful to avoid setting)4 1132 1 1670 6708 t 10 CW f (errpar)2830 6708 w 10 R f (\(2\))3190 6708 w 10 S f (= =)1 55 1 3334 6708 t 10 R f ( the solution is zero at any point)7 1308(0 when)1 294 2 3438 6708 t (in either space or time, or the integration will die for the obvious reasons.)13 2932 1 1670 6828 t cleartomark showpage saveobj restore %%EndPage: 12 12 %%Page: 13 13 /saveobj save def mark 13 pagesetup 10 R f (- 13 -)2 216 1 2772 480 t 10 CW f (HANDLU)1020 840 w 10 R f ( be called by)3 510( user-supplied subroutine that will)4 1366(- A)1 222 3 1520 840 t 10 CW f (TTGU)3644 840 w 10 R f (at the end of each time-step.)5 1130 1 3910 840 t 10 CW f (HANDLU)1670 960 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 960 t ( no output is desired, then the dummy subroutine)8 1974( If)1 118(will be described later.)3 915 3 1670 1080 t 10 CW f (TTGUH)4740 1080 w 10 R f (may be used in place of)5 945 1 1670 1200 t 10 CW f (HANDLU)2640 1200 w 10 R f (.)3000 1200 w (The output from)2 655 1 720 1356 t 10 CW f (TTGU)1400 1356 w 10 R f (is)1665 1356 w 10 CW f (U)1020 1548 w 10 R f ( B-spline coefficients for the)4 1142(- The)1 305 2 1370 1548 t 10 B f (pde)2842 1548 w 10 R f (solution)3023 1548 w 10 B f (u)3371 1548 w 10 R f (at time)1 275 1 3452 1548 t 10 CW f (tstop)3752 1548 w 10 R f (.)4052 1548 w 10 CW f (tstop)1020 1704 w 10 R f ( time at which integration stopped.)5 1433(- The)1 305 2 1370 1704 t 10 CW f (tstop)3237 1704 w 10 R f ( the user-supplied)2 730(may be altered by)3 739 2 3571 1704 t (subroutine)1520 1824 w 10 CW f (HANDLU)1972 1824 w 10 R f ( exists on return \(see Appendix 2\),)6 1402( an error state)3 560(. If)1 146 3 2332 1824 t 10 CW f (tstop)4504 1824 w 10 R f (is set)1 207 1 4833 1824 t ( Thus,)1 295(to the last instant in time when the solution was known accurately.)11 2879 2 1520 1944 t 10 CW f (tstop)4740 1944 w 10 R f (should be a variable,)3 826 1 1520 2064 t 10 I f ( t)1 0(n no ot)2 128 2 2371 2064 t 10 R f (a constant, in the user's call to)6 1212 1 2524 2064 t 10 CW f (TTGU)3761 2064 w 10 R f (.)4001 2064 w 10 CW f (dt)1020 2220 w 10 R f ( final value of the "optimal" time-step.)6 1539(- The)1 305 2 1370 2220 t 10 B f (Static Problems)1 674 1 720 2460 t 10 R f (For static problems, where)3 1065 1 970 2616 t 10 I f (t t)1 28 1 2060 2616 t 10 R f (,)2088 2616 w 10 I f (u u)1 50 1 2138 2616 t 7 I f (t t)1 20 1 2199 2636 t 10 R f (,)2227 2616 w 10 I f (u u)1 50 1 2277 2616 t 7 I f ( t)1 0(x xt)1 51 2 2338 2636 t 10 R f (and)2422 2616 w 10 I f (u u)1 50 1 2591 2616 t 7 I f ( t)1 0(y yt)1 51 2 2652 2636 t 10 R f (do not appear in the)4 793 1 2736 2616 t 10 B f (pde)3554 2616 w 10 R f (,)3710 2616 w 10 CW f (tstart)3795 2616 w 10 R f (,)4155 2616 w 10 CW f (tstop)4205 2616 w 10 R f (and)4530 2616 w 10 CW f (dt)4699 2616 w 10 R f (must)4845 2616 w ( example,)1 388( For)1 189(be chosen consistently but they may be otherwise arbitrary.)8 2371 3 720 2736 t 10 CW f (tstart = 0)2 600 1 2160 3096 t ( 1)1 120(tstop =)1 480 2 2160 3216 t ( tstop)1 360(dt =)1 480 2 2160 3336 t 10 R f (is a fine choice.)3 626 1 720 3696 t (Choosing the initial)2 790 1 970 3852 t 10 CW f (dt)1785 3852 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 3852 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 3972 t 10 CW f (TTGU)720 4092 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 4092 t (than once.)1 410 1 720 4212 t (A "restart" in)2 549 1 970 4368 t 10 CW f (HANDLU)1553 4368 w 10 R f (, see below, for a static problem is a disaster)9 1856 1 1913 4368 t 10 S f (-)3804 4368 w 10 R f (the user should)2 625 1 3894 4368 t 10 B f (STOP)4554 4368 w 10 R f (right)4851 4368 w (there.)720 4488 w 10 CW f (TTGU)1067 4488 w 10 R f (thinks that by lowering)3 934 1 1335 4488 t 10 CW f (dt)2297 4488 w 10 R f ( make the problem easier, a correct assumption if the prob-)10 2373(it can)1 222 2 2445 4488 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 4608 t (Newton's method cannot converge from the initial conditions, or the initial guess in this case.)14 3741 1 720 4728 t 10 B f (Scratch Space Used.)2 863 1 720 4968 t 10 R f ( of scratch space used on the dynamic stack of the Port Library [9] is, neglecting lower order)17 3833(The amount)1 487 2 720 5124 t (terms, when the default setting are used,)6 1608 1 720 5244 t 10 I f (n n)1 50 1 796 5488 t 7 I f (u u)1 35 1 857 5508 t 10 R f (\()908 5488 w 7 I f (i i)1 20 1 1025 5558 t 10 R f (max \()1 213 1 949 5488 t 10 I f (n n)1 50 1 1170 5488 t 7 I f (x x)1 31 1 1231 5508 t 7 R f (,)1267 5508 w 7 I f (i i)1 20 1 1290 5508 t 10 I f (n n)1 50 1 1326 5488 t 7 I f (y y)1 31 1 1387 5508 t 7 R f (,)1423 5508 w 7 I f (i i)1 20 1 1446 5508 t 10 R f (\( 3)1 91 1 1482 5488 t 10 I f (H H)1 72 1 1581 5488 t 7 I f (i i)1 20 1 1664 5508 t 10 S f (- -)1 55 1 1708 5488 t 10 R f (1 \) \) \))3 173 1 1779 5488 t 10 S f (+ +)1 55 1 1968 5488 t 10 R f (2)2039 5488 w 10 I f (n n)1 50 1 2097 5488 t 7 I f (u u)1 35 1 2158 5508 t 7 R f (2)2206 5448 w 10 R f (\()2257 5488 w 7 I f (i i)1 20 1 2298 5588 t 7 S f (= =)1 39 1 2334 5588 t 7 R f (1)2384 5588 w 15 S f (S)2314 5518 w 7 I f (n n)1 35 1 2328 5374 t 4 I f (r r)1 16 1 2369 5388 t 10 I f (n n)1 50 1 2419 5488 t 7 I f (x x)1 31 1 2480 5508 t 7 R f (,)2516 5508 w 7 I f (i i)1 20 1 2539 5508 t 10 I f (n n)1 50 1 2575 5488 t 7 I f (y y)1 31 1 2636 5508 t 7 R f (,)2672 5508 w 7 I f (i i)1 20 1 2695 5508 t 10 R f (\()2731 5488 w 10 I f (n n)1 50 1 2772 5488 t 7 I f (x x)1 31 1 2833 5508 t 7 R f (,)2869 5508 w 7 I f (i i)1 20 1 2892 5508 t 10 S f (- -)1 55 1 2936 5488 t 10 I f (k k)1 44 1 3007 5488 t 7 I f (x x)1 31 1 3062 5508 t 7 R f (,)3098 5508 w 7 I f (i i)1 20 1 3121 5508 t 10 S f (+ +)1 55 1 3165 5488 t 10 I f (n n)1 50 1 3236 5488 t 7 I f (y y)1 31 1 3297 5508 t 7 R f (,)3333 5508 w 7 I f (i i)1 20 1 3356 5508 t 10 S f (- -)1 55 1 3400 5488 t 10 I f (k k)1 44 1 3471 5488 t 7 I f (y y)1 31 1 3526 5508 t 7 R f (,)3562 5508 w 7 I f (i i)1 20 1 3585 5508 t 10 R f (\))3621 5488 w 10 S f (+ +)1 55 1 3670 5488 t 10 R f (\()3741 5488 w 7 I f (i i)1 20 1 3782 5588 t 7 S f (= =)1 39 1 3818 5588 t 7 R f (1)3868 5588 w 15 S f (S)3798 5518 w 7 I f (n n)1 35 1 3812 5374 t 4 I f (r r)1 16 1 3853 5388 t 10 R f (\()3911 5488 w 10 I f (n n)1 50 1 3952 5488 t 7 I f (x x)1 31 1 4013 5508 t 7 R f (,)4049 5508 w 7 I f (i i)1 20 1 4072 5508 t 10 S f (- -)1 55 1 4116 5488 t 10 I f (k k)1 44 1 4187 5488 t 7 I f (x x)1 31 1 4242 5508 t 7 R f (,)4278 5508 w 7 I f (i i)1 20 1 4301 5508 t 10 S f (+ +)1 55 1 4345 5488 t 10 I f (n n)1 50 1 4416 5488 t 7 I f (y y)1 31 1 4477 5508 t 7 R f (,)4513 5508 w 7 I f (i i)1 20 1 4536 5508 t 10 S f (- -)1 55 1 4580 5488 t 10 I f (k k)1 44 1 4651 5488 t 7 I f (y y)1 31 1 4706 5508 t 7 R f (,)4742 5508 w 7 I f (i i)1 20 1 4765 5508 t 10 R f (\) \))1 74 1 4801 5488 t 7 R f (2)4880 5448 w 10 R f (\))4931 5488 w (Real words \(storage units\), where)4 1392 1 720 5784 t 10 I f (H H)1 72 1 2149 5784 t 7 I f (i i)1 20 1 2232 5804 t 10 S f (\272)2301 5784 w 10 I f (n n)1 50 1 2397 5784 t 7 I f (u u)1 35 1 2458 5804 t 10 R f (\()2533 5784 w 10 I f (k k)1 44 1 2598 5784 t 7 I f (x x)1 31 1 2653 5804 t 7 R f (,)2689 5804 w 7 I f (i i)1 20 1 2712 5804 t 10 S f (+ +)1 55 1 2780 5784 t 10 R f (\()2875 5784 w 10 I f (n n)1 50 1 2940 5784 t 7 I f (x x)1 31 1 3001 5804 t 7 R f (,)3037 5804 w 7 I f (i i)1 20 1 3060 5804 t 10 S f (- -)1 55 1 3128 5784 t 10 I f (k k)1 44 1 3223 5784 t 7 I f (x x)1 31 1 3278 5804 t 7 R f (,)3314 5804 w 7 I f (i i)1 20 1 3337 5804 t 10 R f (\) \()1 106 1 3397 5784 t 10 I f (k k)1 44 1 3511 5784 t 7 I f (y y)1 31 1 3566 5804 t 7 R f (,)3602 5804 w 7 I f (i i)1 20 1 3625 5804 t 10 S f (- -)1 55 1 3693 5784 t 10 R f ( of)1 121( is the half-band-width)3 943( \))1 73(1 \))1 115 4 3788 5784 t (the Jacobian, on the)3 799 1 720 5904 t 10 I f (i i)1 28 1 1546 5904 t 7 I f ( h)1 0(t th)1 55 2 1585 5864 t 10 R f (rectangle,)1675 5904 w 10 I f (n n)1 50 1 2092 5904 t 7 I f (u u)1 35 1 2153 5924 t 10 R f (is the number of)3 658 1 2223 5904 t 10 B f (pde)2908 5904 w 10 R f (s,)3064 5904 w 10 I f (n n)1 50 1 3155 5904 t 7 I f (x x)1 31 1 3216 5924 t 7 R f (,)3252 5924 w 7 I f (i i)1 20 1 3275 5924 t 10 R f (is the number of points in the spatial mesh)8 1710 1 3330 5904 t (for)720 6024 w 10 I f (x x)1 44 1 863 6024 t 10 R f (,)907 6024 w 10 I f (k k)1 44 1 959 6024 t 7 I f (x x)1 31 1 1014 6044 t 7 R f (,)1050 6044 w 7 I f (i i)1 20 1 1073 6044 t 10 R f (is the B-spline order for the mesh)6 1349 1 1128 6024 t 10 I f (x x)1 44 1 2504 6024 t 10 R f (,)2548 6024 w 10 I f (n n)1 50 1 2600 6024 t 7 I f (y y)1 31 1 2661 6044 t 7 R f (,)2697 6044 w 7 I f (i i)1 20 1 2720 6044 t 10 R f (is the number of points in the spatial mesh for)9 1853 1 2775 6024 t 10 I f (y y)1 44 1 4655 6024 t 10 R f (and)4726 6024 w 10 I f (k k)1 44 1 4898 6024 t 7 I f (y y)1 31 1 4953 6044 t 7 R f (,)4989 6044 w 7 I f (i i)1 20 1 5012 6044 t 10 R f (is the B-spline order for the mesh in)7 1440 1 720 6144 t 10 I f (y y)1 44 1 2185 6144 t 10 R f (for the)1 263 1 2254 6144 t 10 I f (i i)1 28 1 2542 6144 t 7 I f ( h)1 0(t th)1 55 2 2581 6104 t 10 R f (rectangle.)2669 6144 w (All scratch space for)3 840 1 970 6300 t 10 CW f (TTGU)1841 6300 w 10 R f ( Solving)1 368(is taken from the stack.)4 953 2 2112 6300 t 10 B f (pde)3464 6300 w 10 R f ( non-trivial process, requiring)3 1208(s is a)2 212 2 3620 6300 t ( virtually all problems solved by)5 1307( For)1 191(substantial work space.)2 933 3 720 6420 t 10 CW f (TTGU)3178 6420 w 10 R f ( and initial-)2 463(the user will have to declare)5 1132 2 3445 6420 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 6540 t (trated in the first example of Appendix 1.)7 1653 1 720 6660 t cleartomark showpage saveobj restore %%EndPage: 13 13 %%Page: 14 14 /saveobj save def mark 14 pagesetup 10 R f (- 14 -)2 216 1 2772 480 t 10 B f (Run-time.)720 840 w 10 R f (The run-time of)2 632 1 970 1060 t 10 CW f (TTGU)1627 1060 w 10 R f (is proportional to)2 691 1 1893 1060 t 10 I f (n n)1 50 1 2610 1060 t 7 I f (u u)1 35 1 2671 1080 t (i i)1 20 1 2722 1160 t 7 S f (= =)1 39 1 2758 1160 t 7 R f (1)2808 1160 w 15 S f (S)2738 1090 w 7 I f (n n)1 35 1 2752 946 t 4 I f (r r)1 16 1 2793 960 t 10 I f (n n)1 50 1 2875 1060 t 7 I f (x x)1 31 1 2936 1080 t 7 R f (,)2972 1080 w 7 I f (i i)1 20 1 2995 1080 t 10 I f (n n)1 50 1 3055 1060 t 7 I f (y y)1 31 1 3116 1080 t 7 R f (,)3152 1080 w 7 I f (i i)1 20 1 3175 1080 t 10 R f (\()3235 1060 w 10 I f (H H)1 72 1 3300 1060 t 7 I f (i i)1 20 1 3383 1080 t 10 S f (- -)1 55 1 3451 1060 t 10 R f (1 \))1 115 1 3546 1060 t 7 R f (2)3666 1020 w 10 R f ( sec-)1 186( See)1 195(for the default settings.)3 924 3 3735 1060 t (tion 6 for ways to make)5 946 1 720 1260 t 10 CW f (TTGU)1691 1260 w 10 R f (run much faster.)2 651 1 1956 1260 t (The storage and run-time of)4 1134 1 970 1416 t 10 CW f (TTGU)2135 1416 w 10 R f (are far from optimal with the default settings, being on the order)11 2634 1 2406 1416 t (of)720 1536 w 10 I f (n n)1 50 1 833 1536 t 7 R f (3)894 1496 w 10 R f (and)967 1536 w 10 I f (n n)1 50 1 1141 1536 t 7 R f (4)1202 1496 w 10 R f (, respectively, for an)3 829 1 1245 1536 t 10 I f (n n)1 50 1 2103 1536 t 10 R f (by)2182 1536 w 10 I f (n n)1 50 1 2311 1536 t 10 R f ( space and time would be)5 1032(grid. Optimal)1 568 2 2390 1536 t 10 I f (O O)1 72 1 4019 1536 t 10 R f (\()4099 1536 w 10 I f (n n)1 50 1 4164 1536 t 7 R f (2)4225 1496 w 10 R f ( reasons for)2 473(\). The)1 267 2 4300 1536 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 1656 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 1776 t (space.)720 1896 w 10 B f (Double Precision Version.)2 1108 1 720 2136 t 10 R f (The Double Precision version of)4 1314 1 720 2292 t 10 CW f (TTGU)2063 2292 w 10 R f (is called)1 334 1 2332 2292 t 10 CW f (DTTGU)2695 2292 w 10 R f ( sequence for)2 541( calling)1 301(. The)1 234 3 2995 2292 t 10 CW f (DTTGU)4101 2292 w 10 R f (is precisely the)2 609 1 4431 2292 t (same as that for)3 704 1 720 2412 t 10 CW f (TTGU)1474 2412 w 10 R f (, with)1 253 1 1714 2412 t 10 I f ( ll l)2 28(a al)1 78 2 2016 2412 t 10 R f (floating-point arguments Double Precision,)3 1804 1 2171 2412 t 10 I f ( t)1 0( pt)1 28(e ex xc ce ep)4 226 3 4024 2412 t 10 CW f (errpar)4362 2412 w 10 R f (, which)1 318 1 4722 2412 t (remains Real.)1 549 1 720 2532 t 10 CW f (AF)720 2772 w 10 B f (and)865 2772 w 10 CW f (BC)1052 2772 w 10 B f (Descriptions.)1197 2772 w 10 R f (The user-supplied subroutines)2 1218 1 970 2928 t 10 CW f (AF)2220 2928 w 10 R f (and)2372 2928 w 10 CW f (BC)2548 2928 w 10 R f (, which define the)3 736 1 2668 2928 t 10 B f (pde)3436 2928 w 10 R f (-)3625 2928 w 10 B f (bc)3658 2928 w 10 R f (problem to be solved, are now)5 1249 1 3791 2928 t ( a distinct invocation for each rectangle .i.e., one is asked for information for only one)15 3525( is)1 99(described. There)1 696 3 720 3048 t ( subroutines)1 489( The)1 208(rectangle at a time.)3 765 3 720 3168 t 10 CW f (AF)2245 3168 w 10 R f (and)2393 3168 w 10 CW f (BC)2565 3168 w 10 R f (have the same calling sequence as they do for TTGR[15].)9 2327 1 2713 3168 t (When)720 3288 w 10 CW f (TTGU)983 3288 w 10 R f (needs to compute)2 699 1 1248 3288 t 10 B f (a)1972 3288 w 10 R f (and)2047 3288 w 10 B f (f)2216 3288 w 10 R f (, it will)2 287 1 2249 3288 t 10 CW f (Call AF\(t,Xe,Ye,nxe,nye,Nu, U,Ut,Ux,Uy,Uyt,Uxt,)2 2820 1 720 3648 t (A,AU,AUt,AUx,AUy,AUxt,AUyt,)1200 3768 w (F,FU,FUt,FUx,FUy,FUxt,FUyt\))1200 3888 w 10 R f (Before)720 4248 w 10 CW f (TTGU)1016 4248 w 10 R f (calls)1281 4248 w 10 CW f (AF)1489 4248 w 10 R f (, it sets to)3 384 1 1609 4248 t 10 B f (0)2018 4248 w 10 R f (the 14 arrays)2 515 1 2093 4248 t 10 CW f (A)2668 4248 w 10 R f (through)2753 4248 w 10 CW f (FUyt)3089 4248 w 10 R f (and provides the)2 660 1 3354 4248 t 10 I f ( t)1 0( np pu ut)3 128(i in)1 78 3 4039 4248 t 10 R f (values)4270 4248 w 10 CW f (t)1020 4440 w 10 R f ( current value of time.)4 884(- The)1 305 2 1520 4440 t 10 CW f (Xe)1020 4596 w 10 R f ( list of points)3 559(- A)1 222 2 1520 4596 t 10 I f (x x)1 44 1 2337 4596 t 10 R f (where)2417 4596 w 10 B f (a)2696 4596 w 10 R f (and)2782 4596 w 10 B f (f)2962 4596 w 10 R f ( This)1 240(are to be evaluated.)3 808 2 3031 4596 t 10 CW f (Xe)4116 4596 w 10 R f (is)4273 4596 w 10 I f ( t)1 0(n no ot)2 128 2 4377 4596 t 10 R f (the B-spline)1 498 1 4542 4596 t ( points)1 297( The)1 232(mesh X.)1 360 3 1670 4716 t 10 CW f (Xe)2611 4716 w 10 R f (at which)1 368 1 2783 4716 t 10 B f (a)3203 4716 w 10 R f (and)3305 4716 w 10 B f (f)3500 4716 w 10 R f (are desired are determined by the)5 1456 1 3584 4716 t (quadrature rule used by)3 939 1 1670 4836 t 10 CW f (TTGU)2634 4836 w 10 R f (to implement Galerkin's method.)3 1327 1 2899 4836 t 10 CW f (nxe)1020 4992 w 10 R f ( length of Xe.)3 549(- The)1 305 2 1520 4992 t 10 CW f (Ye)1020 5148 w 10 R f ( list of points)3 559(- A)1 222 2 1520 5148 t 10 I f (y y)1 44 1 2337 5148 t 10 R f (where)2417 5148 w 10 B f (a)2696 5148 w 10 R f (and)2782 5148 w 10 B f (f)2962 5148 w 10 R f ( This)1 240(are to be evaluated.)3 808 2 3031 5148 t 10 CW f (Ye)4116 5148 w 10 R f (is)4273 5148 w 10 I f ( t)1 0(n no ot)2 128 2 4377 5148 t 10 R f (the B-spline)1 498 1 4542 5148 t ( points)1 297( The)1 232(mesh Y.)1 360 3 1670 5268 t 10 CW f (Ye)2611 5268 w 10 R f (at which)1 368 1 2783 5268 t 10 B f (a)3203 5268 w 10 R f (and)3305 5268 w 10 B f (f)3500 5268 w 10 R f (are desired are determined by the)5 1456 1 3584 5268 t (quadrature rule used by)3 939 1 1670 5388 t 10 CW f (TTGU)2634 5388 w 10 R f (to implement Galerkin's method.)3 1327 1 2899 5388 t 10 CW f (nye)1020 5544 w 10 R f ( length of Ye.)3 549(- The)1 305 2 1520 5544 t 10 CW f (nu)1020 5700 w 10 R f ( number)1 330(- The)1 305 2 1520 5700 t 10 I f (n n)1 50 1 2180 5700 t 7 I f (u u)1 35 1 2241 5720 t 10 R f (of)2309 5700 w 10 B f (pde)2417 5700 w 10 R f (variables)2598 5700 w 10 B f (u)2983 5700 w 10 R f (.)3039 5700 w 10 CW f (U)1020 5856 w 10 R f (- The)1 305 1 1520 5856 t 10 I f ( s)1 0( es)1 39( ue)1 44( lu)1 50( al)1 28(v va)1 94 6 1861 5856 t 10 R f (of)2152 5856 w 10 B f (u)2271 5856 w 10 R f (at the)1 230 1 2363 5856 t 10 CW f (Xe)2630 5856 w 10 R f (\()2750 5856 w 10 I f (p p)1 50 1 2783 5856 t 10 R f (\) and)1 214 1 2833 5856 t 10 CW f (Ye)3084 5856 w 10 R f (\()3204 5856 w 10 I f (q q)1 50 1 3237 5856 t 10 R f (\),)3287 5856 w 10 B f (not)3382 5856 w 10 R f (the B-spline coefficients)2 1000 1 3558 5856 t 10 B f (U)4595 5856 w 10 R f (of \(3.1\).)1 336 1 4704 5856 t 10 CW f (U)1670 5976 w 10 R f (\()1730 5976 w 10 I f (p p)1 50 1 1763 5976 t 10 R f (,)1821 5976 w 10 I f (q q)1 50 1 1854 5976 t 10 R f (,)1912 5976 w 10 I f (j j)1 28 1 1953 5976 t 10 R f (\) =)1 114 1 1981 5976 t 10 I f (u u)1 50 1 2120 5976 t 7 I f (j j)1 20 1 2181 5996 t 10 R f (\(t,)2209 5976 w 10 CW f (Xe)2295 5976 w 10 R f (\()2415 5976 w 10 I f (p p)1 50 1 2448 5976 t 10 R f (\),)2498 5976 w 10 CW f (Ye)2556 5976 w 10 R f (\()2676 5976 w 10 I f (q q)1 50 1 2709 5976 t 10 R f (\)\),)2759 5976 w 10 I f (p p)1 50 1 2875 5976 t 10 S f (= =)1 55 1 2949 5976 t 10 R f (1 ,)1 83 1 3020 5976 t (. . .)2 125 1 3136 5951 t (,)3294 5976 w 10 CW f (nxe)3343 5976 w 10 R f (,)3523 5976 w 10 I f (q q)1 50 1 3573 5976 t 10 S f (= =)1 55 1 3647 5976 t 10 R f (1 ,)1 83 1 3718 5976 t (. . .)2 125 1 3834 5951 t (,)3992 5976 w 10 CW f (nye)4041 5976 w 10 R f (and)4246 5976 w 10 I f (j j)1 28 1 4415 5976 t 10 S f (= =)1 55 1 4459 5976 t 10 R f (1 ,)1 83 1 4530 5976 t (. . .)2 125 1 4646 5951 t (,)4804 5976 w 10 CW f (nu)4853 5976 w 10 R f (.)4973 5976 w 10 CW f (Ut)1020 6132 w 10 R f ( values of)2 388(- The)1 305 2 1520 6132 t 10 B f (u)2238 6132 w 7 I f (t t)1 20 1 2305 6152 t 10 R f (at the)1 219 1 2358 6132 t 10 CW f (Xe)2602 6132 w 10 R f (\()2722 6132 w 10 I f (p p)1 50 1 2755 6132 t 10 R f (\) and)1 202 1 2805 6132 t 10 CW f (Ye)3032 6132 w 10 R f (\()3152 6132 w 10 I f (q q)1 50 1 3185 6132 t 10 R f (\), stored as above.)3 723 1 3235 6132 t 10 CW f (Ux)1020 6288 w 10 R f ( values of)2 388(- The)1 305 2 1520 6288 t 10 B f (u)2238 6288 w 7 I f (x x)1 31 1 2305 6308 t 10 R f (, stored as above.)3 690 1 2344 6288 t 10 CW f (Uy)1020 6444 w 10 R f ( values of)2 388(- The)1 305 2 1520 6444 t 10 B f (u)2238 6444 w 7 I f (y y)1 31 1 2305 6464 t 10 R f (, stored as above.)3 690 1 2344 6444 t 10 CW f (Uxt)1020 6600 w 10 R f ( values of)2 388(- The)1 305 2 1520 6600 t 10 B f (u)2238 6600 w 7 I f ( t)1 0(x xt)1 51 2 2305 6620 t 10 R f (, stored as above.)3 690 1 2364 6600 t 10 CW f (Uyt)1020 6756 w 10 R f ( values of)2 388(- The)1 305 2 1520 6756 t 10 B f (u)2238 6756 w 7 I f ( t)1 0(y yt)1 51 2 2305 6776 t 10 R f (, stored as above.)3 690 1 2364 6756 t 10 CW f (AF)720 6912 w 10 R f (must return as)2 566 1 865 6912 t 10 I f ( t)1 0( pu ut)2 78( tp)1 50(o ou ut)2 128 4 1456 6912 t cleartomark showpage saveobj restore %%EndPage: 14 14 %%Page: 15 15 /saveobj save def mark 15 pagesetup 10 R f (- 15 -)2 216 1 2772 480 t 10 CW f (A)1020 840 w 10 R f ( value of)2 407(- The)1 305 2 1520 840 t 10 B f (a)2286 840 w 10 R f (at the)1 248 1 2390 840 t 10 CW f (Xe)2693 840 w 10 R f (\()2813 840 w 10 I f (p p)1 50 1 2846 840 t 10 R f (\) and)1 232 1 2896 840 t 10 CW f (Xe)3218 840 w 10 R f (\()3338 840 w 10 I f (q q)1 50 1 3371 840 t 10 R f (\).)3421 840 w 10 CW f (A)3629 840 w 10 R f (\()3689 840 w 10 I f (p p)1 50 1 3722 840 t 10 R f (,)3780 840 w 10 I f (q q)1 50 1 3813 840 t 10 R f (,)3871 840 w 10 I f (j j)1 28 1 3912 840 t 10 R f (\) =)1 144 1 3940 840 t 10 I f (a a)1 50 1 4139 840 t 7 I f (j j)1 20 1 4200 860 t 10 R f (\(t,)4228 840 w 10 CW f (Xe)4314 840 w 10 R f (\()4434 840 w 10 I f (p p)1 50 1 4467 840 t 10 R f (\),)4517 840 w 10 CW f (Ye)4575 840 w 10 R f (\()4695 840 w 10 I f (q q)1 50 1 4728 840 t 10 R f (\)\), for)1 262 1 4778 840 t 10 I f (p p)1 50 1 1670 960 t 10 S f (= =)1 55 1 1744 960 t 10 R f (1 ,)1 83 1 1815 960 t (. . .)2 125 1 1931 935 t (,)2089 960 w 10 CW f (nxe)2138 960 w 10 R f (,)2318 960 w 10 I f (q q)1 50 1 2368 960 t 10 S f (= =)1 55 1 2442 960 t 10 R f (1 ,)1 83 1 2513 960 t 10 I f (. .)1 25 1 2604 960 t 10 R f (.. ,)1 83 1 2629 960 t 10 CW f (nye)2736 960 w 10 R f (and)2941 960 w 10 I f (j j)1 28 1 3110 960 t 10 S f (= =)1 55 1 3154 960 t 10 R f (1 ,)1 83 1 3225 960 t (. . .)2 125 1 3341 935 t (,)3499 960 w 10 CW f (nu)3548 960 w 10 R f (.)3668 960 w 10 CW f (AU)1020 1116 w 10 R f ( partial derivatives of)3 860(- The)1 305 2 1520 1116 t 10 B f (a)2713 1116 w 10 R f (with respect to)2 594 1 2791 1116 t 10 B f (u)3413 1116 w 10 R f (at the)1 222 1 3497 1116 t 10 CW f (Xe)3747 1116 w 10 R f (\()3867 1116 w 10 I f (p p)1 50 1 3900 1116 t 10 R f (\) and)1 205 1 3950 1116 t 10 CW f (Ye)4183 1116 w 10 R f (\()4303 1116 w 10 I f (q q)1 50 1 4336 1116 t 10 R f (\).)4386 1116 w 10 CW f (AU)4567 1116 w 10 R f (\()4687 1116 w 10 I f (p p)1 50 1 4720 1116 t 10 R f (,)4778 1116 w 10 I f (q q)1 50 1 4811 1116 t 10 R f (,)4869 1116 w 10 I f (i i)1 28 1 4902 1116 t 10 R f (,)4938 1116 w 10 I f (j j)1 28 1 4979 1116 t 10 R f (\))5007 1116 w (=)1670 1236 w 10 S f (\266)1849 1236 w 10 I f (a a)1 50 1 1906 1236 t 7 I f (i i)1 20 1 1967 1256 t 10 I f (/ /)1 28 1 2003 1236 t 10 S f (\266)2039 1236 w 10 I f (u u)1 50 1 2096 1236 t 7 I f (j j)1 20 1 2157 1256 t 10 R f (\(t,)2308 1236 w 10 CW f (Xe)2516 1236 w 10 R f (\()2636 1236 w 10 I f (p p)1 50 1 2669 1236 t 10 R f (\),)2719 1236 w 10 CW f (Ye)2777 1236 w 10 R f (\()2897 1236 w 10 I f (q q)1 50 1 2930 1236 t 10 R f (\)\), for)1 329 1 2980 1236 t 10 I f (p p)1 50 1 3431 1236 t 10 S f (= =)1 55 1 3505 1236 t 10 R f (1 ,)1 83 1 3576 1236 t (. . .)2 125 1 3692 1211 t (,)3850 1236 w 10 CW f (nxe)3899 1236 w 10 R f (,)4079 1236 w 10 I f (q q)1 50 1 4226 1236 t 10 S f (= =)1 55 1 4300 1236 t 10 R f (1 ,)1 83 1 4371 1236 t 10 I f (. .)1 25 1 4462 1236 t 10 R f (.. ,)1 83 1 4487 1236 t 10 CW f (nye)4594 1236 w 10 R f (and)4896 1236 w 10 I f (i i)1 28 1 1670 1356 t 10 R f (,)1706 1356 w 10 I f (j j)1 28 1 1747 1356 t 10 S f (= =)1 55 1 1791 1356 t 10 R f (1 ,)1 83 1 1862 1356 t (. . .)2 125 1 1978 1331 t (,)2136 1356 w 10 CW f (nu)2185 1356 w 10 R f (.)2305 1356 w 10 CW f (AUt)1020 1512 w 10 R f ( partial derivatives of)3 851(- The)1 305 2 1520 1512 t 10 B f (a)2701 1512 w 10 R f (with respect to)2 588 1 2776 1512 t 10 B f (u)3389 1512 w 7 I f (t t)1 20 1 3456 1532 t 10 R f (, as above.)2 421 1 3509 1512 t 10 CW f (AUx)1020 1668 w 10 R f ( partial derivatives of)3 851(- The)1 305 2 1520 1668 t 10 B f (a)2701 1668 w 10 R f (with respect to)2 588 1 2776 1668 t 10 B f (u)3389 1668 w 7 I f (x x)1 31 1 3456 1688 t 10 R f (, as above.)2 421 1 3495 1668 t 10 CW f (AUy)1020 1824 w 10 R f ( partial derivatives of)3 851(- The)1 305 2 1520 1824 t 10 B f (a)2701 1824 w 10 R f (with respect to)2 588 1 2776 1824 t 10 B f (u)3389 1824 w 7 I f (y y)1 31 1 3456 1844 t 10 R f (, as above.)2 421 1 3495 1824 t 10 CW f (AUxt)1020 1980 w 10 R f ( partial derivatives of)3 851(- The)1 305 2 1520 1980 t 10 B f (a)2701 1980 w 10 R f (with respect to)2 588 1 2776 1980 t 10 B f (u)3389 1980 w 7 I f ( t)1 0(x xt)1 51 2 3456 2000 t 10 R f (, as above.)2 421 1 3515 1980 t 10 CW f (AUyt)1020 2136 w 10 R f ( partial derivatives of)3 851(- The)1 305 2 1520 2136 t 10 B f (a)2701 2136 w 10 R f (with respect to)2 588 1 2776 2136 t 10 B f (u)3389 2136 w 7 I f ( t)1 0(y yt)1 51 2 3456 2156 t 10 R f (, as above.)2 421 1 3515 2136 t 10 CW f (F)1020 2292 w 10 R f ( value of)2 413(- The)1 305 2 1520 2292 t 10 B f (f)2296 2292 w 10 R f (at the)1 252 1 2387 2292 t 10 CW f (Xe)2697 2292 w 10 R f (\()2817 2292 w 10 I f (p p)1 50 1 2850 2292 t 10 R f (\) and)1 235 1 2900 2292 t 10 CW f (Ye)3228 2292 w 10 R f (\()3348 2292 w 10 I f (q q)1 50 1 3381 2292 t 10 R f (\).)3431 2292 w 10 CW f (F)3642 2292 w 10 R f (\()3702 2292 w 10 I f (p p)1 50 1 3735 2292 t 10 R f (,)3793 2292 w 10 I f (q q)1 50 1 3826 2292 t 10 R f (,)3884 2292 w 10 I f (j j)1 28 1 3925 2292 t 10 R f (\) =)1 147 1 3953 2292 t 10 I f (f f)1 28 1 4158 2292 t 7 I f (j j)1 20 1 4197 2312 t 10 R f (\(t,)4225 2292 w 10 CW f (Xe)4311 2292 w 10 R f (\()4431 2292 w 10 I f (p p)1 50 1 4464 2292 t 10 R f (\),)4514 2292 w 10 CW f (Ye)4572 2292 w 10 R f (\()4692 2292 w 10 I f (q q)1 50 1 4725 2292 t 10 R f (\)\), for)1 265 1 4775 2292 t 10 I f (p p)1 50 1 1670 2412 t 10 S f (= =)1 55 1 1744 2412 t 10 R f (1 ,)1 83 1 1815 2412 t (. . .)2 125 1 1931 2387 t (,)2089 2412 w 10 CW f (nxe)2138 2412 w 10 R f (,)2318 2412 w 10 I f (q q)1 50 1 2368 2412 t 10 S f (= =)1 55 1 2442 2412 t 10 R f (1 ,)1 83 1 2513 2412 t 10 I f (. .)1 25 1 2604 2412 t 10 R f (.. ,)1 83 1 2629 2412 t 10 CW f (nye)2736 2412 w 10 R f (and)2941 2412 w 10 I f (j j)1 28 1 3110 2412 t 10 S f (= =)1 55 1 3154 2412 t 10 R f (1 ,)1 83 1 3225 2412 t (. . .)2 125 1 3341 2387 t (,)3499 2412 w 10 CW f (nu)3548 2412 w 10 R f (.)3668 2412 w 10 CW f (FU)1020 2568 w 10 R f ( partial derivatives of)3 863(- The)1 305 2 1520 2568 t 10 B f (f)2717 2568 w 10 R f (with respect to)2 596 1 2779 2568 t 10 B f (u)3404 2568 w 10 R f (at the)1 223 1 3489 2568 t 10 CW f (Xe)3741 2568 w 10 R f (\()3861 2568 w 10 I f (p p)1 50 1 3894 2568 t 10 R f (\) and)1 207 1 3944 2568 t 10 CW f (Ye)4181 2568 w 10 R f (\()4301 2568 w 10 I f (q q)1 50 1 4334 2568 t 10 R f (\).)4384 2568 w 10 CW f (FU)4567 2568 w 10 R f (\()4687 2568 w 10 I f (p p)1 50 1 4720 2568 t 10 R f (,)4778 2568 w 10 I f (q q)1 50 1 4811 2568 t 10 R f (,)4869 2568 w 10 I f (i i)1 28 1 4902 2568 t 10 R f (,)4938 2568 w 10 I f (j j)1 28 1 4979 2568 t 10 R f (\))5007 2568 w (=)1670 2688 w 10 S f (\266)1847 2688 w 10 I f (f f)1 28 1 1912 2688 t 7 I f (i i)1 20 1 1951 2708 t 10 I f (/ /)1 28 1 2011 2688 t 10 S f (\266)2047 2688 w 10 I f (u u)1 50 1 2104 2688 t 7 I f (j j)1 20 1 2165 2708 t 10 R f (\(t,)2314 2688 w 10 CW f (Xe)2521 2688 w 10 R f (\()2641 2688 w 10 I f (p p)1 50 1 2674 2688 t 10 R f (\),)2724 2688 w 10 CW f (Ye)2782 2688 w 10 R f (\()2902 2688 w 10 I f (q q)1 50 1 2935 2688 t 10 R f (\)\), for)1 328 1 2985 2688 t 10 I f (p p)1 50 1 3434 2688 t 10 S f (= =)1 55 1 3508 2688 t 10 R f (1 ,)1 83 1 3579 2688 t (. . .)2 125 1 3695 2663 t (,)3853 2688 w 10 CW f (nxe)3902 2688 w 10 R f (,)4082 2688 w 10 I f (q q)1 50 1 4228 2688 t 10 S f (= =)1 55 1 4302 2688 t 10 R f (1 ,)1 83 1 4373 2688 t 10 I f (. .)1 25 1 4464 2688 t 10 R f (.. ,)1 83 1 4489 2688 t 10 CW f (nye)4596 2688 w 10 R f (and)4896 2688 w 10 I f (i i)1 28 1 1670 2808 t 10 R f (,)1706 2808 w 10 I f (j j)1 28 1 1747 2808 t 10 S f (= =)1 55 1 1791 2808 t 10 R f (1 ,)1 83 1 1862 2808 t (. . .)2 125 1 1978 2783 t (,)2136 2808 w 10 CW f (nu)2185 2808 w 10 R f (.)2305 2808 w 10 CW f (FUt)1020 2964 w 10 R f ( partial derivatives of)3 851(- The)1 305 2 1520 2964 t 10 B f (f)2701 2964 w 10 R f (with respect to)2 588 1 2759 2964 t 10 B f (u)3372 2964 w 7 I f (t t)1 20 1 3439 2984 t 10 R f (, as above.)2 421 1 3492 2964 t 10 CW f (FUx)1020 3120 w 10 R f ( partial derivatives of)3 851(- The)1 305 2 1520 3120 t 10 B f (f)2701 3120 w 10 R f (with respect to)2 588 1 2759 3120 t 10 B f (u)3372 3120 w 7 I f (x x)1 31 1 3439 3140 t 10 R f (, as above.)2 421 1 3478 3120 t 10 CW f (FUy)1020 3276 w 10 R f ( partial derivatives of)3 851(- The)1 305 2 1520 3276 t 10 B f (f)2701 3276 w 10 R f (with respect to)2 588 1 2759 3276 t 10 B f (u)3372 3276 w 7 I f (y y)1 31 1 3439 3296 t 10 R f (, as above.)2 421 1 3478 3276 t 10 CW f (FUxt)1020 3432 w 10 R f ( partial derivatives of)3 851(- The)1 305 2 1520 3432 t 10 B f (f)2701 3432 w 10 R f (with respect to)2 588 1 2759 3432 t 10 B f (u)3372 3432 w 7 I f ( t)1 0(x xt)1 51 2 3439 3452 t 10 R f (, as above.)2 421 1 3498 3432 t 10 CW f (FUyt)1020 3588 w 10 R f ( partial derivatives of)3 851(- The)1 305 2 1520 3588 t 10 B f (f)2701 3588 w 10 R f (with respect to)2 588 1 2759 3588 t 10 B f (u)3372 3588 w 7 I f ( t)1 0(y yt)1 51 2 3439 3608 t 10 R f (, as above.)2 421 1 3498 3588 t (Although the calling sequence of)4 1320 1 970 3864 t 10 CW f (AF)2315 3864 w 10 R f ( been changed for historical reasons the user may wish to)10 2294(has not)1 286 2 2460 3864 t (know that actually)2 760 1 720 3984 t 10 CW f (nye)1516 3984 w 10 R f (is currently always set to 1 but that)7 1466 1 1731 3984 t 10 CW f (nxe)3232 3984 w 10 R f (gives all the quadrature points in the)6 1514 1 3447 3984 t 10 I f (x x)1 44 1 4996 3984 t 10 R f (direction for any given rectangle. Thus)5 1562 1 720 4104 t 10 CW f (nxe)2344 4104 w 10 R f (is always larger the)3 782 1 2551 4104 t 10 CW f (nye)3361 4104 w 10 R f ( in the current version the user)6 1232(. Also)1 267 2 3541 4104 t (may determine the current rectangle by inserting the common statement)9 2870 1 720 4224 t 10 CW f (common /ttguc/ ir)2 1020 1 1080 4584 t 10 R f (The variable)1 501 1 720 4944 t 10 CW f (ir)1246 4944 w 10 R f (indicates the rectangle under current investigation and should not be changed by the user.)13 3571 1 1391 4944 t (When)970 5100 w 10 CW f (TTGU)1233 5100 w 10 R f (needs the boundary conditions it will)5 1480 1 1498 5100 t 10 CW f (Call BC\(t,Xe,nxe,Ye,nye,Lx,Rx,Ly,Ry,)1 2160 1 720 5460 t (U,Ut,Ux,Uy,Uxt,Uyt,nu,)1200 5580 w (B,BU,BUt,BUx,BUy,BUxt,BUyt\))1200 5700 w 10 R f (Before)720 5976 w 10 CW f (TTGU)1016 5976 w 10 R f (calls)1281 5976 w 10 CW f (BC)1489 5976 w 10 R f (, it sets to)3 384 1 1609 5976 t 10 B f (0)2018 5976 w 10 R f (the 7 arrays)2 465 1 2093 5976 t 10 CW f (B)2618 5976 w 10 R f (through)2703 5976 w 10 CW f (BUyt)3039 5976 w 10 R f (, and provides the)3 710 1 3279 5976 t 10 I f ( t)1 0( np pu ut)3 128(i in)1 78 3 4014 5976 t 10 CW f (t)1020 6168 w 10 R f ( current value of time.)4 884(- The)1 305 2 1370 6168 t 10 CW f (Xe)1020 6324 w 10 R f ( list of points)3 535(- A)1 222 2 1370 6324 t 10 I f (x x)1 44 1 2155 6324 t 10 R f (where)2227 6324 w 10 B f (b)2498 6324 w 10 R f ( This)1 232(is to be evaluated.)3 730 2 2582 6324 t 10 CW f (Xe)3573 6324 w 10 R f (is)3722 6324 w 10 I f ( t)1 0(n no ot)2 128 2 3818 6324 t 10 R f ( The)1 209(the B-spline mesh X.)3 856 2 3975 6324 t (points)1520 6444 w 10 CW f (Xe)1794 6444 w 10 R f (at which)1 345 1 1943 6444 t 10 B f (b)2317 6444 w 10 R f ( are determined by the quadrature rule used by)8 1880(is desired)1 384 2 2402 6444 t 10 CW f (TTGU)4694 6444 w 10 R f (to)4962 6444 w (implement Galerkin's method.)2 1224 1 1520 6564 t 10 CW f (nxe)1020 6720 w 10 R f ( length of Xe.)3 549(- The)1 305 2 1370 6720 t 10 CW f (Ye)1020 6876 w 10 R f ( list of points)3 535(- A)1 222 2 1370 6876 t 10 I f (y y)1 44 1 2155 6876 t 10 R f (where)2227 6876 w 10 B f (b)2498 6876 w 10 R f ( This)1 232(is to be evaluated.)3 730 2 2582 6876 t 10 CW f (Ye)3573 6876 w 10 R f (is)3722 6876 w 10 I f ( t)1 0(n no ot)2 128 2 3818 6876 t 10 R f ( The)1 209(the B-spline mesh Y.)3 856 2 3975 6876 t (points)1520 6996 w 10 CW f (Ye)1794 6996 w 10 R f (at which)1 345 1 1943 6996 t 10 B f (b)2317 6996 w 10 R f ( are determined by the quadrature rule used by)8 1880(is desired)1 384 2 2402 6996 t 10 CW f (TTGU)4694 6996 w 10 R f (to)4962 6996 w (implement Galerkin's method.)2 1224 1 1520 7116 t cleartomark showpage saveobj restore %%EndPage: 15 15 %%Page: 16 16 /saveobj save def mark 16 pagesetup 10 R f (- 16 -)2 216 1 2772 480 t 10 CW f (nye)1020 840 w 10 R f ( length of Ye.)3 549(- The)1 305 2 1370 840 t 10 CW f (Lx)1020 996 w 10 R f ( left-hand end-point of the)4 1048(- The)1 305 2 1370 996 t 10 I f (x x)1 44 1 2748 996 t 10 R f (spatial domain.)1 611 1 2817 996 t 10 CW f (Rx)1020 1152 w 10 R f ( right-hand end-point of the)4 1104(- The)1 305 2 1370 1152 t 10 I f (x x)1 44 1 2804 1152 t 10 R f (spatial domain.)1 611 1 2873 1152 t 10 CW f (Ly)1020 1308 w 10 R f ( left-hand end-point of the)4 1048(- The)1 305 2 1370 1308 t 10 I f (y y)1 44 1 2748 1308 t 10 R f (spatial domain.)1 611 1 2817 1308 t 10 CW f (Ry)1020 1464 w 10 R f ( right-hand end-point of the)4 1104(- The)1 305 2 1370 1464 t 10 I f (y y)1 44 1 2804 1464 t 10 R f (spatial domain.)1 611 1 2873 1464 t 10 CW f (U)1020 1620 w 10 R f (-)1370 1620 w 10 CW f (U)1520 1620 w 10 R f (\()1580 1620 w 10 I f (p p)1 50 1 1613 1620 t 10 R f (,)1671 1620 w 10 I f (q q)1 50 1 1704 1620 t 10 R f (,)1762 1620 w 10 I f (i i)1 28 1 1795 1620 t 10 R f (\) =)1 114 1 1823 1620 t 10 I f (u u)1 50 1 1962 1620 t 7 I f (i i)1 20 1 2023 1640 t 10 R f (\(t,)2051 1620 w 10 CW f (Xe)2137 1620 w 10 R f (\()2257 1620 w 10 I f (p p)1 50 1 2290 1620 t 10 R f (\),)2340 1620 w 10 CW f (Ye)2398 1620 w 10 R f (\()2518 1620 w 10 I f (q q)1 50 1 2551 1620 t 10 R f (\)\) for)1 207 1 2601 1620 t 10 I f (i i)1 28 1 2833 1620 t 10 S f (= =)1 55 1 2885 1620 t 10 R f (1 ,)1 83 1 2956 1620 t (. . .)2 125 1 3072 1595 t (,)3230 1620 w 10 CW f (nu)3279 1620 w 10 R f (,)3399 1620 w 10 I f (p p)1 50 1 3449 1620 t 10 S f (= =)1 55 1 3523 1620 t 10 R f (1 ,)1 83 1 3594 1620 t 10 I f (. .)1 25 1 3685 1620 t 10 R f (.. ,)1 83 1 3710 1620 t 10 CW f (nxe)3817 1620 w 10 R f (and)4022 1620 w 10 I f (q q)1 50 1 4191 1620 t 10 S f (= =)1 55 1 4265 1620 t 10 R f (1 ,)1 83 1 4336 1620 t (. . .)2 125 1 4452 1595 t (,)4610 1620 w 10 CW f (nye)4659 1620 w 10 R f (.)4839 1620 w 10 CW f (Ut)1020 1776 w 10 R f (-)1370 1776 w 10 CW f (Ut)1520 1776 w 10 S f (= =)1 55 1 1665 1776 t 10 B f (u)1745 1776 w 7 I f (t t)1 20 1 1812 1796 t 10 R f (, stored as above.)3 690 1 1840 1776 t 10 CW f (Ux)1020 1932 w 10 R f (-)1370 1932 w 10 CW f (Ux)1520 1932 w 10 S f (= =)1 55 1 1665 1932 t 10 B f (u)1745 1932 w 7 I f (x x)1 31 1 1812 1952 t 10 R f (, as above.)2 421 1 1851 1932 t 10 CW f (Uy)1020 2088 w 10 R f (-)1370 2088 w 10 CW f (Uy)1520 2088 w 10 S f (= =)1 55 1 1665 2088 t 10 B f (u)1745 2088 w 7 I f (y y)1 31 1 1812 2108 t 10 R f (, as above.)2 421 1 1851 2088 t 10 CW f (Uxt)1020 2244 w 10 R f (-)1370 2244 w 10 CW f (Uxt)1520 2244 w 10 S f (= =)1 55 1 1725 2244 t 10 B f (u)1805 2244 w 7 I f ( t)1 0(x xt)1 51 2 1872 2264 t 10 R f (, as above.)2 421 1 1931 2244 t 10 CW f (Uyt)1020 2400 w 10 R f (-)1370 2400 w 10 CW f (Uyt)1520 2400 w 10 S f (= =)1 55 1 1725 2400 t 10 B f (u)1805 2400 w 7 I f ( t)1 0(y yt)1 51 2 1872 2420 t 10 R f (, as above.)2 421 1 1931 2400 t 10 CW f (nu)1020 2556 w 10 R f ( number)1 330(- The)1 305 2 1370 2556 t 10 I f (n n)1 50 1 2030 2556 t 7 I f (u u)1 35 1 2091 2576 t 10 R f (of)2159 2556 w 10 B f (pde)2267 2556 w 10 R f (variables)2448 2556 w 10 B f (u)2833 2556 w 10 R f (.)2889 2556 w 10 CW f (BC)720 2712 w 10 R f (must return as)2 566 1 865 2712 t 10 I f ( t)1 0( pu ut)2 78( tp)1 50(o ou ut)2 128 4 1456 2712 t 10 CW f (B)1020 2904 w 10 R f (-)1370 2904 w 10 CW f (B)1520 2904 w 10 R f (\()1580 2904 w 10 I f (p p)1 50 1 1613 2904 t 10 R f (,)1671 2904 w 10 I f (q q)1 50 1 1704 2904 t 10 R f (,)1762 2904 w 10 I f (i i)1 28 1 1795 2904 t 10 R f (\) =)1 114 1 1823 2904 t 10 B f (b)1962 2904 w 10 R f (,)2018 2904 w 10 I f (i i)1 28 1 2068 2904 t 10 S f (= =)1 55 1 2120 2904 t 10 R f (1 ,)1 83 1 2191 2904 t (. . .)2 125 1 2307 2879 t (,)2465 2904 w 10 CW f (nu)2514 2904 w 10 R f (,)2634 2904 w 10 I f (p p)1 50 1 2684 2904 t 10 S f (= =)1 55 1 2758 2904 t 10 R f (1 ,)1 83 1 2829 2904 t 10 I f (. .)1 25 1 2920 2904 t 10 R f (.. ,)1 83 1 2945 2904 t 10 CW f (nxe)3052 2904 w 10 R f (and)3257 2904 w 10 I f (q q)1 50 1 3426 2904 t 10 S f (= =)1 55 1 3500 2904 t 10 R f (1 ,)1 83 1 3571 2904 t (. . .)2 125 1 3687 2879 t (,)3845 2904 w 10 CW f (nye)3894 2904 w 10 R f ( \(2.2\).)1 241(. see)1 202 2 4074 2904 t 10 CW f (BU)1020 3060 w 10 R f (-)1370 3060 w 10 CW f (BU)1520 3060 w 10 R f (\()1640 3060 w 10 I f (p p)1 50 1 1673 3060 t 10 R f (,)1731 3060 w 10 I f (q q)1 50 1 1764 3060 t 10 R f (,)1822 3060 w 10 I f (i i)1 28 1 1855 3060 t 10 R f (,)1891 3060 w 10 I f (j j)1 28 1 1932 3060 t 10 R f (\) =)1 189 1 1960 3060 t 10 S f (\266)2250 3060 w 10 I f (b b)1 50 1 2307 3060 t 7 I f (i i)1 20 1 2368 3080 t 10 I f (/ /)1 28 1 2428 3060 t 10 S f (\266)2464 3060 w 10 I f (u u)1 50 1 2521 3060 t 7 I f (j j)1 20 1 2582 3080 t 10 R f (\()2610 3060 w 10 CW f (Xe)2643 3060 w 10 R f (\()2763 3060 w 10 I f (p p)1 50 1 2796 3060 t 10 R f (\),)2846 3060 w 10 CW f (Ye)2904 3060 w 10 R f (\()3024 3060 w 10 I f (q q)1 50 1 3057 3060 t 10 R f (\),)3107 3060 w 10 I f (i i)1 28 1 3266 3060 t 10 R f (,)3302 3060 w 10 I f (j j)1 28 1 3343 3060 t 10 S f (= =)1 55 1 3387 3060 t 10 R f (1 ,)1 83 1 3458 3060 t (. . .)2 125 1 3574 3035 t (,)3732 3060 w 10 CW f (nu)3781 3060 w 10 R f (and)4002 3060 w 10 I f (p p)1 50 1 4247 3060 t 10 S f (= =)1 55 1 4321 3060 t 10 R f (1 ,)1 83 1 4392 3060 t 10 I f (. .)1 25 1 4483 3060 t 10 R f (.. ,)1 83 1 4508 3060 t 10 CW f (nxe)4615 3060 w 10 R f (and)4896 3060 w 10 I f (q q)1 50 1 1520 3180 t 10 S f (= =)1 55 1 1594 3180 t 10 R f (1 ,)1 83 1 1665 3180 t (. . .)2 125 1 1781 3155 t (,)1939 3180 w 10 CW f (nye)1988 3180 w 10 R f (.)2168 3180 w 10 CW f (BUt)1020 3336 w 10 R f (-)1370 3336 w 10 CW f (BUt)1520 3336 w 10 R f (\()1700 3336 w 10 I f (p p)1 50 1 1733 3336 t 10 R f (,)1791 3336 w 10 I f (q q)1 50 1 1824 3336 t 10 R f (,)1882 3336 w 10 I f (i i)1 28 1 1915 3336 t 10 R f (,)1951 3336 w 10 I f (j j)1 28 1 1992 3336 t 10 R f (\) =)1 114 1 2020 3336 t 10 S f (\266)2159 3336 w 10 I f (b b)1 50 1 2216 3336 t 7 I f (i i)1 20 1 2277 3356 t 10 I f (/ /)1 28 1 2337 3336 t 10 S f (\266)2373 3336 w 10 I f (u u)1 50 1 2430 3336 t 7 I f ( t)1 0( t)1 25(j j)1 20 3 2491 3356 t 10 R f (, as above.)2 421 1 2544 3336 t 10 CW f (BUx)1020 3492 w 10 R f (-)1370 3492 w 10 CW f (BUx)1520 3492 w 10 R f (\()1700 3492 w 10 I f (p p)1 50 1 1733 3492 t 10 R f (,)1791 3492 w 10 I f (q q)1 50 1 1824 3492 t 10 R f (,)1882 3492 w 10 I f (i i)1 28 1 1915 3492 t 10 R f (,)1951 3492 w 10 I f (j j)1 28 1 1992 3492 t 10 R f (\) =)1 114 1 2020 3492 t 10 S f (\266)2159 3492 w 10 I f (b b)1 50 1 2216 3492 t 7 I f (i i)1 20 1 2277 3512 t 10 I f (/ /)1 28 1 2337 3492 t 10 S f (\266)2373 3492 w 10 I f (u u)1 50 1 2430 3492 t 7 I f ( x)1 0( x)1 36(j j)1 20 3 2491 3512 t 10 R f (, as above.)2 421 1 2555 3492 t 10 CW f (BUy)1020 3648 w 10 R f (-)1370 3648 w 10 CW f (BUy)1520 3648 w 10 R f (\()1700 3648 w 10 I f (p p)1 50 1 1733 3648 t 10 R f (,)1791 3648 w 10 I f (q q)1 50 1 1824 3648 t 10 R f (,)1882 3648 w 10 I f (i i)1 28 1 1915 3648 t 10 R f (,)1951 3648 w 10 I f (j j)1 28 1 1992 3648 t 10 R f (\) =)1 114 1 2020 3648 t 10 S f (\266)2159 3648 w 10 I f (b b)1 50 1 2216 3648 t 7 I f (i i)1 20 1 2277 3668 t 10 I f (/ /)1 28 1 2337 3648 t 10 S f (\266)2373 3648 w 10 I f (u u)1 50 1 2430 3648 t 7 I f ( y)1 0( y)1 36(j j)1 20 3 2491 3668 t 10 R f (, as above.)2 421 1 2555 3648 t 10 CW f (BUxt)1020 3804 w 10 R f (-)1370 3804 w 10 CW f (BUxt)1520 3804 w 10 R f (\()1760 3804 w 10 I f (p p)1 50 1 1793 3804 t 10 R f (,)1851 3804 w 10 I f (q q)1 50 1 1884 3804 t 10 R f (,)1942 3804 w 10 I f (i i)1 28 1 1975 3804 t 10 R f (,)2011 3804 w 10 I f (j j)1 28 1 2052 3804 t 10 R f (\) =)1 114 1 2080 3804 t 10 S f (\266)2219 3804 w 10 I f (b b)1 50 1 2276 3804 t 7 I f (i i)1 20 1 2337 3824 t 10 I f (/ /)1 28 1 2397 3804 t 10 S f (\266)2433 3804 w 10 I f (u u)1 50 1 2490 3804 t 7 I f ( x)1 0( tx)1 31( t)1 25(j j)1 20 4 2551 3824 t 10 R f (, as above.)2 421 1 2635 3804 t 10 CW f (BUyt)1020 3960 w 10 R f (-)1370 3960 w 10 CW f (BUyt)1520 3960 w 10 R f (\()1760 3960 w 10 I f (p p)1 50 1 1793 3960 t 10 R f (,)1851 3960 w 10 I f (q q)1 50 1 1884 3960 t 10 R f (,)1942 3960 w 10 I f (i i)1 28 1 1975 3960 t 10 R f (,)2011 3960 w 10 I f (j j)1 28 1 2052 3960 t 10 R f (\) =)1 114 1 2080 3960 t 10 S f (\266)2219 3960 w 10 I f (b b)1 50 1 2276 3960 t 7 I f (i i)1 20 1 2337 3980 t 10 I f (/ /)1 28 1 2397 3960 t 10 S f (\266)2433 3960 w 10 I f (u u)1 50 1 2490 3960 t 7 I f ( y)1 0( ty)1 31( t)1 25(j j)1 20 4 2551 3980 t 10 R f (, as above.)2 421 1 2635 3960 t 10 CW f (HANDLU)720 4200 w 10 B f (Description.)1105 4200 w 10 R f (The user-supplied output and control subroutine)5 1958 1 720 4356 t 10 CW f (HANDLU)2710 4356 w 10 R f ( numerical solution at an)4 1016( The)1 212(is now described.)2 710 3 3102 4356 t ( lengthy and complex calculations involving trying several)7 2359(instant in time is obtained only after some rather)8 1961 2 720 4476 t ( When)1 296(small sub-steps in time.)3 966 2 720 4596 t 10 CW f (TTGU)2015 4596 w 10 R f (has finally come up with a solution as accurate as requested by the)12 2752 1 2288 4596 t ( the end of each time-step,)5 1053( At)1 150(user, it just has to tell the user the good news.)10 1821 3 720 4716 t 10 CW f (TTGU)3769 4716 w 10 R f (will)4034 4716 w 10 CW f (Call HANDLU\(t0,U0,t,U,lU,dt,tstop\))1 2040 1 1440 5076 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 5436 t ( only the solution at time)5 1008(put at the end of each time-step is not desired, and)10 2035 2 720 5556 t 10 CW f (tstop)3790 5556 w 10 R f (is needed, the "Return-)3 923 1 4117 5556 t (End")720 5676 w 10 CW f (HANDLU)947 5676 w 10 R f (subroutine)1332 5676 w 10 CW f (TTGUH)1779 5676 w 10 R f (may be used.)2 524 1 2104 5676 t 10 CW f (TTGU)970 5832 w 10 R f (also invokes)1 501 1 1239 5832 t 10 CW f (HANDLU)1769 5832 w 10 R f (whenever it tries to take a time step and fails to obtain the user desired)14 2882 1 2158 5832 t ( may be caused by the time step)7 1287(accuracy. This)1 608 2 720 5952 t 10 CW f (dt)2642 5952 w 10 R f ( it may mean that there is something)7 1466( Or)1 157(being too large.)2 628 3 2789 5952 t ("funny" going on near time)4 1100 1 720 6072 t 10 CW f (t)1847 6072 w 10 R f ( case, the user may want to know that)8 1524( either)1 254(. In)1 160 3 1907 6072 t 10 CW f (TTGU)3873 6072 w 10 R f (failed at time)2 533 1 4141 6072 t 10 CW f (t)4702 6072 w 10 R f (. Such)1 278 1 4762 6072 t (things are called "restarts" and are typically expensive and worth knowing about.)11 3237 1 720 6192 t (The input provided by)3 891 1 970 6348 t 10 CW f (TTGU)1886 6348 w 10 R f (to)2151 6348 w 10 CW f (HANDLU)2254 6348 w 10 R f (is)2639 6348 w 10 CW f (t0)1020 6540 w 10 R f ( at the beginning of the time-step just completed.)8 1957(- Time)1 361 2 1370 6540 t 10 CW f (U0)1020 6696 w 10 R f (-)1370 6696 w 10 B f (pde)1520 6696 w 10 R f (solution)1708 6696 w 10 B f (u)2063 6696 w 10 R f ( given by B-spline coefficients)4 1258(at time t0 is)3 491 2 2151 6696 t 10 CW f (U0)3933 6696 w 10 R f ( array is)2 337(. This)1 261 2 4053 6696 t 10 CW f (Real)4684 6696 w 10 R f (of)4957 6696 w (length)1520 6816 w 10 CW f (lU)1832 6816 w 10 S f (= =)1 55 1 2014 6816 t 10 I f (n n)1 50 1 2118 6816 t 7 I f (x x)1 31 1 2179 6836 t 10 I f (n n)1 50 1 2250 6816 t 7 I f (y y)1 31 1 2311 6836 t 10 I f (n n)1 50 1 2382 6816 t 7 I f (u u)1 35 1 2443 6836 t 10 R f ( are stored as if)4 753( coefficients)1 527(. The)1 267 3 2486 6816 t 10 CW f (U0)4094 6816 w 10 R f (were dimensioned)1 765 1 4275 6816 t 10 CW f (\(nx,ny,Nu\))1520 6936 w 10 R f (, where the)2 440 1 2120 6936 t 10 I f (x x)1 44 1 2585 6936 t 10 R f (grid has)1 319 1 2654 6936 t 10 CW f (nx)2998 6936 w 10 R f (points, similarly for)2 792 1 3143 6936 t 10 I f (y y)1 44 1 3960 6936 t 10 R f (and there are)2 514 1 4029 6936 t 10 CW f (Nu)4568 6936 w 10 B f (pde)4713 6936 w 10 R f (s.)4869 6936 w cleartomark showpage saveobj restore %%EndPage: 16 16 %%Page: 17 17 /saveobj save def mark 17 pagesetup 10 R f (- 17 -)2 216 1 2772 480 t 10 CW f (t)1020 840 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 840 t 10 CW f (U)1020 996 w 10 R f (-)1370 996 w 10 B f (pde)1520 996 w 10 R f (solution)1704 996 w 10 B f (u)2055 996 w 10 R f ( given by B-spline coefficients)4 1242(at time t is)3 429 2 2139 996 t 10 CW f (U)3839 996 w 10 R f (. If)1 145 1 3899 996 t 10 CW f (t0)4073 996 w 10 R f (=)4222 996 w 10 CW f (t)4307 996 w 10 R f (, then a restart is)4 673 1 4367 996 t (in progress and the values in)5 1140 1 1520 1116 t 10 CW f (U)2685 1116 w 10 R f (are meaningless.)1 665 1 2770 1116 t 10 CW f (lU)1020 1272 w 10 R f ( length of the array)4 759(- The)1 305 2 1370 1272 t 10 CW f (U)2459 1272 w 10 R f (.)2519 1272 w 10 CW f (dt)1020 1428 w 10 R f ( current "optimal" value of)4 1069(- The)1 305 2 1370 1428 t 10 CW f (dt)2769 1428 w 10 R f (.)2889 1428 w 10 CW f (tstop)1020 1584 w 10 R f ( current final value for time.)5 1125(- The)1 305 2 1370 1584 t (The use of)2 425 1 720 1740 t 10 CW f (lU)1172 1740 w 10 R f ( use of)2 272(above is a botch required by the)6 1287 2 1319 1740 t 10 CW f (IODE)2906 1740 w 10 R f (to solve the spatially discretized problem - the)7 1866 1 3174 1740 t ( Appendix 1, 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 1860 t 10 CW f (nx)4746 1860 w 10 R f (,)4866 1860 w 10 CW f (ny)4920 1860 w 10 R f (and)720 1980 w 10 CW f (Nu)938 1980 w 10 R f (are obtained, magically, from)3 1220 1 1097 1980 t 10 CW f (Common)2356 1980 w 10 R f (regions internal to)2 755 1 2755 1980 t 10 CW f (TTGU)3549 1980 w 10 R f ( be)1 134( botch will probably)3 850(. This)1 267 3 3789 1980 t (fixed in the next edition of)5 1063 1 720 2100 t 10 CW f (TTGU)1808 2100 w 10 R f (.)2048 2100 w (The output from)2 655 1 720 2256 t 10 CW f (HANDLU)1400 2256 w 10 R f (is)1785 2256 w 10 CW f (t)1020 2448 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 2448 t 10 CW f (U)1020 2604 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 2604 t (non-negative or monotone.)2 1079 1 1520 2724 t 10 CW f (dt)1020 2880 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 2880 t 10 CW f (dt)3952 2880 w 10 R f (so that some particular)3 934 1 4106 2880 t (value of time is achieved on the next time-step.)8 1889 1 1520 3000 t 10 CW f (tstop)1020 3156 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 3156 t 10 B f (u)4852 3156 w 7 I f (t t)1 20 1 4919 3176 t 10 R f (is)4973 3156 w ("small enough" and then stop.)4 1201 1 1520 3276 t 10 B f (Evaluating the Solution.)2 1033 1 720 3516 t 10 R f (For each rectangle to evaluate the solution created by)8 2128 1 970 3672 t 10 CW f (TTGU)3123 3672 w 10 R f (, simply)1 323 1 3363 3672 t 10 CW f ( k, t, it, nt, a, e, ie, ne, m, f\) .)11 2160(Call TSD1\(p,)1 780 2 840 4032 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 4392 t (distributed with)1 636 1 720 4512 t 10 CW f (TTGU)1386 4512 w 10 R f ( 1 in Appendix 1)4 688( Example)1 409(to make evaluation easier for both users and the authors.)9 2287 3 1656 4512 t (shows)720 4632 w 10 CW f (TSD1)996 4632 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 4632 t 10 CW f (p)4980 4632 w 10 R f ( input to)2 334( The)1 205(as 2.)1 183 3 720 4752 t 10 CW f (TSD1)1467 4752 w 10 R f (is)1732 4752 w 10 CW f (p)770 4908 w 10 R f ( number of coordinates, that is, 2.)6 1340(- The)1 305 2 1270 4908 t 10 CW f (k)770 5064 w 10 R f ( that)1 179( Note)1 248( order of the tensor product spline.)6 1402(- The)1 305 4 1270 5064 t 10 CW f (k)3434 5064 w 10 R f (,)3494 5064 w 10 CW f (it)3549 5064 w 10 R f (,)3669 5064 w 10 CW f (nt)3724 5064 w 10 R f (,)3844 5064 w 10 CW f (ie)3899 5064 w 10 R f (,)4019 5064 w 10 CW f (ne)4074 5064 w 10 R f (and)4224 5064 w 10 CW f (m)4398 5064 w 10 R f (are vectors of)2 552 1 4488 5064 t (length)1420 5184 w 10 CW f (p)1727 5184 w 10 R f ( of)1 139( Think)1 320( coordinate.)1 502(with an independent value for each)5 1559 4 1844 5184 t 10 CW f (k\(1\)=kx)4420 5184 w 10 R f (and)4896 5184 w 10 CW f (k\(2\)=ky)1420 5304 w 10 R f (.)1840 5304 w 10 CW f (t)770 5460 w 10 R f ( each spatial coordinate.)3 1435( array containing the meshes for)5 2063(- An)1 272 3 1270 5460 t 10 CW f (t\(it\(1\)\),...,t\(it\(1\)-1+nt\(1\)\))1420 5580 w 10 R f ( for the first coordinate \()5 1013(contains the mesh)2 732 2 3193 5580 t 10 I f (x x)1 44 1 4938 5580 t 10 R f (\),)4982 5580 w 10 CW f (t\(it\(2\)\),...,t\(it\(2\)-1+nt\(2\)\))1420 5700 w 10 R f (for the second \()3 623 1 3210 5700 t 10 I f (y y)1 44 1 3833 5700 t 10 R f (\), and so on.)3 491 1 3877 5700 t 10 CW f (it)770 5856 w 10 R f ( pointers to the meshes in each coordinate, as used above.)10 2301(- The)1 305 2 1270 5856 t 10 CW f (nt)770 6012 w 10 R f ( of mesh points in each dimension.)6 1391(- Number)1 477 2 1270 6012 t 10 CW f (a)770 6168 w 10 R f ( dimensioned)1 549( coefficients, stored as if)4 1026(- B-spline)1 489 3 1270 6168 t 10 CW f (\( nt\(1\)-k\(1\),...,nt\(p\)-k\(p\))1 1633 1 3407 6168 t (\))1420 6288 w 10 R f (.)1480 6288 w 10 CW f (e)770 6444 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 1270 6444 t 10 CW f (t)3826 6444 w 10 R f (,)3886 6444 w 10 CW f (it)3936 6444 w 10 R f (and)4081 6444 w 10 CW f (nt)4250 6444 w 10 R f (.)4370 6444 w 10 CW f (ie)770 6600 w 10 R f ( indices in each dimension, as with)6 1396(- Evaluation)1 583 2 1270 6600 t 10 CW f (it)3274 6600 w 10 R f (above.)3419 6600 w 10 CW f (ne)770 6756 w 10 R f ( number of evaluation points in each dimension.)7 1926(- The)1 305 2 1270 6756 t 10 CW f (m)770 6912 w 10 R f ( of the partial derivatives. See)5 1192(- Order)1 382 2 1270 6912 t 10 CW f (f)2869 6912 w 10 R f (below.)2954 6912 w (The output of)2 544 1 720 7068 t 10 CW f (TSD1)1289 7068 w 10 R f (is)1554 7068 w cleartomark showpage saveobj restore %%EndPage: 17 17 %%Page: 18 18 /saveobj save def mark 18 pagesetup 10 R f (- 18 -)2 216 1 2772 480 t 10 CW f (f)770 840 w 10 R f ( values, stored as if dimensioned)5 1324(- Derivative)1 571 2 1270 840 t 10 CW f (\( ne\(1\),...,ne\(p\) \))2 1148 1 3229 840 t 10 R f ( derivative)1 429(. The)1 234 2 4377 840 t (is of order)2 412 1 1420 960 t 10 CW f (m\(1\))1858 960 w 10 R f ( first coordinate,)2 657(with respect to the)3 738 2 2124 960 t 10 CW f (m\(2\))3544 960 w 10 R f (with respect to the second, and)5 1231 1 3809 960 t ( is,)1 119( That)1 235(so on.)1 241 3 1420 1130 t 10 CW f (f\(i,j\))2042 1130 w 10 S f (= =)1 55 1 2429 1130 t (\266)2558 1210 w 10 I f (x x)1 44 1 2615 1210 t 7 I f (m m)1 50 1 2670 1170 t 7 R f (\( 1 \))2 91 1 2725 1170 t 10 S f (\266)2587 1070 w 7 I f (m m)1 50 1 2641 1030 t 7 R f (\( 1 \))2 91 1 2696 1030 t 10 S1 f (_ _____)1 296 1 2543 1100 t 10 S f (\266)2923 1210 w 10 I f (y y)1 44 1 2980 1210 t 7 I f (m m)1 50 1 3035 1170 t 7 R f (\( 2 \))2 91 1 3090 1170 t 10 S f (\266)2952 1070 w 7 I f (m m)1 50 1 3006 1030 t 7 R f (\( 2 \))2 91 1 3061 1030 t 10 S1 f (_ _____)1 296 1 2908 1100 t 10 I f (s s)1 39 1 3230 1130 t 10 R f (\()3277 1130 w 10 I f (x x)1 44 1 3318 1130 t 7 I f (i i)1 20 1 3373 1150 t 10 R f (,)3409 1130 w 10 I f (y y)1 44 1 3466 1130 t 7 I f (j j)1 20 1 3521 1150 t 10 R f (\), where)1 328 1 3557 1130 t 10 I f (s s)1 39 1 3913 1130 t 10 R f (is the spline whose coeffi-)4 1060 1 3980 1130 t (cients are in)2 540 1 1420 1310 t 10 CW f (a)2014 1310 w 10 R f (, for)1 195 1 2074 1310 t 10 CW f (i)2323 1310 w 10 S f (= =)1 55 1 2424 1310 t 10 R f (1 ,)1 83 1 2528 1310 t 10 I f (. .)1 25 1 2619 1310 t 10 R f (.. ,)1 83 1 2644 1310 t 10 CW f (ne\(1\))2727 1310 w 10 R f (and)3081 1310 w 10 CW f (j)3279 1310 w 10 S f (= =)1 55 1 3380 1310 t 10 R f (1 ,)1 83 1 3484 1310 t 10 I f (. .)1 25 1 3575 1310 t 10 R f (.. ,)1 83 1 3600 1310 t 10 CW f (ne\(2\))3683 1310 w 10 R f ( means that)2 513(. This)1 282 2 3983 1310 t 10 CW f (m =)1 208 1 4832 1310 t (\(0,0\))1420 1430 w 10 R f (gives the function value, for example.)5 1511 1 1745 1430 t (The double precision version of)4 1270 1 970 1586 t 10 CW f (TSD1)2265 1586 w 10 R f (is)2530 1586 w 10 CW f (DTSD1)2622 1586 w 10 R f (with all Real arguments typed double precision.)6 1912 1 2947 1586 t 10 B f (Obtaining Initial Conditions.)2 1240 1 720 1826 t 10 R f (If your initial conditions are constant, setting the initial values for)10 2783 1 970 1982 t 10 B f (U)3794 1982 w 10 R f ( easy: simply set)3 715(in \(3.1\) is)2 418 2 3907 1982 t 10 I f (U U)1 72 1 720 2102 t 7 I f (q qp p)2 70 1 803 2122 t 10 S f (\272)922 2102 w 10 R f ( your initial data is not constant, then you can)9 1821(constant. If)1 474 2 1035 2102 t 10 CW f (Call ICON\(u, nu, nr, kxr, x, nxr, ixb, kyr, y, nyr, iyb, ic\))12 3600 1 1140 2462 t 10 R f ( nu, nr, kxr, x, nxr, ixb, kyr, y, nyr, and iyb have the same meaning as they do)18 3212(The input parameters)2 858 2 970 2738 t (on the invocation of)3 814 1 720 2858 t 10 CW f (TTGU)1563 2858 w 10 R f ( passed directly into)3 811(. The output parameter u may be)6 1324 2 1803 2858 t 10 CW f (TTGU)3966 2858 w 10 R f (without any modifi-)2 806 1 4234 2858 t ( input parameter)2 670(cation. The)1 482 2 720 2978 t 10 CW f (ic)1905 2978 w 10 R f ( user written subroutine which must declared external in)8 2318(is the name of a)4 664 2 2058 2978 t (the program calling)2 786 1 720 3098 t 10 CW f (ICON)1533 3098 w 10 R f (. The user written subroutine)4 1159 1 1773 3098 t 10 CW f (ic)2959 3098 w 10 R f (is called once per rectangle and supplies the ini-)8 1934 1 3106 3098 t ( at specified points in the rectangle. When)7 1694(tial values of the solution)4 1015 2 720 3218 t 10 CW f (ICON)3456 3218 w 10 R f (needs to compute the initial con-)5 1317 1 3723 3218 t (ditions it will)2 535 1 720 3338 t 10 CW f (CALL IC\(nu, ir, xq, nxq, yq, nyq, ui\))7 2220 1 1620 3698 t 10 R f (and provides as input values)4 1132 1 720 4058 t 10 CW f (nu)770 4214 w 10 R f ( number)1 330(- The)1 305 2 1270 4214 t 10 I f (n n)1 50 1 1930 4214 t 7 I f (u u)1 35 1 1991 4234 t 10 R f (of)2059 4214 w 10 B f (pde)2167 4214 w 10 R f (variables)2348 4214 w 10 B f (u)2733 4214 w 10 R f (.)2789 4214 w 10 CW f (ir)770 4370 w 10 R f ( rectangle on which data is to be supplied)8 1653(- The)1 305 2 1270 4370 t 10 CW f (xq)770 4526 w 10 R f ( list of points)3 553(- A)1 222 2 1270 4526 t 10 I f (x x)1 44 1 2079 4526 t 10 R f (where)2157 4526 w 10 I f (u u)1 50 1 2434 4526 t 10 R f ( This)1 237(is to be evaluated.)3 748 2 2518 4526 t 10 CW f (xq)3537 4526 w 10 R f (is)3691 4526 w 10 I f ( t)1 0(n no ot)2 128 2 3792 4526 t 10 R f ( The)1 215(the B-spline mesh X.)3 871 2 3954 4526 t (points)1420 4646 w 10 CW f (xq)1700 4646 w 10 R f (at which)1 351 1 1855 4646 t 10 I f (u u)1 50 1 2241 4646 t 10 R f (is desired are determined by the quadrature rule used by)9 2326 1 2326 4646 t 10 CW f (ICON)4687 4646 w 10 R f (to)4962 4646 w (implement Galerkin's method.)2 1224 1 1420 4766 t 10 CW f (nxq)770 4922 w 10 R f ( length of xq.)3 533(- The)1 305 2 1270 4922 t 10 CW f (yq)770 5078 w 10 R f ( list of points)3 553(- A)1 222 2 1270 5078 t 10 I f (y y)1 44 1 2079 5078 t 10 R f (where)2157 5078 w 10 I f (u u)1 50 1 2434 5078 t 10 R f ( This)1 237(is to be evaluated.)3 748 2 2518 5078 t 10 CW f (yq)3537 5078 w 10 R f (is)3691 5078 w 10 I f ( t)1 0(n no ot)2 128 2 3792 5078 t 10 R f ( The)1 215(the B-spline mesh Y.)3 871 2 3954 5078 t (points)1420 5198 w 10 CW f (yq)1697 5198 w 10 R f (at which)1 348 1 1849 5198 t 10 I f (u u)1 50 1 2229 5198 t 10 R f ( quadrature rule used by)4 988(are desired are determined by the)5 1361 2 2311 5198 t 10 CW f (ICON)4691 5198 w 10 R f (to)4962 5198 w (implement Galerkin's method.)2 1224 1 1420 5318 t 10 CW f (nyq)770 5474 w 10 R f ( length of yq.)3 533(- The)1 305 2 1270 5474 t 10 CW f (IC)720 5630 w 10 R f (must return as)2 566 1 865 5630 t 10 I f ( t)1 0( pu ut)2 78( tp)1 50(o ou ut)2 128 4 1456 5630 t 10 CW f (ui)1020 5822 w 10 R f ( value of)2 349(- The)1 305 2 1520 5822 t 10 I f (u u)1 50 1 2200 5822 t 10 R f (initially at the)2 558 1 2276 5822 t 10 CW f (xq)2860 5822 w 10 R f (\()2980 5822 w 10 I f (p p)1 50 1 3013 5822 t 10 R f (\) and)1 203 1 3063 5822 t 10 CW f (yq)3327 5822 w 10 R f (\()3447 5822 w 10 I f (q q)1 50 1 3480 5822 t 10 R f (\).)3530 5822 w 10 CW f (ui)3709 5822 w 10 R f (\()3829 5822 w 10 I f (p p)1 50 1 3862 5822 t 10 R f (,)3920 5822 w 10 I f (q q)1 50 1 3953 5822 t 10 R f (,)4011 5822 w 10 I f (j j)1 28 1 4052 5822 t 10 R f (\) =)1 115 1 4080 5822 t 10 I f (u u)1 50 1 4221 5822 t 7 I f (j j)1 20 1 4282 5842 t 10 R f (\()4310 5822 w 10 CW f (xq)4343 5822 w 10 R f (\()4463 5822 w 10 I f (p p)1 50 1 4496 5822 t 10 R f (\),)4546 5822 w 10 CW f (yq)4604 5822 w 10 R f (\()4724 5822 w 10 I f (q q)1 50 1 4757 5822 t 10 R f (\)\), for)1 233 1 4807 5822 t 10 I f (p p)1 50 1 1670 5942 t 10 S f (= =)1 55 1 1744 5942 t 10 R f (1 ,)1 83 1 1815 5942 t (. . .)2 125 1 1931 5917 t (,)2089 5942 w 10 CW f (nxq)2138 5942 w 10 R f (,)2318 5942 w 10 I f (q q)1 50 1 2368 5942 t 10 S f (= =)1 55 1 2442 5942 t 10 R f (1 ,)1 83 1 2513 5942 t 10 I f (. .)1 25 1 2604 5942 t 10 R f (.. ,)1 83 1 2629 5942 t 10 CW f (nyq)2736 5942 w 10 R f (and)2941 5942 w 10 I f (j j)1 28 1 3110 5942 t 10 S f (= =)1 55 1 3154 5942 t 10 R f (1 ,)1 83 1 3225 5942 t (. . .)2 125 1 3341 5917 t (,)3499 5942 w 10 CW f (nu)3548 5942 w 10 R f (.)3668 5942 w (The double precision version of)4 1294 1 1220 6098 t 10 CW f (ICON)2545 6098 w 10 R f (is)2816 6098 w 10 CW f (DICON)2914 6098 w 10 R f (with all Real arguments typed double preci-)6 1795 1 3245 6098 t (sion.)970 6218 w 10 B f (Other Ways to Use and Speed-Up)5 1436 1 720 6458 t 10 CW f (TTGU)2181 6458 w 10 B f (.)2421 6458 w 10 R f (There are many "knobs" in)4 1074 1 970 6614 t 10 CW f (TTGU)2069 6614 w 10 R f (; section 6 describes their use.)5 1198 1 2309 6614 t cleartomark showpage saveobj restore %%EndPage: 18 18 %%Page: 19 19 /saveobj save def mark 19 pagesetup 10 R f (- 19 -)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 (TTGU)2350 1116 w 10 R f (through the named Common region)4 1427 1 2615 1116 t 10 CW f (Common / TTGUF / Failed ; Logical Failed .)8 2520 1 720 1476 t 10 R f (Before)720 1752 w 10 CW f (TTGU)1020 1752 w 10 R f (calls)1289 1752 w 10 CW f (AF)1501 1752 w 10 R f (or)1650 1752 w 10 CW f (BC)1762 1752 w 10 R f (, it sets)2 289 1 1882 1752 t 10 CW f (Failed)2200 1752 w 10 R f (=)2589 1752 w 10 CW f (.False.)2674 1752 w 10 R f ( if the user doesn't use or even know of)9 1613(. Thus,)1 304 2 3123 1752 t (the existence of)2 634 1 720 1872 t 10 CW f (TTGUF)1382 1872 w 10 R f (,)1682 1872 w 10 CW f (TTGU)1770 1872 w 10 R f (assumes that everything has been correctly computed on return from those)10 3002 1 2038 1872 t ( however, the user has a problem, uses)7 1580(subroutines. If,)1 633 2 720 1992 t 10 CW f (TTGUF)2965 1992 w 10 R f (, and sets)2 383 1 3265 1992 t 10 CW f (Failed)3680 1992 w 10 R f (=)4072 1992 w 10 CW f (.True.)4160 1992 w 10 R f (,)4520 1992 w 10 CW f (TTGU)4612 1992 w 10 R f (will)4884 1992 w ( in an attempt to obtain a more accurate, and hence more reasonable,)12 2929(automatically lower the time-step)3 1391 2 720 2112 t (numerical solution so that)3 1036 1 720 2232 t 10 CW f (AF)1781 2232 w 10 R f (and)1926 2232 w 10 CW f (BC)2095 2232 w 10 R f (can do their job.)3 649 1 2240 2232 t (The Double Precision version of)4 1298 1 970 2388 t 10 CW f (TTGUF)2293 2388 w 10 R f (is)2618 2388 w 10 CW f (DTTGUF)2710 2388 w 10 R f (.)3070 2388 w 10 B f (Mapping Software)1 797 1 720 2628 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 2784 t ( and is not strictly speaking necessary when solving)8 2182(ever, conformal mapping is a tricky business)6 1879 2 720 2904 t 10 B f (pde)4820 2904 w 10 R f (s.)4976 2904 w (Much simpler non-conformal maps can often be used as the documentation for)11 3147 1 720 3024 t 10 CW f (TTGR)3927 3024 w 10 R f ([15] indicates.)1 571 1 4192 3024 t 10 B f ( for the Sporting User)4 932(6. Advice)1 419 2 720 3264 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 3420 t (and subprograms used in)3 1011 1 720 3540 t 10 CW f (TTGU)1761 3540 w 10 R f ( examples of knob twiddling that can increase the speed of)10 2390(. Several)1 379 2 2001 3540 t 10 CW f (TTGU)4800 3540 w 10 R f (considerably are also given.)3 1114 1 720 3660 t (An exceedingly brief outline of the organization of)7 2032 1 970 3816 t 10 CW f (TTGU)3027 3816 w 10 R f (is, in pseudo-English,)2 867 1 3292 3816 t 10 CW f (t0 = tstart)2 660 1 720 4176 t ( Time-step loop.)2 960( #)1 540(While \( t0 != tstop \))5 1260 3 720 4296 t ({)960 4416 w ( Loop to build extrapolation tableau.)5 2220( #)1 540(Do m = 1, ..., mmax)5 1140 3 960 4536 t ({)1200 4656 w ( Sub-steps loop.)2 960( #)1 540(Do istep = 1, ..., N\(m\))5 1380 3 1200 4776 t ({)1440 4896 w ( Newton loop.)2 780( #)1 540(Do iter = 1, ..., maxit)5 1380 3 1440 5016 t ({)1680 5136 w (Solve the)1 540 1 1680 5256 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 2280 5256 t 10 CW f (Galerkin Equations; Update solution)3 2100 1 2740 5256 t (Check ERROR for "convergence"; Check Convergence Rate)6 3180 1 1680 5376 t (})1680 5496 w (})1440 5616 w (Extrapolate and check the ERROR.)4 1920 1 1200 5736 t (})1200 5856 w (Get_Optimal_dt for next time-step.)3 2040 1 960 5976 t (Output the results for this time-step \( HANDLE \))8 2880 1 960 6096 t (t0 = t0 + dt)4 720 1 960 6216 t (})960 6336 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 6696 t (obtained by)1 469 1 720 6816 t cleartomark showpage saveobj restore %%EndPage: 19 19 %%Page: 39 20 /saveobj save def mark 20 pagesetup 10 R f (- 39 -)2 216 1 2772 480 t (Get integrals for matrix \()4 998 1 1130 900 t 10 CW f (AF)2153 900 w 10 R f (\), mesh interval by mesh interval,)5 1340 1 2298 900 t (Get the)1 291 1 1130 1020 t 10 B f (bc)1446 1020 w 10 R f (s \()1 97 1 1546 1020 t 10 CW f (BC)1668 1020 w 10 R f (\).)1813 1020 w (Get)1130 1140 w 10 B f (pde)1299 1140 w 10 R f (\()1480 1140 w 10 CW f (AF)1538 1140 w 10 R f (\) on the boundary.)3 732 1 1683 1140 t (Solve the)1 375 1 1130 1260 t 10 B f (pde)1530 1260 w 10 R f (system.)1711 1260 w ( These)1 303( many variables and procedures that define the above algorithm.)9 2696(This section describes the)3 1071 3 970 1476 t (include the maximum number of levels of extrapolation to allow \()10 2636 1 720 1596 t 10 CW f (mmax)3381 1596 w 10 R f (\), the sequence of sub-steps to take)6 1394 1 3646 1596 t (\()720 1716 w 10 CW f (N)784 1716 w 10 R f (\(1\),)844 1716 w 10 CW f (N)1017 1716 w 10 R f (\(2\),)1077 1716 w (. . .)2 125 1 1275 1691 t (\), how to solve the linearized Galerkin equations, and the maximum number of Newton)13 3583 1 1457 1716 t (iterations to allow \()3 780 1 720 1836 t 10 CW f (maxit)1525 1836 w 10 R f (\).)1850 1836 w 10 B f (The Linear System)2 811 1 720 2076 t 10 R f ( linear sys-)2 444( The)1 207(At the bottom level we must solve the linearized Galerkin equations, a linear system.)13 3419 3 970 2232 t (tem for each rectangle is a banded system. If one interchanges the)11 2703 1 720 2352 t 10 I f (x x)1 44 1 3455 2352 t 10 R f (and)3531 2352 w 10 I f (y y)1 44 1 3707 2352 t 10 R f ( a rectangle, the)3 649(coordinates for)1 608 2 3783 2352 t ( choose that orientation that would)5 1415(resulting system would still be banded, and in general we would like to)12 2905 2 720 2472 t ( the linear systems for several rectangles, sometimes the)8 2283(minimize the bandwidth. When one knits together)6 2037 2 720 2592 t ( while processing)2 703(band structure of the composite system is preserved; other times it is destroyed. In general)14 3617 2 720 2712 t ( we will take advantage of its band structure and we will arrange the order in which)16 3395(an individual rectangle)2 925 2 720 2832 t ( structure and still keep narrow band widths. Sometimes the)9 2473(the rectangles are processed to preserve band)6 1847 2 720 2952 t ( system is such that we are forced to form a matrix for the equations on the interfaces)17 3656(geometry of the)2 664 2 720 3072 t (between rectangles and treat it as a dense matrix.)8 1953 1 720 3192 t ( More)1 274(On the interface between rectangles, one would like a multiple knot and the B-splines "add".)14 3796 2 970 3348 t (explicitly, assume rectangles 1 and 2 meet at)7 1786 1 720 3468 t 10 I f (x x)1 44 1 2531 3468 t 10 S f (= =)1 55 1 2599 3468 t 10 R f ( B splines for rectangle 1 for)6 1142(4. The)1 280 2 2670 3468 t 10 I f (k k)1 44 1 4117 3468 t 10 S f (= =)1 55 1 4185 3468 t 10 R f (3 would look like)3 703 1 4256 3468 t 1926 3630 1926 5070 Dl 4086 3630 1926 3630 Dl 4086 5070 4086 3630 Dl 1926 5070 4086 5070 Dl 2058 5142 2058 5070 Dl (0)2033 5252 w 2532 5142 2532 5070 Dl (1)2507 5252 w 3006 5142 3006 5070 Dl (2)2981 5252 w 3479 5142 3479 5070 Dl (3)3454 5252 w 3953 5142 3953 5070 Dl (4)3928 5252 w 1854 4981 1926 4981 Dl (0)1779 5001 w 1854 4350 1926 4350 Dl (0.5)1704 4370 w 1854 3718 1926 3718 Dl (1)1779 3738 w 2105 3957 2058 3718 Dl 2153 4172 2106 3958 Dl 2200 4361 2153 4172 Dl 2248 4526 2201 4362 Dl 2294 4665 2247 4527 Dl 2342 4779 2295 4666 Dl 2389 4867 2342 4779 Dl 2437 4930 2390 4867 Dl 2484 4969 2437 4931 Dl 2531 4981 2484 4969 Dl 2105 4747 2058 4981 Dl 2153 4553 2106 4748 Dl 2200 4395 2153 4552 Dl 2248 4273 2201 4393 Dl 2294 4192 2247 4274 Dl 2342 4149 2295 4192 Dl 2389 4141 2342 4147 Dl 2437 4172 2390 4141 Dl 2484 4241 2437 4172 Dl 2531 4349 2484 4242 Dl 2579 4470 2532 4350 Dl 2626 4577 2579 4470 Dl 2674 4672 2627 4577 Dl 2721 4753 2674 4671 Dl 2768 4822 2721 4753 Dl 2816 4879 2769 4823 Dl 2863 4923 2816 4880 Dl 2911 4955 2864 4924 Dl 2957 4974 2910 4956 Dl 3005 4980 2958 4974 Dl 2105 4975 2058 4981 Dl 2153 4956 2106 4974 Dl 2200 4925 2153 4956 Dl 2248 4881 2201 4924 Dl 2294 4824 2247 4880 Dl 2342 4754 2295 4823 Dl 2389 4671 2342 4753 Dl 2437 4576 2390 4671 Dl 2484 4470 2437 4577 Dl 2531 4350 2484 4470 Dl 2579 4237 2532 4350 Dl 2626 4148 2579 4236 Dl 2674 4084 2627 4147 Dl 2721 4047 2674 4085 Dl 2768 4034 2721 4046 Dl 2816 4045 2769 4033 Dl 2863 4084 2816 4046 Dl 2911 4148 2864 4085 Dl 2957 4235 2910 4147 Dl 3005 4349 2958 4236 Dl 3053 4470 3006 4350 Dl 3100 4577 3053 4470 Dl 3148 4672 3101 4577 Dl 3194 4753 3147 4671 Dl 3242 4822 3195 4753 Dl 3289 4879 3242 4823 Dl 3337 4923 3290 4880 Dl 3384 4955 3337 4924 Dl 3431 4974 3384 4956 Dl 3479 4980 3432 4974 Dl 2579 4975 2532 4981 Dl 2626 4956 2579 4974 Dl 2674 4925 2627 4956 Dl 2721 4881 2674 4924 Dl 2768 4824 2721 4880 Dl 2816 4754 2769 4823 Dl 2863 4671 2816 4753 Dl 2911 4576 2864 4671 Dl 2957 4470 2910 4577 Dl 3005 4350 2958 4470 Dl 3053 4237 3006 4350 Dl 3100 4148 3053 4236 Dl 3148 4084 3101 4147 Dl 3194 4047 3147 4085 Dl 3242 4034 3195 4046 Dl 3289 4045 3242 4033 Dl 3337 4084 3290 4046 Dl 3384 4148 3337 4085 Dl 3431 4235 3384 4147 Dl 3479 4349 3432 4236 Dl 3526 4470 3479 4350 Dl 3574 4577 3527 4470 Dl 3621 4672 3574 4577 Dl 3668 4753 3621 4671 Dl 3716 4822 3669 4753 Dl 3763 4879 3716 4823 Dl 3811 4923 3764 4880 Dl 3857 4955 3810 4924 Dl 3905 4974 3858 4956 Dl 3953 4980 3906 4974 Dl 3053 4975 3006 4981 Dl 3100 4956 3053 4974 Dl 3148 4925 3101 4956 Dl 3194 4881 3147 4924 Dl 3242 4824 3195 4880 Dl 3289 4754 3242 4823 Dl 3337 4671 3290 4753 Dl 3384 4576 3337 4671 Dl 3431 4470 3384 4577 Dl 3479 4350 3432 4470 Dl 3526 4243 3479 4350 Dl 3574 4173 3527 4242 Dl 3621 4141 3574 4172 Dl 3668 4147 3621 4141 Dl 3716 4190 3669 4147 Dl 3763 4274 3716 4192 Dl 3811 4394 3764 4274 Dl 3857 4550 3810 4393 Dl 3905 4747 3858 4552 Dl 3953 4982 3906 4748 Dl 3526 4969 3479 4981 Dl 3574 4931 3527 4969 Dl 3621 4868 3574 4931 Dl 3668 4779 3621 4867 Dl 3716 4666 3669 4779 Dl 3763 4528 3716 4666 Dl 3811 4363 3764 4527 Dl 3857 4173 3810 4362 Dl 3905 3958 3858 4172 Dl 3953 3719 3906 3958 Dl (and for rectangle 2 would look like)6 1403 1 720 5502 t cleartomark showpage saveobj restore %%EndPage: 39 20 %%Page: 40 21 /saveobj save def mark 21 pagesetup 10 R f (- 40 -)2 216 1 2772 480 t 1926 840 1926 2280 Dl 4086 840 1926 840 Dl 4086 2280 4086 840 Dl 1926 2280 4086 2280 Dl 2058 2352 2058 2280 Dl (4)2033 2462 w 2532 2352 2532 2280 Dl (5)2507 2462 w 3006 2352 3006 2280 Dl (6)2981 2462 w 3479 2352 3479 2280 Dl (7)3454 2462 w 3953 2352 3953 2280 Dl (8)3928 2462 w 1854 2191 1926 2191 Dl (0)1779 2211 w 1854 1560 1926 1560 Dl (0.5)1704 1580 w 1854 928 1926 928 Dl (1)1779 948 w 2105 1167 2058 928 Dl 2153 1382 2106 1168 Dl 2200 1571 2153 1382 Dl 2248 1736 2201 1572 Dl 2294 1875 2247 1737 Dl 2342 1989 2295 1876 Dl 2389 2077 2342 1989 Dl 2437 2140 2390 2077 Dl 2484 2179 2437 2141 Dl 2531 2191 2484 2179 Dl 2105 1957 2058 2191 Dl 2153 1763 2106 1958 Dl 2200 1605 2153 1762 Dl 2248 1483 2201 1603 Dl 2294 1402 2247 1484 Dl 2342 1359 2295 1402 Dl 2389 1351 2342 1357 Dl 2437 1382 2390 1351 Dl 2484 1451 2437 1382 Dl 2531 1559 2484 1452 Dl 2579 1680 2532 1560 Dl 2626 1787 2579 1680 Dl 2674 1882 2627 1787 Dl 2721 1963 2674 1881 Dl 2768 2032 2721 1963 Dl 2816 2089 2769 2033 Dl 2863 2133 2816 2090 Dl 2911 2165 2864 2134 Dl 2957 2184 2910 2166 Dl 3005 2190 2958 2184 Dl 2105 2185 2058 2191 Dl 2153 2166 2106 2184 Dl 2200 2135 2153 2166 Dl 2248 2091 2201 2134 Dl 2294 2034 2247 2090 Dl 2342 1964 2295 2033 Dl 2389 1881 2342 1963 Dl 2437 1786 2390 1881 Dl 2484 1680 2437 1787 Dl 2531 1560 2484 1680 Dl 2579 1447 2532 1560 Dl 2626 1358 2579 1446 Dl 2674 1294 2627 1357 Dl 2721 1257 2674 1295 Dl 2768 1244 2721 1256 Dl 2816 1255 2769 1243 Dl 2863 1294 2816 1256 Dl 2911 1358 2864 1295 Dl 2957 1445 2910 1357 Dl 3005 1559 2958 1446 Dl 3053 1680 3006 1560 Dl 3100 1787 3053 1680 Dl 3148 1882 3101 1787 Dl 3194 1963 3147 1881 Dl 3242 2032 3195 1963 Dl 3289 2089 3242 2033 Dl 3337 2133 3290 2090 Dl 3384 2165 3337 2134 Dl 3431 2184 3384 2166 Dl 3479 2190 3432 2184 Dl 2579 2185 2532 2191 Dl 2626 2166 2579 2184 Dl 2674 2135 2627 2166 Dl 2721 2091 2674 2134 Dl 2768 2034 2721 2090 Dl 2816 1964 2769 2033 Dl 2863 1881 2816 1963 Dl 2911 1786 2864 1881 Dl 2957 1680 2910 1787 Dl 3005 1560 2958 1680 Dl 3053 1447 3006 1560 Dl 3100 1358 3053 1446 Dl 3148 1294 3101 1357 Dl 3194 1257 3147 1295 Dl 3242 1244 3195 1256 Dl 3289 1255 3242 1243 Dl 3337 1294 3290 1256 Dl 3384 1358 3337 1295 Dl 3431 1445 3384 1357 Dl 3479 1559 3432 1446 Dl 3526 1680 3479 1560 Dl 3574 1787 3527 1680 Dl 3621 1882 3574 1787 Dl 3668 1963 3621 1881 Dl 3716 2032 3669 1963 Dl 3763 2089 3716 2033 Dl 3811 2133 3764 2090 Dl 3857 2165 3810 2134 Dl 3905 2184 3858 2166 Dl 3953 2190 3906 2184 Dl 3053 2185 3006 2191 Dl 3100 2166 3053 2184 Dl 3148 2135 3101 2166 Dl 3194 2091 3147 2134 Dl 3242 2034 3195 2090 Dl 3289 1964 3242 2033 Dl 3337 1881 3290 1963 Dl 3384 1786 3337 1881 Dl 3431 1680 3384 1787 Dl 3479 1560 3432 1680 Dl 3526 1453 3479 1560 Dl 3574 1383 3527 1452 Dl 3621 1351 3574 1382 Dl 3668 1357 3621 1351 Dl 3716 1400 3669 1357 Dl 3763 1484 3716 1402 Dl 3811 1604 3764 1484 Dl 3857 1760 3810 1603 Dl 3905 1957 3858 1762 Dl 3953 2192 3906 1958 Dl 3526 2179 3479 2191 Dl 3574 2141 3527 2179 Dl 3621 2078 3574 2141 Dl 3668 1989 3621 2077 Dl 3716 1876 3669 1989 Dl 3763 1738 3716 1876 Dl 3811 1573 3764 1737 Dl 3857 1383 3810 1572 Dl 3905 1168 3858 1382 Dl 3953 929 3906 1168 Dl ( with a multiple knot of order 2 at)8 1405(The B-splines representing the union of the rectangles)7 2202 2 720 2712 t 10 I f (x x)1 44 1 4359 2712 t 10 S f (= =)1 55 1 4427 2712 t 10 R f (4 would look)2 542 1 4498 2712 t (like:)720 2832 w 1926 2994 1926 4434 Dl 4086 2994 1926 2994 Dl 4086 4434 4086 2994 Dl 1926 4434 4086 4434 Dl 2058 4506 2058 4434 Dl (0)2033 4616 w 2532 4506 2532 4434 Dl (2)2507 4616 w 3006 4506 3006 4434 Dl (4)2981 4616 w 3479 4506 3479 4434 Dl (6)3454 4616 w 3953 4506 3953 4434 Dl (8)3928 4616 w 1854 4345 1926 4345 Dl (0)1779 4365 w 1854 3714 1926 3714 Dl (0.5)1704 3734 w 1854 3082 1926 3082 Dl (1)1779 3102 w 2081 3321 2058 3082 Dl 2105 3536 2082 3322 Dl 2129 3725 2106 3536 Dl 2152 3890 2129 3726 Dl 2176 4029 2153 3891 Dl 2200 4143 2177 4030 Dl 2224 4231 2201 4143 Dl 2247 4294 2224 4231 Dl 2270 4333 2247 4295 Dl 2294 4345 2271 4333 Dl 2081 4111 2058 4345 Dl 2105 3917 2082 4112 Dl 2129 3759 2106 3916 Dl 2152 3637 2129 3757 Dl 2176 3556 2153 3638 Dl 2200 3513 2177 3556 Dl 2224 3505 2201 3511 Dl 2247 3536 2224 3505 Dl 2270 3605 2247 3536 Dl 2294 3713 2271 3606 Dl 2318 3834 2295 3714 Dl 2342 3941 2319 3834 Dl 2365 4036 2342 3941 Dl 2389 4117 2366 4035 Dl 2413 4186 2390 4117 Dl 2437 4243 2414 4187 Dl 2460 4287 2437 4244 Dl 2483 4319 2460 4288 Dl 2507 4338 2484 4320 Dl 2531 4344 2508 4338 Dl 2081 4339 2058 4345 Dl 2105 4320 2082 4338 Dl 2129 4289 2106 4320 Dl 2152 4245 2129 4288 Dl 2176 4188 2153 4244 Dl 2200 4118 2177 4187 Dl 2224 4035 2201 4117 Dl 2247 3940 2224 4035 Dl 2270 3834 2247 3941 Dl 2294 3714 2271 3834 Dl 2318 3601 2295 3714 Dl 2342 3512 2319 3600 Dl 2365 3448 2342 3511 Dl 2389 3411 2366 3449 Dl 2413 3398 2390 3410 Dl 2437 3409 2414 3397 Dl 2460 3448 2437 3410 Dl 2483 3512 2460 3449 Dl 2507 3599 2484 3511 Dl 2531 3713 2508 3600 Dl 2555 3834 2532 3714 Dl 2579 3941 2556 3834 Dl 2602 4036 2579 3941 Dl 2626 4117 2603 4035 Dl 2650 4186 2627 4117 Dl 2674 4243 2651 4187 Dl 2697 4287 2674 4244 Dl 2720 4319 2697 4288 Dl 2744 4338 2721 4320 Dl 2768 4344 2745 4338 Dl 2318 4339 2295 4345 Dl 2342 4320 2319 4338 Dl 2365 4289 2342 4320 Dl 2389 4245 2366 4288 Dl 2413 4188 2390 4244 Dl 2437 4118 2414 4187 Dl 2460 4035 2437 4117 Dl 2483 3940 2460 4035 Dl 2507 3834 2484 3941 Dl 2531 3714 2508 3834 Dl 2555 3601 2532 3714 Dl 2579 3512 2556 3600 Dl 2602 3448 2579 3511 Dl 2626 3411 2603 3449 Dl 2650 3398 2627 3410 Dl 2674 3409 2651 3397 Dl 2697 3448 2674 3410 Dl 2720 3512 2697 3449 Dl 2744 3599 2721 3511 Dl 2768 3713 2745 3600 Dl 2792 3834 2769 3714 Dl 2815 3941 2792 3834 Dl 2839 4036 2816 3941 Dl 2863 4117 2840 4035 Dl 2887 4186 2864 4117 Dl 2910 4243 2887 4187 Dl 2933 4287 2910 4244 Dl 2957 4319 2934 4288 Dl 2981 4338 2958 4320 Dl 3005 4344 2982 4338 Dl 2555 4339 2532 4345 Dl 2579 4320 2556 4338 Dl 2602 4289 2579 4320 Dl 2626 4245 2603 4288 Dl 2650 4188 2627 4244 Dl 2674 4118 2651 4187 Dl 2697 4035 2674 4117 Dl 2720 3940 2697 4035 Dl 2744 3834 2721 3941 Dl 2768 3714 2745 3834 Dl 2792 3607 2769 3714 Dl 2815 3537 2792 3606 Dl 2839 3505 2816 3536 Dl 2863 3511 2840 3505 Dl 2887 3554 2864 3511 Dl 2910 3638 2887 3556 Dl 2933 3758 2910 3638 Dl 2957 3914 2934 3757 Dl 2981 4111 2958 3916 Dl 3005 4346 2982 4112 Dl 3029 4111 3006 4345 Dl 3052 3917 3029 4112 Dl 3076 3759 3053 3916 Dl 3100 3637 3077 3757 Dl 3124 3556 3101 3638 Dl 3147 3513 3124 3556 Dl 3170 3505 3147 3511 Dl 3194 3536 3171 3505 Dl 3218 3605 3195 3536 Dl 3242 3713 3219 3606 Dl 3265 3834 3242 3714 Dl 3289 3941 3266 3834 Dl 3313 4036 3290 3941 Dl 3337 4117 3314 4035 Dl 3360 4186 3337 4117 Dl 3383 4243 3360 4187 Dl 3407 4287 3384 4244 Dl 3431 4319 3408 4288 Dl 3455 4338 3432 4320 Dl 3479 4344 3456 4338 Dl 2792 4333 2769 4345 Dl 2815 4295 2792 4333 Dl 2839 4232 2816 4295 Dl 2863 4143 2840 4231 Dl 2887 4030 2864 4143 Dl 2910 3892 2887 4030 Dl 2933 3727 2910 3891 Dl 2957 3537 2934 3726 Dl 2981 3322 2958 3536 Dl 3005 3083 2982 3322 Dl 3029 3321 3006 3082 Dl 3052 3536 3029 3322 Dl 3076 3725 3053 3536 Dl 3100 3890 3077 3726 Dl 3124 4029 3101 3891 Dl 3147 4143 3124 4030 Dl 3170 4231 3147 4143 Dl 3194 4294 3171 4231 Dl 3218 4333 3195 4295 Dl 3242 4345 3219 4333 Dl 3029 4339 3006 4345 Dl 3052 4320 3029 4338 Dl 3076 4289 3053 4320 Dl 3100 4245 3077 4288 Dl 3124 4188 3101 4244 Dl 3147 4118 3124 4187 Dl 3170 4035 3147 4117 Dl 3194 3940 3171 4035 Dl 3218 3834 3195 3941 Dl 3242 3714 3219 3834 Dl 3265 3601 3242 3714 Dl 3289 3512 3266 3600 Dl 3313 3448 3290 3511 Dl 3337 3411 3314 3449 Dl 3360 3398 3337 3410 Dl 3383 3409 3360 3397 Dl 3407 3448 3384 3410 Dl 3431 3512 3408 3449 Dl 3455 3599 3432 3511 Dl 3479 3713 3456 3600 Dl 3502 3834 3479 3714 Dl 3526 3941 3503 3834 Dl 3550 4036 3527 3941 Dl 3574 4117 3551 4035 Dl 3597 4186 3574 4117 Dl 3620 4243 3597 4187 Dl 3644 4287 3621 4244 Dl 3668 4319 3645 4288 Dl 3692 4338 3669 4320 Dl 3715 4344 3692 4338 Dl 3265 4339 3242 4345 Dl 3289 4320 3266 4338 Dl 3313 4289 3290 4320 Dl 3337 4245 3314 4288 Dl 3360 4188 3337 4244 Dl 3383 4118 3360 4187 Dl 3407 4035 3384 4117 Dl 3431 3940 3408 4035 Dl 3455 3834 3432 3941 Dl 3479 3714 3456 3834 Dl 3502 3601 3479 3714 Dl 3526 3512 3503 3600 Dl 3550 3448 3527 3511 Dl 3574 3411 3551 3449 Dl 3597 3398 3574 3410 Dl 3620 3409 3597 3397 Dl 3644 3448 3621 3410 Dl 3668 3512 3645 3449 Dl 3692 3599 3669 3511 Dl 3715 3713 3692 3600 Dl 3739 3834 3716 3714 Dl 3763 3941 3740 3834 Dl 3787 4036 3764 3941 Dl 3810 4117 3787 4035 Dl 3833 4186 3810 4117 Dl 3857 4243 3834 4187 Dl 3881 4287 3858 4244 Dl 3905 4319 3882 4288 Dl 3929 4338 3906 4320 Dl 3952 4344 3929 4338 Dl 3502 4339 3479 4345 Dl 3526 4320 3503 4338 Dl 3550 4289 3527 4320 Dl 3574 4245 3551 4288 Dl 3597 4188 3574 4244 Dl 3620 4118 3597 4187 Dl 3644 4035 3621 4117 Dl 3668 3940 3645 4035 Dl 3692 3834 3669 3941 Dl 3715 3714 3692 3834 Dl 3739 3607 3716 3714 Dl 3763 3537 3740 3606 Dl 3787 3505 3764 3536 Dl 3810 3511 3787 3505 Dl 3833 3554 3810 3511 Dl 3857 3638 3834 3556 Dl 3881 3758 3858 3638 Dl 3905 3914 3882 3757 Dl 3929 4111 3906 3916 Dl 3952 4346 3929 4112 Dl 3739 4333 3716 4345 Dl 3763 4295 3740 4333 Dl 3787 4232 3764 4295 Dl 3810 4143 3787 4231 Dl 3833 4030 3810 4143 Dl 3857 3892 3834 4030 Dl 3881 3727 3858 3891 Dl 3905 3537 3882 3726 Dl 3929 3322 3906 3536 Dl 3952 3083 3929 3322 Dl (Algebraically, the linear system of rectangle 1 either looks like or can be permuted to look like)16 3785 1 970 4902 t 10 S f (\354)2518 5080 w (\357)2518 5180 w (\356)2518 5280 w 10 I f (C C)1 67 1 2567 5247 t 10 B f (a)2575 5107 w (d)2684 5247 w (b)2684 5107 w 10 S f (\374)2740 5080 w (\357)2740 5180 w (\376)2740 5280 w (\354)2805 5080 w (\357)2805 5180 w (\356)2805 5280 w 10 B f (u)2854 5227 w 7 R f (2)2921 5247 w 10 B f (u)2854 5107 w 7 R f (1)2921 5127 w 10 S f (\374)2964 5080 w (\357)2964 5180 w (\376)2964 5280 w (= =)1 55 1 3029 5167 t (\354)3108 5130 w (\356)3108 5230 w 10 B f (f)3162 5227 w (e)3157 5127 w 10 S f (\374)3201 5130 w (\376)3201 5230 w 10 R f (\(6.1\))4849 5167 w (and the one for rectangle 2 can be permuted to look like)11 2231 1 720 5452 t 10 S f (\354)2471 5630 w (\357)2471 5730 w (\356)2471 5830 w 10 I f (J J)1 44 1 2534 5797 t (G G)1 72 1 2520 5657 t (K K)1 67 1 2644 5797 t (H H)1 72 1 2642 5657 t 10 S f (\374)2714 5630 w (\357)2714 5730 w (\376)2714 5830 w (\354)2779 5630 w (\357)2779 5730 w (\356)2779 5830 w 10 B f (u)2828 5777 w 7 R f (4)2895 5797 w 10 B f (u)2828 5657 w 7 R f (3)2895 5677 w 10 S f (\374)2938 5630 w (\357)2938 5730 w (\376)2938 5830 w (= =)1 55 1 3003 5717 t (\354)3082 5680 w (\356)3082 5780 w 10 B f (n)3144 5777 w (m)3131 5677 w 10 S f (\374)3214 5680 w (\376)3214 5780 w 10 I f (. .)1 25 1 3271 5717 t 10 R f (\(6.2\))4849 5717 w (The linear system representing the union would look like)8 2281 1 720 6002 t 10 S f (\354)2286 6180 w (\357)2286 6280 w (\357)2286 6380 w (\356)2286 6480 w 10 I f (C C)1 67 1 2335 6327 t 10 B f (a)2343 6187 w 10 I f (J J)1 44 1 2549 6467 t (D D)1 72 1 2452 6327 t 10 S f (+ +)1 55 1 2548 6327 t 10 I f (G G)1 72 1 2619 6327 t 10 B f (b)2543 6187 w 10 I f (K K)1 67 1 2743 6467 t (H H)1 72 1 2741 6327 t 10 S f (\374)2813 6180 w (\357)2813 6280 w (\357)2813 6380 w (\376)2813 6480 w (\354)2878 6180 w (\357)2878 6280 w (\357)2878 6380 w (\356)2878 6480 w 10 B f (u)2927 6437 w 7 R f (4)2994 6457 w 10 B f (u)2927 6317 w 7 R f (2)2994 6337 w 10 B f (u)2927 6197 w 7 R f (1)2994 6217 w 10 S f (\374)3037 6180 w (\357)3037 6280 w (\357)3037 6380 w (\376)3037 6480 w (= =)1 55 1 3102 6317 t (\354)3181 6230 w (\357)3181 6330 w (\356)3181 6430 w 10 B f (n)3303 6427 w (f)3230 6327 w 10 S f (+)3279 6327 w 10 B f (m)3350 6327 w (e)3309 6227 w 10 S f (\374)3433 6230 w (\357)3433 6330 w (\376)3433 6430 w 10 R f (\(6.3\))4849 6317 w (and)720 6652 w 10 B f (u)889 6652 w 7 R f (3)956 6672 w 10 R f (is set to)2 306 1 1024 6652 t 10 B f (u)1355 6652 w 7 R f (2)1422 6672 w 10 R f (, i.e. the solution is forced to match on the interface.)10 2080 1 1465 6652 t ( know that eventually we wish to form the LU)9 1998( We)1 205( explicitly form \(6.3\).)3 914(In our code we do not)5 953 4 970 6808 t ( LU decomposition of \(6.3\))4 1116( The)1 210( that the blocks that make up \(6.3\) are banded.)9 1894(decomposition of \(6.3\) and)3 1100 4 720 6928 t (is given by)2 439 1 720 7048 t 10 S f (\354)2195 7226 w (\357)2195 7326 w (\357)2195 7426 w (\357)2195 7526 w (\356)2195 7626 w 10 I f (C C)1 67 1 2265 7413 t (L L)1 56 1 2244 7253 t 7 B f (a)2311 7273 w 10 I f (J J)1 44 1 2456 7573 t (L L)1 56 1 2404 7413 t 7 I f (G G)1 50 1 2471 7433 t 7 S f (\242 \242)1 18 1 2526 7433 t 10 I f (L L)1 56 1 2602 7573 t 7 I f (K K)1 47 1 2669 7593 t 7 S f (\242 \242)1 18 1 2721 7593 t 10 S f (\374)2747 7226 w (\357)2747 7326 w (\357)2747 7426 w (\357)2747 7526 w (\376)2747 7626 w (\354)2812 7226 w (\357)2812 7326 w (\357)2812 7426 w (\357)2812 7526 w (\356)2812 7626 w 10 B f (u)2861 7253 w 7 B f (a)2928 7273 w 10 B f (u)3056 7423 w 7 I f (G G)1 50 1 3123 7443 t 7 S f (\242 \242)1 18 1 3178 7443 t 10 I f (L L)1 56 1 3021 7253 t 7 B f (a)3082 7272 w 7 S f (- -)1 39 1 3082 7213 t 7 R f (1)3132 7213 w 10 B f (b)3183 7253 w (u)3333 7583 w 7 I f (K K)1 47 1 3400 7603 t 7 S f (\242 \242)1 18 1 3452 7603 t 10 I f (L L)1 56 1 3289 7423 t 7 I f (G G)1 50 1 3350 7442 t 7 S f (\242 \242)1 18 1 3405 7442 t (- -)1 39 1 3350 7383 t 7 R f (1)3400 7383 w 10 I f (H H)1 72 1 3451 7423 t 10 S f (\374)3523 7226 w (\357)3523 7326 w (\357)3523 7426 w (\357)3523 7526 w (\376)3523 7626 w 10 R f (\(6.4\))4849 7413 w cleartomark showpage saveobj restore %%EndPage: 40 21 %%Page: 41 22 /saveobj save def mark 22 pagesetup 10 R f (- 41 -)2 216 1 2772 480 t (where)720 840 w 10 I f (L L)1 56 1 2697 1020 t 7 B f (a)2764 1040 w 10 B f (u)2815 1020 w 7 B f (a)2882 1040 w 10 S f (= =)1 55 1 2941 1020 t 10 B f (a)3012 1020 w 10 R f (\(6.5\))4849 1020 w 10 I f (L L)1 56 1 2299 1200 t 7 I f (G G)1 50 1 2366 1220 t 7 S f (\242 \242)1 18 1 2421 1220 t 10 B f (u)2455 1200 w 7 I f (G G)1 50 1 2522 1220 t 7 S f (\242 \242)1 18 1 2577 1220 t 10 S f (= =)1 55 1 2619 1200 t 10 I f (G G)1 72 1 2690 1200 t 10 S f ( \272)1 63(\242 \242)1 25 2 2770 1200 t 10 B f (d)2866 1200 w 10 S f (+ +)1 55 1 2938 1200 t 10 I f (G G)1 72 1 3009 1200 t 10 S f (+ +)1 55 1 3105 1200 t 10 I f ( L)1 0(C CL)1 123 2 3176 1200 t 7 B f (a)3304 1219 w 7 S f (- -)1 39 1 3304 1160 t 7 R f (1)3354 1160 w 10 B f (b)3405 1200 w 10 R f (\(6.6\))4849 1200 w 10 I f (L L)1 56 1 2451 1380 t 7 I f (K K)1 47 1 2518 1400 t 7 S f (\242 \242)1 18 1 2570 1400 t 10 B f (u)2604 1380 w 7 I f (K K)1 47 1 2671 1400 t 7 S f (\242 \242)1 18 1 2723 1400 t 10 S f (= =)1 55 1 2765 1380 t 10 I f (K K)1 67 1 2836 1380 t 10 S f (+ +)1 55 1 2927 1380 t 10 I f ( L)1 0(J JL)1 100 2 2998 1380 t 7 I f (G G)1 50 1 3103 1399 t 7 S f (\242 \242)1 18 1 3158 1399 t (- -)1 39 1 3103 1340 t 7 R f (1)3153 1340 w 10 I f ( .)1 0( .)1 33(H H)1 72 3 3204 1380 t 10 R f (We can formed the LU decomposition of the banded matrix)9 2387 1 720 1560 t 10 B f (a)3132 1560 w 10 R f (and then form)2 560 1 3232 1560 t 10 I f (G G)1 72 1 2566 1745 t 11 R f (\303)2589 1715 w 10 S f (= =)1 55 1 2662 1745 t 10 B f (d)2733 1745 w 10 S f (+ +)1 55 1 2805 1745 t 10 I f ( L)1 0(C CL)1 123 2 2876 1745 t 7 B f (a)3004 1764 w 7 S f (- -)1 39 1 3004 1705 t 7 R f (1)3054 1705 w 10 B f (b)3105 1745 w 10 I f (. .)1 25 1 3169 1745 t 10 R f (\(6.7\))4849 1745 w (The matrix)1 441 1 720 1930 t 10 I f (G G)1 72 1 1186 1930 t 11 R f (\303)1209 1900 w 10 R f (will be dense but the matrix)5 1113 1 1283 1930 t 10 S f (\354)2654 2108 w (\357)2654 2208 w (\356)2654 2308 w 10 I f (J J)1 44 1 2800 2287 t (G G)1 72 1 2703 2147 t 11 R f (\303)2726 2117 w 10 S f (+ +)1 55 1 2799 2147 t 10 I f (G G)1 72 1 2870 2147 t (K K)1 67 1 2994 2287 t (H H)1 72 1 2992 2147 t 10 S f (\374)3064 2108 w (\357)3064 2208 w (\376)3064 2308 w 10 R f (\(6.8\))4849 2195 w (will have the same band width as \(6.2\) and we treat it the same way.)14 2728 1 720 2480 t ( is valid if)3 412( banded form of \(6.3\))4 864( The)1 207(We would like to permute our matrices to minimize band width.)10 2587 4 970 2636 t (rectangles 1 and 2 look like)5 1101 1 720 2756 t 1440 2918 1440 3278 Dl 2880 2918 1440 2918 Dl 2880 3278 2880 2918 Dl 1440 3278 2880 3278 Dl (1)2135 3118 w 2880 2918 2880 3278 Dl 4320 2918 2880 2918 Dl 4320 3278 4320 2918 Dl 2880 3278 4320 3278 Dl (2)3575 3118 w (If, in fact, the rectangles looked like)6 1441 1 970 3494 t 2088 3656 2088 4016 Dl 2232 3656 2088 3656 Dl 2232 4016 2232 3656 Dl 2088 4016 2232 4016 Dl (1)2135 3856 w 2232 3656 2232 4016 Dl 3672 3656 2232 3656 Dl 3672 4016 3672 3656 Dl 2232 4016 3672 4016 Dl (2)2927 3856 w ( linear system corresponding to rectangle 1 to minimize band width which)11 2988(then we may wish to permute the)6 1332 2 720 4196 t ( interface will not be at the bottom)7 1400(would mean that the unknown variables representing the solution on the)10 2920 2 720 4316 t ( complicates the programming but the)5 1598( This)1 244( throughout the matrix.)3 970(of the matrix but regularly scattered)5 1508 4 720 4436 t ( may wish. However if)4 912(resulting matrix of \(6.8\) still has a band structure with as narrow a band width as one)16 3408 2 720 4556 t (the union of rectangles had the form)6 1447 1 720 4676 t 2520 4838 2520 6278 Dl 2880 4838 2520 4838 Dl 2880 6278 2880 4838 Dl 2520 6278 2880 6278 Dl (1)2675 5578 w 2880 4838 2880 6278 Dl 3240 4838 2880 4838 Dl 3240 6278 3240 4838 Dl 2880 6278 3240 6278 Dl (2)3035 5578 w ( both matrices and for the union of rectangles we would arrive at a system)14 3067(then we would like to permute)5 1253 2 720 6458 t (that looks like)2 567 1 720 6578 t 10 S f (\354)2591 6756 w (\357)2591 6856 w (\357)2591 6956 w (\356)2591 7056 w 10 I f (C C)1 67 1 2640 7043 t 10 B f (a)2648 6763 w 10 I f (H H)1 72 1 2757 7043 t (K K)1 67 1 2759 6903 t 10 B f (d)2879 7043 w 10 S f (+ +)1 55 1 2951 7043 t 10 I f (G G)1 72 1 3022 7043 t (J J)1 44 1 2964 6903 t 10 B f (b)2958 6763 w 10 S f (\374)3094 6756 w (\357)3094 6856 w (\357)3094 6956 w (\376)3094 7056 w 10 I f (. .)1 25 1 3151 6893 t 10 R f (\(6.9\))4849 6893 w (One could use a banded LU decomposition routine to find the LU decompositions of)13 3440 1 720 7228 t 10 B f (a)4189 7228 w 10 R f (and)4268 7228 w 10 I f (K K)1 67 1 4442 7228 t 10 R f (, but in order)3 531 1 4509 7228 t cleartomark showpage saveobj restore %%EndPage: 41 22 %%Page: 42 23 /saveobj save def mark 23 pagesetup 10 R f (- 42 -)2 216 1 2772 480 t (to find the LU decomposition of \(6.9\) one would also need the LU decomposition of)14 3377 1 720 840 t 10 I f (D D)1 72 1 2388 1020 t 10 S f (+ +)1 55 1 2484 1020 t 10 I f (G G)1 72 1 2555 1020 t 10 S f (+ +)1 55 1 2651 1020 t 10 I f ( L)1 0(C CL)1 123 2 2722 1020 t 7 B f (a)2850 1039 w 7 S f (- -)1 39 1 2850 980 t 7 R f (1)2900 980 w 10 B f (b)2951 1020 w 10 S f (+ +)1 55 1 3023 1020 t 10 I f ( L)1 0(H HL)1 128 2 3094 1020 t 7 I f (K K)1 47 1 3227 1039 t 7 S f (- -)1 39 1 3227 980 t 7 R f (1)3277 980 w 10 I f (J J)1 44 1 3328 1020 t 10 R f (\(6.10\))4799 1020 w (which is not likely to be banded.)6 1302 1 720 1200 t (Now let us consider a more complicated geometry. Consider)8 2421 1 970 1356 t 1980 1518 1980 2238 Dl 2340 1518 1980 1518 Dl 2340 2238 2340 1518 Dl 1980 2238 2340 2238 Dl (1)2135 1898 w 1980 2238 1980 2958 Dl 2340 2238 1980 2238 Dl 2340 2958 2340 2238 Dl 1980 2958 2340 2958 Dl (2)2135 2618 w 2340 2238 2340 2958 Dl 3780 2238 2340 2238 Dl 3780 2958 3780 2238 Dl 2340 2958 3780 2958 Dl (3)3035 2618 w (Processing the above boxes in the order given would mean that we would always be working with banded)17 4320 1 720 3138 t ( "3" first, we would either be forced to separate out the inter-)12 2429( if we chose to process box)6 1087(matrices. However,)1 804 3 720 3258 t ( banded but would be of wide)6 1259(face variables or permute the equations for box "2" so that they would be)13 3061 2 720 3378 t ( to)1 105( of this, we first perform an analysis to determine a good ordering of the rectangles)15 3355( Because)1 385(band width.)1 475 4 720 3498 t ( we find that)3 511( When)1 291( and to keep banded structure throughout the computation.)8 2357(preserve narrow band widths)3 1161 4 720 3618 t ( bandwidth banded systems is impossible we construct a general matrix as in \(6.10\) to handle)15 3759(keeping small)1 561 2 720 3738 t (all those variables on the interface that do not fit in the banded structure.)13 2895 1 720 3858 t 10 B f (Twiddling Procedure Knobs.)2 1238 1 720 4098 t 10 R f (Control over the procedure)3 1123 1 970 4254 t 10 CW f (ERROR)2133 4254 w 10 R f (is given by)2 469 1 2473 4254 t 10 CW f (TTGUR)2983 4254 w 10 R f ( routine allows the user to override)6 1488(. This)1 269 2 3283 4254 t (some of the default routines of)5 1223 1 720 4374 t 10 CW f (TTGU)1968 4374 w 10 R f (.)2208 4374 w 10 CW f (TTGUR)2353 4374 w 10 R f (is invoked by)2 539 1 2678 4374 t 10 CW f (Call TTGUR\(U,Nu,kx,x,nx, ky,y,ny,)2 1980 1 840 4734 t (tstart,tstop,dt,)1500 4854 w (AF,BC,)1500 4974 w (ERROR,errpar,)1500 5094 w (HANDLE\))1500 5214 w 10 R f (The extra argument)2 787 1 720 5490 t 10 CW f (ERROR)1535 5490 w 10 R f ( direct user control over the accuracy of the integra-)9 2104(in this subroutine provides)3 1073 2 1863 5490 t (tion process.)1 505 1 720 5610 t (The default routine, used by)4 1123 1 970 5766 t 10 CW f (TTGU)2118 5766 w 10 R f (, is)1 117 1 2358 5766 t 10 CW f (TTGUE)2500 5766 w 10 R f (for)2825 5766 w 10 CW f (ERROR)2966 5766 w 10 R f (.)3266 5766 w 10 B f (Error Options)1 614 1 720 6006 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 6162 t (gram)720 6282 w 10 CW f (ERROR)955 6282 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 6282 t (yet been implemented because of using)5 1622 1 720 6402 t 10 CW f (IODE)2377 6402 w 10 R f ( can roll their own)4 772(raw. Users)1 461 2 2652 6402 t 10 CW f (ERROR)3920 6402 w 10 R f (routines, but)1 510 1 4255 6402 t 10 CW f (TTGU)4800 6402 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 6522 t (trolled by the switch)3 819 1 720 6642 t 10 CW f (erputs)1564 6642 w 10 R f (, and implemented by the subprogram)5 1515 1 1924 6642 t 10 CW f (TTGUE)3464 6642 w 10 R f (.)3764 6642 w (The error control provided in the subroutine)6 1836 1 970 6798 t 10 CW f (TTGUE)2879 6798 w 10 R f ( of the variables.)3 707(is based on the local value)5 1116 2 3217 6798 t (That is, the error acceptable in)5 1213 1 720 6918 t 10 I f (u u)1 50 1 1958 6918 t 7 I f (i i)1 20 1 2019 6938 t 10 R f (\()2055 6918 w 10 I f (t t)1 28 1 2096 6918 t 10 R f (,)2132 6918 w 10 I f (x x)1 44 1 2165 6918 t 10 R f (,)2217 6918 w 10 I f (y y)1 44 1 2250 6918 t 10 R f (\) is)1 125 1 2302 6918 t 10 CW f (errpar)1080 7098 w 10 R f (\(1\) *)1 191 1 1440 7098 t 10 S f ( \357)1 0( \357)1 24(\357 \357)1 49 3 1656 7115 t 10 B f (U)1761 7098 w 7 I f (. .)1 18 1 1844 7118 t 7 R f (.)1862 7118 w 7 I f (i i)1 20 1 1885 7118 t 10 S f ( \357)1 0( \357)1 24(\357 \357)1 49 3 1946 7115 t 7 S f (\245)2008 7118 w 10 R f (+)2100 7098 w 10 CW f (errpar)2156 7098 w 10 R f (\(2\))2516 7098 w (where)720 7278 w 10 B f (U)992 7278 w 7 I f (. .)1 18 1 1075 7298 t 7 R f (.)1093 7298 w 7 I f (i i)1 20 1 1116 7298 t 10 R f ( block of B-spline coefficients for)5 1375(denotes the)1 456 2 1173 7278 t 10 I f (u u)1 50 1 3034 7278 t 7 I f (i i)1 20 1 3095 7298 t 10 R f (, which depends only upon the current value of)8 1917 1 3123 7278 t cleartomark showpage saveobj restore %%EndPage: 42 23 %%Page: 43 24 /saveobj save def mark 24 pagesetup 10 R f (- 43 -)2 216 1 2772 480 t 10 B f (u)720 840 w 10 R f (.)776 840 w ( in the subroutine)3 715(The error control provided)3 1076 2 970 996 t 10 CW f (TTGUE)2792 996 w 10 R f (is an)1 192 1 3123 996 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 996 t 10 R f ( can be)2 294(criterion. This)1 597 2 4149 996 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 1116 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 1236 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 1236 t ( p)1 0( ep)1 50( te)1 44(s st)1 67 4 720 1356 t 10 R f ( making the error tolerance in)5 1183(criterion. By)1 530 2 906 1356 t 10 I f (u u)1 50 1 2644 1356 t 7 I f (i i)1 20 1 2705 1376 t 10 R f (\()2741 1356 w 10 I f (t t)1 28 1 2782 1356 t 10 R f (,)2818 1356 w 10 I f (x x)1 44 1 2851 1356 t 10 R f (,)2903 1356 w 10 I f (y y)1 44 1 2936 1356 t 10 R f (\) look like)2 411 1 2988 1356 t 10 S f (\357 \357)1 49 1 1105 1553 t 10 CW f (dt)1178 1536 w 10 S f (\357 \357)1 49 1 1323 1553 t 10 R f (* \()1 108 1 1421 1536 t 10 CW f (errpar)1554 1536 w 10 R f (\(1\) *)1 191 1 1914 1536 t 10 S f ( \357)1 0( \357)1 24(\357 \357)1 49 3 2130 1553 t 10 B f (U)2235 1536 w 7 I f (. .)1 18 1 2318 1556 t 7 R f (.)2336 1556 w 7 I f (i i)1 20 1 2359 1556 t 10 S f ( \357)1 0( \357)1 24(\357 \357)1 49 3 2420 1553 t 7 S f (\245)2482 1556 w 10 R f (+)2574 1536 w 10 CW f (errpar)2630 1536 w 10 R f (\(2\) \) ,)2 224 1 2990 1536 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 1716 t ( argument holds - when)4 967(time-step criterion, the reverse)3 1240 2 720 1836 t 10 CW f (dt)2958 1836 w 10 R f ( error per)2 382( The)1 211(is large, so is the error tolerance.)6 1338 3 3109 1836 t (time-step versus unit-time-step option is controlled by the switch)8 2599 1 720 1956 t 10 CW f (erputs)3344 1956 w 10 R f (as described below.)2 784 1 3729 1956 t (The output from)2 655 1 970 2112 t 10 CW f (TTGUR)1650 2112 w 10 R f (is the same as that for)5 868 1 1975 2112 t 10 CW f (TTGU)2868 2112 w 10 R f ( that the user may)4 714(with the additional possibility)3 1193 2 3133 2112 t (alter)720 2232 w 10 CW f (errpar)922 2232 w 10 R f (through his subprogram)2 955 1 1307 2232 t 10 CW f (ERROR)2287 2232 w 10 R f (.)2587 2232 w (The double precision version of)4 1278 1 970 2388 t 10 CW f (TTGUR)2275 2388 w 10 R f (is)2602 2388 w 10 CW f (DTTGUR)2697 2388 w 10 R f (, with all Real arguments typed double precision,)7 1983 1 3057 2388 t (except)720 2508 w 10 CW f (errpar)1005 2508 w 10 R f ( for)1 141( Similarly)1 423(, which remains Real.)3 868 3 1365 2508 t 10 CW f (TTGUE)2822 2508 w 10 R f (and)3147 2508 w 10 CW f (TTGUP)3316 2508 w 10 R f (.)3616 2508 w 10 B f (Twiddling non-Procedure Knobs.)2 1433 1 720 2748 t 10 R f (The main knob-twiddling routine is)4 1430 1 970 2904 t 10 CW f (TTGUV)2427 2904 w 10 R f ( internal variable of)3 793( the user wants to tinker some)6 1205(. When)1 315 3 2727 2904 t 10 CW f (TTGU)720 3024 w 10 R f (the)985 3024 w 10 CW f (Call TTGUV\(j,f,r,i,l\))1 1260 1 840 3384 t 10 R f (just before calling)2 731 1 720 3744 t 10 CW f (TTGU)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 (TTGUV)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 (TTGUV)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 (TTGUV)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 (TTGUV)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 (TTGU)1903 4776 w 10 R f (and Double Precision for)3 1001 1 2168 4776 t 10 CW f (DTTGU)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 (TTGUV)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 (TTGUV)3172 5556 w 10 R f (.)3472 5556 w 10 CW f (theta)770 5712 w 10 R f ( When)1 290( time-discretization parameter, see section 3 and Appendix 2 of [15].)10 2764(- The)1 305 3 1270 5712 t 10 CW f (theta)4656 5712 w 10 R f (=)4984 5712 w ( For)1 192( Backwards-Euler formula is used.)4 1395(1 the extremely stable, first order accurate)6 1705 3 1420 5832 t 10 CW f (theta)4740 5832 w 10 R f ( If)1 122(= 1/2, the second-order Crank-Nicholson scheme is used.)7 2330 2 1420 5952 t 10 CW f (theta)3903 5952 w 10 S f (\271)4234 5952 w 10 R f (1/2, then)1 356 1 4320 5952 t 10 CW f (N)4707 5952 w 10 S f (= =)1 55 1 4798 5952 t 10 R f ({ 1,)1 155 1 4885 5952 t (2, 3, 4, 6,)3 378 1 1420 6112 t (. . .)2 125 1 1849 6087 t (} and)1 218 1 2025 6112 t 10 CW f (gamma)2269 6112 w 10 S f (= =)1 55 1 2595 6112 t 10 R f (1. If)1 192 1 2676 6112 t 10 CW f (theta)2894 6112 w 10 S f (= =)1 55 1 3220 6112 t 10 R f (2)3349 6182 w (1)3349 6052 w 10 S1 f (_ _)1 80 1 3334 6082 t 10 R f (, then)1 223 1 3424 6112 t 10 CW f (N)3673 6112 w 10 S f (= =)1 55 1 3759 6112 t 10 R f ({ 2, 4, 6,)3 351 1 3840 6112 t (. . .)2 125 1 4242 6087 t (} and)1 217 1 4418 6112 t 10 CW f (gamma)4660 6112 w 10 S f (= =)1 55 1 4985 6112 t 10 R f (2. 0)1 175 1 1420 6282 t 10 S f (<)1620 6282 w 10 R f (=)1675 6282 w 10 CW f (theta)1756 6282 w 10 S f (<)2081 6282 w 10 R f ( Default:)1 377(= 1 is required.)3 605 2 2136 6282 t 10 CW f (theta)3143 6282 w 10 R f (= 1.)1 156 1 3468 6282 t 10 CW f (j)3744 6282 w 10 R f (= 1.)1 156 1 3829 6282 t 10 CW f (theta)4010 6282 w 10 R f (=)4335 6282 w 10 CW f (f)4416 6282 w 10 R f (.)4476 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 %%EndPage: 43 24 %%Page: 44 25 /saveobj save def mark 25 pagesetup 10 R f (- 44 -)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( \()1 84( solve the nonlinear equations to a tolerance of)8 1871(cally, it will try to)4 720 4 1420 1632 t (to)1420 1752 w 10 CW f (TTGU)1534 1752 w 10 R f (\) /)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 (TTGU)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 (TTGU)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 (Default:)1420 7116 w 10 CW f (LA)1772 7116 w 10 R f (= +1.)1 212 1 1917 7116 t 10 CW f (j)2249 7116 w 10 R f (= 2010.)1 306 1 2334 7116 t 10 CW f (LA)2665 7116 w 10 R f (=)2810 7116 w 10 CW f (i)2891 7116 w 10 R f (.)2951 7116 w cleartomark showpage saveobj restore %%EndPage: 44 25 %%Page: 45 26 /saveobj save def mark 26 pagesetup 10 R f (- 45 -)2 216 1 2772 480 t 10 CW f (xpoly)770 840 w 10 R f ( variable indicating whether polynomial \(True\) or rational \(False\) extrapolation)9 3214( Logical)1 334(- A)1 222 3 1270 840 t ( Default:)1 377(is to be used.)3 522 2 1420 960 t 10 CW f (xpoly)2344 960 w 10 R f (= False.)1 317 1 2669 960 t 10 CW f (j)3106 960 w 10 R f (= 3001.)1 306 1 3191 960 t 10 CW f (xpoly)3522 960 w 10 R f (=)3847 960 w 10 CW f (l)3928 960 w 10 R f (.)3988 960 w 10 CW f (erputs)770 1116 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 1116 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 1236 t 10 CW f (erputs)4591 1236 w 10 R f (=)4984 1236 w (False.)1420 1356 w 10 CW f (j)1776 1356 w 10 R f (= 3002.)1 306 1 1861 1356 t 10 CW f (erputs)2192 1356 w 10 R f (=)2577 1356 w 10 CW f (l)2658 1356 w 10 R f (.)2718 1356 w 10 CW f (N)770 1512 w 10 R f ( of length)2 385( Integer array)2 536(- An)1 272 3 1270 1512 t 10 CW f (mmax)2489 1512 w 10 R f (giving the number of sub-steps to be used in the extrapo-)10 2285 1 2755 1512 t (lation.)1420 1632 w 10 CW f (N)1795 1632 w 10 R f (must be strictly monotone increasing and positive.)6 2025 1 1882 1632 t 10 CW f (N)4029 1632 w 10 R f (\(i\) is set by)3 450 1 4089 1632 t 10 CW f (j)4565 1632 w 10 S f (= =)1 55 1 4651 1632 t 10 R f (4000)4732 1632 w 10 S f (+ +)1 55 1 4932 1632 t 10 R f (i.)4987 1632 w (If)1420 1752 w 10 CW f (N)1514 1752 w 10 R f (\(i\) is set, then)3 553 1 1574 1752 t 10 CW f (N)2155 1752 w 10 R f (\(i+1\) is set to 0 by default; so be sure to set)11 1757 1 2215 1752 t 10 CW f (N)4001 1752 w 10 R f (in increasing order of i.)4 950 1 4090 1752 t (For any)1 315 1 1420 1872 t 10 CW f (N)1767 1872 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 1872 t 10 CW f (N)4599 1872 w 10 R f (\(1\) is set,)2 381 1 4659 1872 t (then the user wants)3 804 1 1420 1998 t 10 CW f (N)2261 1998 w 10 R f (\(i\))2321 1998 w 10 S f (\272)2452 1998 w 10 R f (\()2544 1998 w 11 S f (\326` `)1 127 1 2609 1998 t 10 R f (2 \))1 131 1 2678 1998 t 7 I f (i i)1 20 1 2820 1928 t 7 S f (- -)1 39 1 2856 1928 t 7 R f (1)2906 1928 w 10 CW f (N)2986 1998 w 10 R f ( only)1 216(\(1\). If)1 269 2 3046 1998 t 10 CW f (N)3569 1998 w 10 R f (\(1\) and)1 298 1 3629 1998 t 10 CW f (N)3965 1998 w 10 R f (\(2\) are set, then)3 659 1 4025 1998 t 10 CW f (N)4722 1998 w 10 R f (\(i\))4782 1998 w 10 S f (= =)1 55 1 4914 1998 t 10 R f (\()5007 1998 w 10 CW f (N)1420 2118 w 10 R f (\(2\)/)1480 2118 w 10 CW f (N)1624 2118 w 10 R f (\(1\) \))1 175 1 1684 2118 t 10 CW f (N)1885 2118 w 10 R f (\(i-1\) for any)2 489 1 1945 2118 t 10 CW f (N)2460 2118 w 10 R f (\(i\))2520 2118 w 10 S f (= =)1 55 1 2640 2118 t 10 R f (0. If)1 192 1 2744 2118 t 10 CW f (N)2961 2118 w 10 R f (\(3\) is also set, then)4 752 1 3021 2118 t 10 CW f (N)3798 2118 w 10 R f (\(i\) = 2 *)3 325 1 3858 2118 t 10 CW f (N)4208 2118 w 10 R f (\(i-2\), for any)2 512 1 4268 2118 t 10 CW f (N)4805 2118 w 10 R f (\(i\) =)1 175 1 4865 2118 t ( the)1 147(0. See)1 269 2 1420 2238 t 10 CW f (theta)1861 2238 w 10 R f (description above for some default)4 1386 1 2186 2238 t 10 CW f (N)3597 2238 w 10 R f (values.)3682 2238 w 10 CW f (j)4082 2238 w 10 R f (= 4000 + i.)3 440 1 4167 2238 t 10 CW f (N)4632 2238 w 10 R f (\(i\) =)1 175 1 4692 2238 t 10 CW f (i)4892 2238 w 10 R f (.)4952 2238 w (The following table summarizes the values that can be set by)10 2435 1 720 2394 t 10 CW f (TTGUV)3180 2394 w 10 S f (_ __________________________________)1 1725 1 2017 2474 t 10 R f ( to)1 103( Set)1 279( Default)1 577(Name j)1 605 4 2128 2594 t 10 S f (_ __________________________________)1 1725 1 2017 2614 t (_ __________________________________)1 1725 1 2017 2634 t 10 CW f (theta)2097 2814 w 10 R f (1 1)1 492 1 2694 2814 t 10 CW f (f)3546 2814 w (beta)2127 2934 w 10 R f (2 1)1 492 1 2694 2934 t 10 CW f (f)3546 2934 w (gamma)2097 3054 w 10 R f (3 1)1 492 1 2694 3054 t 10 CW f (f)3546 3054 w (delta)2097 3174 w 10 R f (4 0)1 492 1 2694 3174 t 10 CW f (f)3546 3174 w 10 S f (_ __________________________________)1 1725 1 2017 3194 t 10 CW f (hfract)2067 3314 w 10 R f (1001 1)1 567 1 2619 3314 t 10 CW f (r)3546 3314 w (egive)2097 3434 w 10 R f (1002 100)1 617 1 2619 3434 t 10 CW f (r)3546 3434 w 10 S f (_ __________________________________)1 1725 1 2017 3454 t 10 CW f (kj)2187 3574 w 10 R f (2001 0)1 567 1 2619 3574 t 10 CW f (i)3546 3574 w (minit)2097 3694 w 10 R f (2002 10)1 592 1 2619 3694 t 10 CW f (i)3546 3694 w (maxit)2097 3814 w 10 R f (2003 50)1 592 1 2619 3814 t 10 CW f (i)3546 3814 w (kmax)2127 3934 w 10 R f (2004 10)1 592 1 2619 3934 t 10 CW f (i)3546 3934 w (kinit)2097 4054 w 10 R f (2005 2)1 567 1 2619 4054 t 10 CW f (i)3546 4054 w (mmax)2127 4174 w 10 R f (2006 15)1 592 1 2619 4174 t 10 CW f (i)3546 4174 w (mxq)2157 4294 w 10 R f (2008)2619 4294 w 10 CW f (0 i)1 475 1 3131 4294 t (myq)2157 4414 w 10 R f (2009)2619 4414 w 10 CW f (0 i)1 475 1 3131 4414 t (la)2187 4534 w 10 R f (2010 1)1 567 1 2619 4534 t 10 CW f (i)3546 4534 w 10 S f (_ __________________________________)1 1725 1 2017 4554 t 10 CW f (xpoly)2097 4674 w 10 R f (3001)2619 4674 w 10 CW f (False l)1 595 1 3011 4674 t (erputs)2067 4794 w 10 R f (3002)2619 4794 w 10 CW f (False l)1 595 1 3011 4794 t 10 S f (_ __________________________________)1 1725 1 2017 4814 t 10 CW f (N)2170 4934 w 10 R f (\(i\) 4000+i)1 631 1 2230 4934 t 10 S f (- -)1 55 1 3133 4934 t 10 CW f (i)3546 4934 w 10 S f ( \347)1 -1725(_ __________________________________)1 1725 2 2017 4954 t (\347)2017 4874 w (\347)2017 4774 w (\347)2017 4674 w (\347)2017 4574 w (\347)2017 4474 w (\347)2017 4374 w (\347)2017 4274 w (\347)2017 4174 w (\347)2017 4074 w (\347)2017 3974 w (\347)2017 3874 w (\347)2017 3774 w (\347)2017 3674 w (\347)2017 3574 w (\347)2017 3474 w (\347)2017 3374 w (\347)2017 3274 w (\347)2017 3174 w (\347)2017 3074 w (\347)2017 2974 w (\347)2017 2874 w (\347)2017 2774 w (\347)2017 2674 w (\347)2017 2574 w (\347)2502 4954 w (\347)2502 4874 w (\347)2502 4774 w (\347)2502 4674 w (\347)2502 4574 w (\347)2502 4474 w (\347)2502 4374 w (\347)2502 4274 w (\347)2502 4174 w (\347)2502 4074 w (\347)2502 3974 w (\347)2502 3874 w (\347)2502 3774 w (\347)2502 3674 w (\347)2502 3574 w (\347)2502 3474 w (\347)2502 3374 w (\347)2502 3274 w (\347)2502 3174 w (\347)2502 3074 w (\347)2502 2974 w (\347)2502 2874 w (\347)2502 2774 w (\347)2502 2674 w (\347)2502 2574 w (\347)2936 4954 w (\347)2936 4874 w (\347)2936 4774 w (\347)2936 4674 w (\347)2936 4574 w (\347)2936 4474 w (\347)2936 4374 w (\347)2936 4274 w (\347)2936 4174 w (\347)2936 4074 w (\347)2936 3974 w (\347)2936 3874 w (\347)2936 3774 w (\347)2936 3674 w (\347)2936 3574 w (\347)2936 3474 w (\347)2936 3374 w (\347)2936 3274 w (\347)2936 3174 w (\347)2936 3074 w (\347)2936 2974 w (\347)2936 2874 w (\347)2936 2774 w (\347)2936 2674 w (\347)2936 2574 w (\347)3386 4954 w (\347)3386 4874 w (\347)3386 4774 w (\347)3386 4674 w (\347)3386 4574 w (\347)3386 4474 w (\347)3386 4374 w (\347)3386 4274 w (\347)3386 4174 w (\347)3386 4074 w (\347)3386 3974 w (\347)3386 3874 w (\347)3386 3774 w (\347)3386 3674 w (\347)3386 3574 w (\347)3386 3474 w (\347)3386 3374 w (\347)3386 3274 w (\347)3386 3174 w (\347)3386 3074 w (\347)3386 2974 w (\347)3386 2874 w (\347)3386 2774 w (\347)3386 2674 w (\347)3386 2574 w (\347)3742 4954 w (\347)3742 4874 w (\347)3742 4774 w (\347)3742 4674 w (\347)3742 4574 w (\347)3742 4474 w (\347)3742 4374 w (\347)3742 4274 w (\347)3742 4174 w (\347)3742 4074 w (\347)3742 3974 w (\347)3742 3874 w (\347)3742 3774 w (\347)3742 3674 w (\347)3742 3574 w (\347)3742 3474 w (\347)3742 3374 w (\347)3742 3274 w (\347)3742 3174 w (\347)3742 3074 w (\347)3742 2974 w (\347)3742 2874 w (\347)3742 2774 w (\347)3742 2674 w (\347)3742 2574 w 10 R f (Note that)1 369 1 720 5170 t 10 CW f (mxq)1114 5170 w 10 S f (= =)1 55 1 1319 5170 t 10 R f (0 means that)2 505 1 1414 5170 t 10 CW f (kx)1944 5170 w 10 R f (points will be used in)4 856 1 2089 5170 t 10 I f (x x)1 44 1 2970 5170 t 10 R f ( for)1 141( Similarly)1 423(, by default.)2 477 3 3014 5170 t 10 CW f (myq)4080 5170 w 10 R f (.)4260 5170 w ( of)1 119(The Double precision version)3 1214 2 970 5326 t 10 CW f (TTGUV)2339 5326 w 10 R f (is)2675 5326 w 10 CW f (DTTGUV)2778 5326 w 10 R f (, with all Real arguments typed Double preci-)7 1902 1 3138 5326 t (sion, except)1 477 1 720 5446 t 10 CW f (hfract)1222 5446 w 10 R f (and)1607 5446 w 10 CW f (egive)1776 5446 w 10 R f (which remain Real.)2 779 1 2101 5446 t 10 B f (Run-time Statistics.)1 838 1 720 5686 t 10 R f (A subroutine is provided to print run-time statistics for)8 2188 1 970 5842 t 10 CW f (TTGU)3183 5842 w 10 R f ( statement)1 408(. The)1 230 2 3423 5842 t (Call TTGUX)1 530 1 870 6082 t (will print a line of the form:)6 1116 1 720 6442 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 5 780 6802 t (The fields of this line refer to)6 1920 1 720 7162 t cleartomark showpage saveobj restore %%EndPage: 45 26 %%Page: 46 27 /saveobj save def mark 27 pagesetup 10 R f (- 46 -)2 216 1 2772 480 t 10 CW f (j)770 840 w 10 R f ( number of Jacobian evaluations.)4 1317(- The)1 305 2 1270 840 t 10 CW f (f)770 996 w 10 R f ( number of factorizations of the Jacobian.)6 1660(- The)1 305 2 1270 996 t 10 CW f (ts)770 1152 w 10 R f ( number of time-steps.)3 899(- The)1 305 2 1270 1152 t 10 CW f (ss)770 1308 w 10 R f ( number of sub-steps.)3 860(- The)1 305 2 1270 1308 t 10 CW f (nit)770 1464 w 10 R f ( number of Newton iterations.)4 1201(- The)1 305 2 1270 1464 t 10 CW f (nd)770 1620 w 10 R f ( number of predicted Newton failures \( error increasing \).)9 2293(- The)1 305 2 1270 1620 t 10 CW f (nf)770 1776 w 10 R f ( number of Newton failures \( more than maxit iterations \).)10 2321(- The)1 305 2 1270 1776 t 10 CW f (r)770 1932 w 10 R f ( number of restarts.)3 776(- The)1 305 2 1270 1932 t (If)970 2088 w 10 CW f (TTGUX)1063 2088 w 10 R f (is invoked by the user while inside)6 1400 1 1390 2088 t 10 CW f (TTGU)2818 2088 w 10 R f (, the statistics reported will be the current values.)8 1982 1 3058 2088 t (If invoked outside)2 727 1 720 2208 t 10 CW f (TTGU)1472 2208 w 10 R f (, the statistics will be those of the last call to)10 1769 1 1712 2208 t 10 CW f (TTGU)3506 2208 w 10 R f (.)3746 2208 w 10 B f (Initial Conditions)1 755 1 720 2448 t 10 R f (The subroutine)1 617 1 970 2604 t 10 CW f (ICON)1627 2604 w 10 R f (determines the initial values of)4 1292 1 1907 2604 t 10 CW f (U)3239 2604 w 10 R f ( in)1 119(by calling the underlying static solver)5 1582 2 3339 2604 t 10 CW f (TTGU)720 2724 w 10 R f (with the problem)2 683 1 985 2724 t 10 I f (u u)1 50 1 2569 2904 t 10 S f (- -)1 55 1 2643 2904 t 10 R f (u \( data \))3 306 1 2714 2904 t 10 S f (= =)1 55 1 3036 2904 t 10 R f (0)3107 2904 w 10 I f (. .)1 25 1 3165 2904 t 10 R f (For one rectangle calling)3 1025 1 720 3084 t 10 CW f (TSL2W)1815 3084 w 10 R f ( if one)2 277( However,)1 451( suggested in [15], is appropriate.)5 1389(by E.H. Grosse, as)3 773 4 2150 3084 t (called)720 3204 w 10 CW f (TSL2W)984 3204 w 10 R f ( conditions would agree along)4 1204(for more than one rectangle, it is highly unlikely that the initial)11 2526 2 1310 3204 t ( the bottom level subroutines)4 1161( Calling)1 345(interfaces. One really has to consider the total geometry of the system.)11 2814 3 720 3324 t (of)720 3444 w 10 CW f (TTGU)831 3444 w 10 R f ( work is done in setting up the under-)8 1509(with a problem that has no derivative means that some extra)10 2432 2 1099 3444 t ( con-)1 210( most time)2 437( the)1 179(lying matrix problem, but unless the order of the approximation is say greater than 4,)14 3494 4 720 3564 t ( involves solving the linear system and not constructing the system. Thus)11 2927(suming portion of the computation)4 1393 2 720 3684 t (using)720 3804 w 10 CW f (TTGU)971 3804 w 10 R f ( the solution is an easy, not too)7 1309(to determine the initial values of the B-spline coefficients of)9 2486 2 1245 3804 t ( rectan-)1 305(time-consuming solution. In fact, this is a good way of generally approximating data on a union of)16 4015 2 720 3924 t (gles.)720 4044 w cleartomark showpage saveobj restore %%EndPage: 46 27 %%Page: 1 28 /saveobj save def mark 28 pagesetup 10 B f (Bibliography)2598 840 w 10 R f ( deBoor,)1 344([1] C.)1 342 2 720 1596 t 10 B f (A Practical Guide to Splines,)4 1231 1 1431 1596 t 10 R f (Springer, New York, Applied Math. Sciences 27, 1978.)7 2219 1 2687 1596 t ( de Boor, "On Uniform Approximation by Splines,")7 2073([2] C.)1 342 2 720 1752 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 1752 t 10 B f (1,)3779 1752 w 10 R f (219-235\(1968\).)3879 1752 w ( de Boor, "On Calculating with B-splines,")6 1715([3] C.)1 342 2 720 1908 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 1908 t 10 B f (6,)3421 1908 w 10 R f (50-62\(1972\).)3521 1908 w ( Ver-)1 210( Bulirsch and J. Stoer, "Fehlerabschatzungen und Extrapolation mit rationalen Funktionen bei)11 3768([4] R.)1 342 3 720 2064 t (fahren vom Richardson-Typus,")2 1286 1 970 2184 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 2184 t 10 B f (6,)2889 2184 w 10 R f (413-427\(1964\).)2989 2184 w ( and J. Stoer, "Numerical Treatment of Ordinary Differential Equations by Extrapolation)11 3609( Bulirsch)1 369([5] R.)1 342 3 720 2340 t (Methods,")970 2460 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 2460 t 10 B f (8,)2019 2460 w 10 R f (1-13\(1966\).)2119 2460 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 2616 t (ods,")970 2736 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 2736 t 10 B f (8,)1808 2736 w 10 R f (93-104\(1966\).)1908 2736 w ( Dahlquist, "A Special Stability Problem for Linear Multistep Methods,")9 2901([7] G.)1 347 2 720 2892 t 10 I f ( T)1 0( IT)1 56(B BI)1 94 3 3993 2892 t 10 B f (3,)4168 2892 w 10 R f (27-43\(1963\).)4268 2892 w ( Dahlquist, "Stability Questions for Some Numerical Methods for Ordinary Differential Equa-)11 3973([8] G.)1 347 2 720 3048 t (tions,")970 3168 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 3168 t 10 B f (15,)2530 3168 w 10 R f (147-158\(1963\).)2680 3168 w ( and N.L. Schryer, "The PORT Mathematical Subroutine Library,")8 2787( Fox, A.D. Hall)3 664([9] P.A.)1 428 3 720 3324 t 10 I f ( ,)1 0( S,)1 25( MS)1 50( OM)1 83(T TO)1 128 5 4639 3324 t 10 B f (4,)4965 3324 w 10 R f (104-126\(1978\).)970 3444 w ( Automatic Integration of Ordinary Differential Equations,")6 2438( Gear, "The)2 480([10] C.W.)1 461 3 720 3600 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 3600 t 10 B f (14,)4698 3600 w 10 R f (176-)4857 3600 w (179\(1971\).)970 3720 w ( "Repeated Extrapolation to the Limit in the Numerical Solution of Ordinary Differential)12 3560( Gragg,)1 299([11] W.B.)1 461 3 720 3876 t (Equations," Thesis, UCLA \(1963\).)3 1390 1 970 3996 t ( Gragg, "On Extrapolation Algorithms for Ordinary Initial Value Problems")9 3034([12] W.B.)1 461 2 720 4152 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 4152 t 10 B f (2,)970 4272 w 10 R f (384-403\(1965\).)1070 4272 w ( "Lecture Notes on Extrapolation Methods," presented at the SIAM National Meeting,)11 3551( Gragg,)1 308([13] W.B.)1 461 3 720 4428 t ( Conference on Ordinary Differential Equations, Dundee, Scot-)7 2621(Washington, June 1971, and at the)5 1449 2 970 4548 t (land, August, 1971.)2 786 1 970 4668 t ( H. Grosse, "Tensor Spline Approximation,")5 1865([14] E.)1 336 2 720 4824 t 10 B f (Linear Algebra and its Applications)4 1607 1 2964 4824 t 10 R f (, 34, 29-41)2 469 1 4571 4824 t (\(1980\).)970 4944 w ( Solving Partial Differential Equations in Two)6 1897( Kaufman and N.L. Schryer "TTGR-A Package for)7 2087([15] L.)1 336 3 720 5100 t (Space Variables," Bell Laboratories Computing Science Technical Report #135, 1985.)9 3458 1 970 5220 t ( Richtmeyer and K.W. Morton,)4 1275([16] R.D.)1 439 2 720 5376 t 10 B f (Difference Methods for Initial Value Problems,)5 2038 1 2465 5376 t 10 R f (Interscience,)4534 5376 w (New York, 1967.)2 693 1 970 5496 t ( Schryer, "An Extrapolation Step-Size and Order Monitor for use in Solving Differential Equa-)13 3887([17] N.L.)1 433 2 720 5652 t (tions," Proceedings ACM National Meeting, San Diego, 1974.)7 2498 1 970 5772 t ( Schryer, "An Extrapolation Step-Size and Order Monitor for use in Solving Differential Equa-)13 3887([18] N.L.)1 433 2 720 5928 t (tions," in preparation.)2 868 1 970 6048 t ( B-splines, for Solving Differential Equa-)5 1749( Schryer, "A Tutorial on Galerkin's Method, using)7 2138([19] N.L.)1 433 3 720 6204 t (tions," Bell Laboratories Computing Science Technical Report #52, 1976.)8 2958 1 970 6324 t ( Differential Equations in One Space Variable", Bell Laboratories Computing)9 3184( Schryer, "Partial)2 703([20] N.L.)1 433 3 720 6480 t (Science Technical Report #115, 1984.)4 1525 1 970 6600 t ( Schultz, ")2 462([21] M.)1 364 2 720 6756 t 10 I f (L L)1 56 1 1546 6756 t 7 S f (\245)1613 6716 w 10 R f (-)1672 6756 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 6756 t 10 R f (",)3184 6756 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 6756 t 10 B f (6)4965 6756 w 10 R f (,)5015 6756 w (161-183\(1969\).)970 6876 w ( der Houven, "Algorithm 621. Software with Low Storage Require-)9 2809( P. Sommeijer and P. J. van)6 1169([22] B.)1 342 3 720 7032 t (ments for Two-Dimensional Parabolic Differential Equations,")5 2570 1 970 7152 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 7152 t 10 B f (10,)4915 7152 w 10 R f (378-396\(1985\).)970 7272 w cleartomark showpage saveobj restore %%EndPage: 1 28 %%Page: 2 29 /saveobj save def mark 29 pagesetup 10 R f (\261 B-2 \261)2 300 1 2730 480 t ( Two-Dimensional Nonlinear Partial Differ-)4 1771( K. Melgaard and R.F. Sincovec "General Software for)8 2202([23] D.)1 347 3 720 840 t (ential Equations,")1 713 1 970 960 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 960 t 10 B f (7,)2995 960 w 10 R f (106-125\(1981\).)3095 960 w ( Expansions for the Error of Discretization Algorithms for Non-Linear)9 3021( Stetter, "Asymptotic)2 888([24] H.J.)1 411 3 720 1116 t (Functional Equations,")1 919 1 970 1236 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 1236 t 10 B f (7,)2522 1236 w 10 R f (18-31\(1965\).)2622 1236 w ( Strang and G. Fix,)4 761([25] G.)1 347 2 720 1392 t 10 B f (An Analysis of the Finite Element Method,)6 1819 1 1853 1392 t 10 R f (Prentice-Hall, New York, 1973.)3 1275 1 3697 1392 t (Appendix 1)1 469 1 2770 1512 t 10 B f (Examples - Programs)2 921 1 2544 1752 t 10 R f ( use of)2 276(The program examples given below are intended to both illustrate the)10 2819 2 970 2148 t 10 CW f (TTGU)4095 2148 w 10 R f (and provide pro-)2 675 1 4365 2148 t ( contemplating using)2 843( Anyone)1 368(totypes for a prospective user.)4 1212 3 720 2268 t 10 CW f (TTGU)3170 2268 w 10 R f (would be well advised to pick an exam-)7 1603 1 3437 2268 t (ple program that invokes those capabilities of)6 1874 1 720 2388 t 10 CW f (TTGU)2628 2388 w 10 R f ( type it in \(or)4 562(the intended problem will require, and)5 1576 2 2902 2388 t ( running the example and confirming the correct-)7 1972( After)1 261( code from the author\).)4 921(obtain a copy of the example)5 1166 4 720 2508 t (ness of the program, the)4 986 1 720 2628 t 10 CW f (AF)1737 2628 w 10 R f (and)1888 2628 w 10 CW f (BC)2063 2628 w 10 R f (subroutines specifying the)2 1061 1 2214 2628 t 10 B f (pde)3306 2628 w 10 R f (-)3462 2628 w 10 B f (bc)3495 2628 w 10 R f (may simply be altered to solve the)6 1413 1 3627 2628 t ( produce a correct pro-)4 911( progression makes it much more likely that the user will easily)11 2556( This)1 230(user's problem.)1 623 4 720 2748 t (gram unit for the problem at hand.)6 1373 1 720 2868 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 3024 t (ning on small machines, like Vaxen, not too onerous a chore for folks installing and testing the package.)17 4167 1 720 3144 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 3300 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 3420 t ( section 5 describing the)4 991(user must have read)3 807 2 720 3540 t 10 CW f (TTGU)2548 3540 w 10 R f (software before proceeding in this appendix, otherwise)6 2222 1 2818 3540 t (the reading will be dark and obscure.)6 1477 1 720 3660 t (Before invoking)1 652 1 970 3816 t 10 CW f (TTGU)1647 3816 w 10 R f (the user must)2 533 1 1912 3816 t 10 S f (\267)970 3972 w 10 R f (Make a B-spline mesh.)3 921 1 1041 3972 t 10 S f (\267)970 4128 w 10 R f (Make initial conditions for the B-spline coefficients)6 2070 1 1041 4128 t 10 B f (U)3136 4128 w 10 R f (in \(3.1\).)1 319 1 3233 4128 t 10 S f (\267)970 4284 w 10 R f (Write subroutines)1 713 1 1041 4284 t 10 S f (\267)1220 4440 w 10 CW f (AF)1326 4440 w 10 R f (- to evaluate)2 493 1 1471 4440 t 10 B f (a)1989 4440 w 10 R f (and)2064 4440 w 10 B f (f)2233 4440 w 10 R f (in \(2.1\).)1 319 1 2291 4440 t 10 S f (\267)1220 4596 w 10 CW f (BC)1326 4596 w 10 R f (- to evaluate)2 493 1 1471 4596 t 10 B f (b)1989 4596 w 10 R f (in \(2.2\).)1 319 1 2070 4596 t 10 S f (\267)1220 4752 w 10 CW f (HANDLE)1326 4752 w 10 R f (- to output \(print\) the solution results.)6 1503 1 1711 4752 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 4908 t (ditions \()1 331 1 720 5028 t 10 B f (ic)1076 5028 w 10 R f (s \) for)2 238 1 1148 5028 t 10 B f (u)1411 5028 w 10 R f (are now briefly described.)3 1041 1 1492 5028 t 10 B f (Mesh Making)1 592 1 720 5268 t 10 R f ( is)1 95( There)1 285(The PORT Library [9] has several B-spline mesh generation subroutines available.)10 3338 3 970 5424 t 10 CW f (UMB)4716 5424 w 10 R f (for)4924 5424 w ( meshes that are the union of)6 1154( creating B-spline)2 714( For)1 191(generating uniform B-spline meshes on a given interval.)7 2261 4 720 5544 t ( intervals there are)3 751(uniform meshes over basic contiguous)4 1549 2 720 5664 t 10 CW f (LUMB)3049 5664 w 10 R f (and)3318 5664 w 10 CW f (PUMB)3491 5664 w 10 R f (.)3731 5664 w 10 CW f (LUMB)3880 5664 w 10 R f (uses the same number)3 891 1 4149 5664 t (of mesh points in each basic interval and)7 1628 1 720 5784 t 10 CW f (PUMB)2408 5784 w 10 R f (allows that number to vary by interval.)6 1551 1 2673 5784 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 5940 t (using)720 6060 w 10 CW f (LUMB)967 6060 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 6060 t ( a uniform mesh, but it would require roughly)8 1845(example 5 below could be solved completely on)7 1935 2 720 6180 t 10 B f (105,625)4527 6180 w 10 R f (grid)4879 6180 w (points to do it for)4 703 1 720 6300 t 10 I f (k k)1 44 1 1450 6300 t 10 S f (= =)1 55 1 1543 6300 t 10 R f ( uniform mesh scheme given for)5 1300( the combination of non-uniform mesh and)6 1727(2. Using)1 366 3 1647 6300 t ( that the run-time and memory requirements of)7 1886( Remember)1 490( the job nicely.)3 600(that problem, 3125 grid points do)5 1344 4 720 6420 t 10 CW f (TTGU)720 6540 w 10 R f ( little thought given to mesh con-)6 1352( A)1 126( used.)1 237(are proportional to a power of the number of mesh points)10 2335 4 990 6540 t (struction can save a lot of computer run-time and memory.)9 2346 1 720 6660 t cleartomark showpage saveobj restore %%EndPage: 2 29 %%Page: 3 30 /saveobj save def mark 30 pagesetup 10 R f (\261 B-3 \261)2 300 1 2730 480 t 10 B f (Initial Conditions for u.)3 1013 1 720 840 t 10 R f (There are)1 388 1 970 996 t 10 B f (ic)1393 996 w 10 R f (s for)1 190 1 1465 996 t 10 B f (u)1690 996 w 10 R f (and these must be converted into)5 1362 1 1781 996 t 10 B f (ic)3178 996 w 10 R f (s for)1 191 1 3250 996 t 10 B f (U)3477 996 w 10 R f (, the B-spline coefficients for)4 1211 1 3549 996 t 10 B f (u)4796 996 w 10 R f (, see)1 188 1 4852 996 t ( case is when the)4 680( simplest)1 361(\(3.1\). The)1 423 3 720 1116 t 10 B f (ic)2210 1116 w 10 R f (s for)1 181 1 2282 1116 t 10 B f (u)2489 1116 w 10 R f ( simply setting all the spline coefficients to)7 1726( By)1 168(are a constant.)2 575 3 2571 1116 t (that constant, the spline)3 944 1 720 1236 t 10 B f (is)1689 1236 w 10 R f (the constant and we are done.)5 1180 1 1781 1236 t (If the)1 213 1 970 1392 t 10 B f (ic)1208 1392 w 10 R f (s for)1 180 1 1280 1392 t 10 I f (u u)1 50 1 1485 1392 t 10 R f (are not constant, then there is)5 1170 1 1560 1392 t 10 CW f (ICON)2755 1392 w 10 R f (available, see section 4.)3 945 1 3020 1392 t 10 B f (The Examples)1 609 1 720 1632 t 10 R f ( arithmetic, under the)3 863(All examples reported here were run on a VAX 11/8550 using double-precision)11 3207 2 970 1788 t 9 R f (UNIX)720 1908 w 10 S f (\322)945 1908 w 10 R f (operating system, Research Tenth Edition.)4 1698 1 1049 1908 t 10 B f (Example 1 - A Simple Heat Equation.)6 1604 1 720 2148 t 10 R f (As a simple example of the use of)7 1356 1 970 2304 t 10 CW f (TTGU)2351 2304 w 10 R f (, consider solving the scalar heat equation)6 1672 1 2591 2304 t 10 I f (a a)1 50 1 1220 2484 t 7 R f (\( 1 \))2 91 1 1281 2444 t 10 S f (= =)1 55 1 1437 2484 t 10 I f (u u)1 50 1 1541 2484 t 10 S f (+ +)1 55 1 1631 2484 t 10 I f (u u)1 50 1 1726 2484 t 7 I f (x x)1 31 1 1787 2504 t 10 S f (+ +)1 55 1 1866 2484 t 10 I f (. .)1 25 1 1961 2484 t 10 R f (1)1994 2484 w 10 I f (u u)1 50 1 2076 2484 t 7 I f (y y)1 31 1 2137 2504 t 10 R f (,)2184 2484 w 10 I f (a a)1 50 1 1228 2664 t 7 R f (\( 2 \))2 91 1 1289 2624 t 10 S f (= =)1 55 1 1445 2664 t 10 I f (u u)1 50 1 1549 2664 t 10 S f (+ +)1 55 1 1639 2664 t 10 I f (u u)1 50 1 1734 2664 t 7 I f (y y)1 31 1 1795 2684 t 10 S f (+ +)1 55 1 1874 2664 t 10 I f (. .)1 25 1 1969 2664 t 10 R f (1)2002 2664 w 10 I f (u u)1 50 1 2084 2664 t 7 I f (x x)1 31 1 2145 2684 t 10 R f (, \(A.1\))1 2848 1 2192 2664 t 10 I f (f f)1 28 1 1236 2844 t 10 S f (= =)1 55 1 1329 2844 t 10 I f (u u)1 50 1 1433 2844 t 7 I f (t t)1 20 1 1494 2864 t 10 S f (+ +)1 55 1 1562 2844 t 10 I f (u u)1 50 1 1657 2844 t 7 I f (x x)1 31 1 1718 2864 t 10 S f (+ +)1 55 1 1797 2844 t 10 I f (u u)1 50 1 1892 2844 t 7 I f (y y)1 31 1 1953 2864 t 10 S f (- -)1 55 1 2032 2844 t 10 I f (g g)1 50 1 2127 2844 t 10 R f (\()2185 2844 w 10 I f (t t)1 28 1 2226 2844 t 10 R f (,)2262 2844 w 10 I f (x x)1 44 1 2295 2844 t 10 R f (,)2347 2844 w 10 I f (y y)1 44 1 2380 2844 t 10 R f (\))2432 2844 w (on the T-shaped domain)3 968 1 720 3024 t 1800 3186 1800 3906 Dl 2520 3186 1800 3186 Dl 2520 3906 2520 3186 Dl 1800 3906 2520 3906 Dl (-1,1)1642 3206 w (-1,0)1642 3926 w (0,1)2458 3146 w 2520 3186 2520 3906 Dl 3240 3186 2520 3186 Dl 3240 3906 3240 3186 Dl 2520 3906 3240 3906 Dl (1,1)3178 3146 w 3240 3186 3240 3906 Dl 3960 3186 3240 3186 Dl 3960 3906 3960 3186 Dl 3240 3906 3960 3906 Dl (2,1)3960 3206 w (2,0)3960 3926 w 2520 3906 2520 4626 Dl 3240 3906 2520 3906 Dl 3240 4626 3240 3906 Dl 2520 4626 3240 4626 Dl (0,-1 1,-1)1 1036 1 2362 4646 t (0,0 1,0)1 1042 1 2359 3998 t (with)1420 4806 w 10 B f (bc)1623 4806 w 10 R f (s \(4.2\))1 255 1 1723 4806 t 10 B f (b)1220 4986 w 10 S f (= =)1 55 1 1325 4986 t 10 I f (u u)1 50 1 1429 4986 t 10 R f (\()1487 4986 w 10 I f (t t)1 28 1 1528 4986 t 10 R f (,)1564 4986 w 10 I f (x x)1 44 1 1597 4986 t 10 R f (,)1649 4986 w 10 I f (y y)1 44 1 1682 4986 t 10 R f (\))1734 4986 w 10 S f (- -)1 55 1 1815 4986 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 4986 t 10 R f (\(A.2\))4827 4986 w (The solution is)2 601 1 1420 5166 t 10 I f (u u)1 50 1 2049 5166 t 10 S f (\272)2140 5166 w 10 I f ( y)1 0( y)1 76( x)1 0( x)1 76(t t)1 28 5 2236 5166 t 10 R f ( initial conditions are taken to)5 1211( The)1 209(, which can be gotten exactly.)5 1204 3 2416 5166 t (be 0.)1 194 1 1420 5286 t ( unit, solves \(A.1\)-\(A.2\) using)4 1227(The following program)2 940 2 1670 5442 t 10 CW f (TTGU)3867 5442 w 10 R f (, with a linear B-spline)4 933 1 4107 5442 t (\()1420 5562 w 10 I f (k k)1 44 1 1453 5562 t 10 S f (= =)1 55 1 1546 5562 t 10 R f ( of)1 110(2\) over a spatial mesh consisting of 3 equally spaced, distinct points on each side)14 3280 2 1650 5562 t ( evolution carried out to 10)5 1135(each rectangle in the domain with the time)7 1763 2 1420 5682 t 7 S f (- -)1 39 1 4329 5642 t 7 R f (2)4379 5642 w 10 R f (absolute accu-)1 583 1 4457 5682 t (racy.)1420 5802 w (The main program uses several PORT [9] library subprograms.)8 2526 1 1670 5958 t 10 S f (\267)970 6114 w 10 CW f (ISTKIN)1078 6114 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 6114 t (precision items, consistent with the declaration for)6 2029 1 970 6234 t 10 CW f (Ds)3026 6234 w 10 R f ( precision alias of the stack in the)7 1341(in the double)2 526 2 3173 6234 t (common region)1 630 1 970 6354 t 10 CW f (CSTAK)1625 6354 w 10 R f (.)1925 6354 w 10 S f (\267)970 6510 w 10 CW f (ENTER)1080 6510 w 10 R f (and)1409 6510 w 10 CW f (LEAVE)1582 6510 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 6510 t (all allocations made after the)4 1153 1 970 6630 t 10 CW f (ENTER)2148 6630 w 10 R f (but before the)2 554 1 2473 6630 t 10 CW f (LEAVE)3087 6630 w 10 R f (are released by the)3 744 1 3412 6630 t 10 CW f (LEAVE)4181 6630 w 10 R f (.)4481 6630 w 10 S f (\267)970 6786 w 10 CW f (IDUMB)1076 6786 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 1401 6786 t 10 S f (\267)970 6942 w 10 CW f (ISTKGT)1121 6942 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 6942 t 10 CW f (iU =)1 286 1 4754 6942 t (ISTKGT\(L,4\))970 7062 w 10 R f ( pointer)1 318(sets the)1 308 2 1666 7062 t 10 CW f (iU)2327 7062 w 10 R f (so that locations)2 670 1 2482 7062 t 10 CW f (Ws\(iU\), ..., Ws\(iU+L-1\))2 1400 1 3222 7062 t 10 R f (are avail-)1 383 1 4657 7062 t (able for the B-spline coefficients of the solution.)7 1936 1 970 7182 t cleartomark showpage saveobj restore %%EndPage: 3 30 %%Page: 4 31 /saveobj save def mark 31 pagesetup 10 R f (\261 B-4 \261)2 300 1 2730 480 t 10 S f (\267)970 840 w 10 CW f (SETD)1079 840 w 10 R f (sets an array to a given Double precision constant.)8 2034 1 1347 840 t 10 CW f (SETD)3504 840 w 10 R f (provides the constant)2 855 1 3772 840 t 10 B f (ic)4655 840 w 10 R f ('s \(A.3\))1 313 1 4727 840 t ( a constant, then)3 670(via the B-spline coefficients \(3.1\), since if all the B-spline coefficients are equal to)13 3400 2 970 960 t (the B-spline itself is identically equal to that constant \(see Appendix 1\).)11 2864 1 970 1080 t 10 S f (\267)970 1236 w 10 CW f (WRAPUP)1076 1236 w 10 R f (checks that a run has terminated without errors and prints out the stack space used.)14 3302 1 1461 1236 t ( we have the)3 501( Because)1 382(The PORT Library stack is used in a rather interesting way to setup the meshes.)14 3187 3 970 1392 t ( the)1 161(array IXB,)1 441 2 720 1512 t 10 I f (x x)1 44 1 1361 1512 t 10 R f (meshes for all the rectangles do not have to occupy consecutive locations in one long)14 3596 1 1444 1512 t ( for the first rectangle in an array, skip some spaces, insert the mesh for the sec-)16 3197(array. One can place a mesh)5 1123 2 720 1632 t ( uniform meshes in the PORT stack and returns as its)10 2246(ond rectangle, etc. The subroutine IDUMB makes)6 2074 2 720 1752 t (value a pointer to the mesh in the stack. In our example with 8 uniform meshes we can use 8 calls to)21 4320 1 720 1872 t ( their beginning locations in IXB and IYB. Since our meshes are now)12 2825(IDUMB to make the meshes and put)6 1495 2 720 1992 t ( name of the PORT stack, namely)6 1363(in the stack, when we call DTTGU we use as the X array and Y array, the)16 2957 2 720 2112 t (WS.)720 2232 w ( at the corners, center, and midpoint of each)8 1775(At the end of each time-step the solution is printed out at)11 2295 2 970 2388 t ( main program is)3 680( The)1 205(side for each rectangle in the domain.)6 1499 3 720 2508 t 10 CW f ( program)1 480(c main)1 420 2 720 2748 t (c to solve the heat equation with solution u == t*x*y,)10 3240 1 720 2868 t ( . \( u + ux + .1 * uy, u + uy + .1 * ux \) = ut + ux + uy +g\(x,t\))24 3840(c grad)1 480 2 720 2988 t (common /cstak/ ds)2 1020 1 1080 3108 t (double precision ds\(350000\))2 1620 1 1080 3228 t (integer ixb\(4\), iyb\(4\), nxr\(4\), nyr\(4\), kxr\(4\), kyr\(4\))6 3240 1 1080 3348 t (external handlu, bc, af)3 1380 1 1080 3468 t (integer ndx, ndy, istkgt, is\(1000\), iu, nu, kx, ky, idumb)9 3420 1 1080 3588 t (real errpar\(2\), rs\(1000\))2 1440 1 1080 3708 t (logical ls\(1000\))1 960 1 1080 3828 t (complex cs\(500\))1 900 1 1080 3948 t (double precision tstart, dt, tstop, ws\(500\))5 2580 1 1080 4068 t (c the port library stack and its aliases.)7 2460 1 720 4188 t (equivalence \(ds\(1\), cs\(1\), ws\(1\), rs\(1\), is\(1\), ls\(1\)\))6 3240 1 1080 4308 t (c initialize the port library stack length.)6 2580 1 720 4428 t (call istkin\(350000, 4\))2 1320 1 1080 4548 t (call enter\(1\))1 780 1 1080 4668 t (nu = 1)2 360 1 1080 4788 t (kx = 2)2 360 1 1080 4908 t (ky = 2)2 360 1 1080 5028 t (ndx = 3)2 420 1 1080 5148 t (ndy = 3)2 420 1 1080 5268 t (nr=4)1080 5388 w (tstart = 0.d0)2 780 1 1080 5508 t (tstop = 1.d0)2 720 1 1080 5628 t (dt = 1)2 360 1 1080 5748 t (errpar\(1\) = 1e-2)2 960 1 1080 5868 t (errpar\(2\) = 1e-4)2 960 1 1080 5988 t (c uniform grid.)2 900 1 720 6108 t (c)720 6228 w (c make grid for t-shaped region)5 1860 1 720 6348 t (c)720 6468 w (ixb\(1\) = idumb\(-1.0d0, 0.0d0, ndx, kx, nxr\(1\)\))6 2760 1 1080 6588 t (ixb\(2\) = idumb\(0.0d0, 1.0d0, ndx, kx, nxr\(2\)\))6 2700 1 1080 6708 t (ixb\(3\) = idumb\(1.0d0, 2.0d0, ndx, kx, nxr\(3\)\))6 2700 1 1080 6828 t (ixb\(4\) = idumb\(0.0d0, 1.0d0, ndx, kx, nxr\(4\)\))6 2700 1 1080 6948 t (iyb\(1\) = idumb\(0.0d0, 1.0d0, ndy, ky, nyr\(1\)\))6 2700 1 1080 7068 t (iyb\(2\) = idumb\(0.0d0, 1.0d0, ndy, ky, nyr\(2\)\))6 2700 1 1080 7188 t (iyb\(3\) = idumb\(0.0d0, 1.0d0, ndy, ky, nyr\(3\)\))6 2700 1 1080 7308 t cleartomark showpage saveobj restore %%EndPage: 4 31 %%Page: 5 32 /saveobj save def mark 32 pagesetup 10 R f (\261 B-5 \261)2 300 1 2730 480 t 10 CW f (iyb\(4\) = idumb\(-1.0d0, 0.0d0, ndy, ky, nyr\(4\)\))6 2760 1 1080 840 t (nnu =nu*\(\(nxr\(1\)-kx\)*\(nyr\(1\)+nyr\(3\)-2*ky\)+)1 2520 1 1080 960 t (1 \(nxr\(2\)-kx\)*\(nyr\(2\)-ky\)+)1 1620 1 1020 1080 t (4 \(nxr\(4\)-kx\)*\(nyr\(4\)-ky\)\))1 1620 1 1020 1200 t (nr=4)1080 1320 w (c space for the solution.)4 1500 1 720 1440 t (iu = istkgt\(nnu, 4\))3 1140 1 1080 1560 t (do 1 i=1,nr)2 660 1 1080 1680 t (kxr\(i\)=kx)1260 1800 w (kyr\(i\)=ky)1260 1920 w (1 continue)1 840 1 720 2040 t (c initial conditions for u.)4 1620 1 720 2160 t (call setd\(nnu, 0.d0,ws\(iu\)\))2 1620 1 1080 2280 t (c since idumb places the meshes in the port stack, the name of)12 3720 1 720 2400 t (c the port stack, ws, is used as the x and y arrays)12 3060 1 720 2520 t (call dttgu\(ws\(iu\),nu,nr,kxr,ws,nxr,ixb,kyr,ws,nyr,iyb,tstart,)1 3660 1 1080 2640 t ( dt, af, bc, errpar, handlu\))5 1680(1 tstop,)1 600 2 1020 2760 t (call leave)1 600 1 1080 2880 t (call wrapup)1 660 1 1080 3000 t (stop)1080 3120 w (end)1080 3240 w 10 R f ( the various arguments of the)5 1197(The dimension statements for)3 1203 2 720 3516 t 10 CW f (AF)3152 3516 w 10 R f (,)3272 3516 w 10 CW f (BC)3329 3516 w 10 R f (and)3481 3516 w 10 CW f (HANDLE)3657 3516 w 10 R f (subroutines given below)2 991 1 4049 3516 t (are general and thus will not be repeated in subsequent)9 2187 1 720 3636 t 10 B f (pde-bc)2932 3636 w 10 R f (examples.)3246 3636 w (Note that since the arrays)4 1038 1 970 3792 t 10 CW f (A)2039 3792 w 10 R f (, ... ,)2 187 1 2099 3792 t 10 CW f (FUYT)2317 3792 w 10 R f (are set to zero by)4 705 1 2588 3792 t 10 CW f (TTGU)3324 3792 w 10 R f (before it calls)2 555 1 3595 3792 t 10 CW f (AF)4181 3792 w 10 R f ( active)1 270(, only the)2 387 2 4301 3792 t 10 B f (a)4990 3792 w 10 R f (and)720 3912 w 10 B f (f)893 3912 w 10 R f ( in)1 106(terms and their derivatives need be computed)6 1837 2 955 3912 t 10 CW f (AF)2926 3912 w 10 R f ( subroutine)1 450(. The)1 233 2 3046 3912 t 10 CW f (AF)3757 3912 w 10 R f (for specifying the)2 710 1 3905 3912 t 10 B f (pde)4643 3912 w 10 R f (\(A.1\))4827 3912 w (is exactly that given in example 1 for TTGR[15], namely:)9 2308 1 720 4032 t 10 CW f (subroutine af\(t, x, nx, y, ny, nu, u, ut, ux, uy, uxt, uyt)12 3480 1 1080 4272 t ( a, au, aut, aux, auy, auxt, auyt, f, fu, fut, fux, fuy, fuxt,)13 3720(1 ,)1 300 2 1020 4392 t (2 fuyt\))1 540 1 1020 4512 t (integer nu, nx, ny)3 1080 1 1080 4632 t (double precision t, x\(nx\), y\(ny\), u\(nx, ny, nu\), ut\(nx, ny, nu\),)10 3840 1 1080 4752 t ( ny, nu\))2 480(1 ux\(nx,)1 600 2 1020 4872 t (double precision uy\(nx, ny, nu\), uxt\(nx, ny, nu\), uyt\(nx, ny, nu\),)10 3960 1 1080 4992 t ( ny, nu, 2\), au\(nx, ny, nu, nu, 2\), aut\(nx, ny, nu, nu, 2\))13 3480(1 a\(nx,)1 540 2 1020 5112 t (double precision aux\(nx, ny, nu, nu, 2\), auy\(nx, ny, nu, nu, 2\),)11 3840 1 1080 5232 t ( ny, nu, nu, 2\), auyt\(nx, ny, nu, nu, 2\), f\(nx, ny, nu\))12 3300(1 auxt\(nx,)1 720 2 1020 5352 t ( fu\(nx, ny, nu, nu\))4 1140(2 ,)1 300 2 1020 5472 t (double precision fut\(nx, ny, nu, nu\), fux\(nx, ny, nu, nu\), fuy\(nx,)10 3960 1 1080 5592 t ( nu, nu\), fuxt\(nx, ny, nu, nu\), fuyt\(nx, ny, nu, nu\))10 3120(1 ny,)1 420 2 1020 5712 t (integer i, p, q)3 900 1 1080 5832 t ( i = 1, nu)4 600(do 3)1 300 2 1080 5952 t ( q = 1, ny)4 600(do 2)1 300 2 1260 6072 t ( p = 1, nx)4 600(do 1)1 300 2 1440 6192 t (a\(p, q, i, 1\) = ux\(p, q, i\)+.1*uy\(p, q, i\)+u\(p, q, i\))11 3180 1 1620 6312 t (a\(p, q, i, 2\) = uy\(p, q, i\)+.1*ux\(p, q, i\)+u\(p, q, i\))11 3180 1 1620 6432 t (aux\(p, q, i, i, 1\) = 1)6 1320 1 1620 6552 t (auy\(p, q, i, i, 2\) = 1)6 1320 1 1620 6672 t (auy\(p, q, i, i, 1\) = .1)6 1380 1 1620 6792 t (aux\(p, q, i, i, 2\) = .1)6 1380 1 1620 6912 t (au\(p, q, i, i, 1\) = 1)6 1260 1 1620 7032 t (au\(p, q, i, i, 2\) = 1)6 1260 1 1620 7152 t (f\(p, q, i\) = ut\(p, q, i\)+ux\(p, q, i\)+uy\(p, q, i\))10 2880 1 1620 7272 t cleartomark showpage saveobj restore %%EndPage: 5 32 %%Page: 6 33 /saveobj save def mark 33 pagesetup 10 R f (\261 B-6 \261)2 300 1 2730 480 t 10 CW f (fut\(p, q, i, i\) = 1)5 1140 1 1620 840 t (fux\(p, q, i, i\) = 1)5 1140 1 1620 960 t (fuy\(p, q, i, i\) = 1)5 1140 1 1620 1080 t (f\(p, q, i\) = f\(p, q, i\)+.2*t-x\(p\)*y\(q\))6 2280 1 1620 1200 t (1 continue)1 1200 1 900 1320 t (2 continue)1 1020 1 900 1440 t (3 continue)1 840 1 900 1560 t (return)1080 1680 w (end)1080 1800 w 10 R f (Note that since the arrays)4 1034 1 720 2076 t 10 CW f (B)1784 2076 w 10 R f (, ... ,)2 185 1 1844 2076 t 10 CW f (BUyt)2060 2076 w 10 R f (are set to zero by)4 705 1 2331 2076 t 10 CW f (TTGU)3067 2076 w 10 R f (before it calls)2 555 1 3338 2076 t 10 CW f (BC)3924 2076 w 10 R f (, only the active)3 656 1 4044 2076 t 10 B f (b)4731 2076 w 10 R f (terms)4818 2076 w (and their derivatives need be computed in)6 1711 1 720 2196 t 10 CW f (BC)2463 2196 w 10 R f ( subroutine)1 454(. The)1 237 2 2583 2196 t 10 CW f (BC)3306 2196 w 10 R f (for specifying the)2 718 1 3458 2196 t 10 B f (bc)4207 2196 w 10 R f ('s \(A.2\) is exactly)3 733 1 4307 2196 t (the same as in Example 1 for TTGR[15], namely:)8 1983 1 720 2316 t 10 CW f (subroutine bc\(t, x, nx, y, ny, lx, rx, ly, ry, u, ut, ux,)12 3420 1 1080 2556 t ( uxt, uyt, nu, b, bu, but, bux, buy, buxt, buyt\))10 2880(1 uy,)1 420 2 1020 2676 t (integer nu, nx, ny)3 1080 1 1080 2796 t (double precision t, x\(nx\), y\(ny\), lx, rx, ly)7 2640 1 1080 2916 t (double precision ry, u\(nx, ny, nu\), ut\(nx, ny, nu\), ux\(nx, ny, nu\))11 3960 1 1080 3036 t ( uy\(nx, ny, nu\), uxt\(nx, ny, nu\))6 1920(1 ,)1 300 2 1020 3156 t (double precision uyt\(nx, ny, nu\), b\(nx, ny, nu\), bu\(nx, ny, nu,)10 3780 1 1080 3276 t ( but\(nx, ny, nu, nu\), bux\(nx, ny, nu, nu\), buy\(nx, ny, nu)11 3420(1 nu\),)1 480 2 1020 3396 t ( nu\))1 240(2 ,)1 300 2 1020 3516 t (double precision buxt\(nx, ny, nu, nu\), buyt\(nx, ny, nu, nu\))9 3540 1 1080 3636 t (integer i, j)2 720 1 1080 3756 t ( j = 1, ny)4 600(do 2)1 300 2 1080 3876 t ( i = 1, nx)4 600(do 1)1 300 2 1260 3996 t (bu\(i, j, 1, 1\) = 1)5 1080 1 1440 4116 t (b\(i, j, 1\) = u\(i, j, 1\)-t*x\(i\)*y\(j\))6 2100 1 1440 4236 t (1 continue)1 1020 1 900 4356 t (2 continue)1 840 1 900 4476 t (return)1080 4596 w (end)1080 4716 w 10 R f (The following output subroutine simply evaluates and prints)7 2448 1 720 4992 t 10 I f (u u)1 50 1 3198 4992 t 10 R f (\()3256 4992 w 10 I f (t t)1 28 1 3297 4992 t 10 R f (,)3333 4992 w 10 I f (x x)1 44 1 3366 4992 t 10 R f (,)3418 4992 w 10 I f (y y)1 44 1 3451 4992 t 10 R f (\), for three rows and three columns in)7 1537 1 3503 4992 t ( dimension statement for the vari-)5 1356( The)1 206( time-step.)1 423(each rectangle of the domain at the end of each successful)10 2335 4 720 5112 t (ous arguments is for arbitrary input and thus will not be repeated in subsequent examples.)14 3587 1 720 5232 t (Three PORT Library routines are used)5 1538 1 970 5388 t 10 S f (\267)970 5544 w 10 CW f (I1MACH)1076 5544 w 10 R f (determines the standard output unit number,)5 1765 1 1461 5544 t 10 CW f (I1MACH)3251 5544 w 10 R f (\(2\).)3611 5544 w 10 S f (\267)970 5700 w 10 CW f (DTSD1)1076 5700 w 10 R f (evaluates a spline, given the mesh and the coefficients. See section 4.)11 2767 1 1401 5700 t 10 S f (\267)970 5856 w 10 CW f (IDLUMD)1076 5856 w 10 R f ( a basic mesh by inserting a given number of points)10 2069(generates a list of distinct points from)6 1510 2 1461 5856 t (between the basic points.)3 1004 1 970 5976 t (Also, two)1 389 1 720 6132 t 10 CW f (Common)1134 6132 w 10 R f (regions from)1 513 1 1519 6132 t 10 CW f (TTGU)2057 6132 w 10 R f (are used to provide, magically, the meshes for)7 1838 1 2322 6132 t 10 I f (x x)1 44 1 4186 6132 t 10 R f (and)4256 6132 w 10 I f (y y)1 44 1 4426 6132 t 10 R f (and)4496 6132 w 10 CW f (Nu)4666 6132 w 10 R f (. This)1 254 1 4786 6132 t ( the fixed calling sequence for the output routine from)9 2196(is because)1 412 2 720 6252 t 10 CW f (IODE)3357 6252 w 10 R f (doesn't currently allow the passing)4 1414 1 3626 6252 t ( specifically we put the information for the)7 1746( More)1 271( we pass it under the table.)6 1092( So)1 161( information.)1 527(of such extra)2 523 6 720 6372 t ( the PORT Library stack and KXP, IX, NXP, KYP, IY, and)11 2424(KXR, X, NXR, KYR, Y, and NYR arrays into)8 1896 2 720 6492 t ( respective arrays in the stack. NXNYT gives the total number of mesh)12 2896(NYP point to the beginnings of the)6 1424 2 720 6612 t (points in the domain, NR indicates the number of rectangles in the domain and IUP points to a location in)19 4320 1 720 6732 t ( the coefficients)2 655(the stack which starts an array whose elements indicate the position in the U array where)15 3665 2 720 6852 t (are stored for successive rectangles.)4 1431 1 720 6972 t 10 CW f (subroutine handlu\(t0, u0, t, u, nv, dt, tstop\))7 2760 1 1080 7248 t cleartomark showpage saveobj restore %%EndPage: 6 33 %%Page: 7 34 /saveobj save def mark 34 pagesetup 10 R f (\261 B-7 \261)2 300 1 2730 480 t 10 CW f (integer nv)1 600 1 1080 840 t (double precision t0, u0\(nv\), t, u\(nv\), dt, tstop)7 2880 1 1080 960 t (common /d7tgup/ errpar, nu, mxp, myp)5 2160 1 1080 1080 t (integer nu)1 600 1 1080 1200 t (real errpar\(2\))1 840 1 1080 1320 t ( kxp,ix,nxp,kyp,iy,nyp,nxnyt,nr,iup)1 2160(common /d7tgum/)1 900 2 1080 1440 t (integer kx, ix, nx, ky, iy, ny)6 1800 1 1080 1560 t (common /cstak/is)1 960 1 1080 1680 t (integer is\(1000\))1 960 1 1080 1800 t (iwrite=i1mach\(2\))1080 1920 w (if \(t0 .ne. t\) goto 2)5 1260 1 1080 2040 t ( t)1 120( 1\))1 240(write \(iwrite,)1 840 3 1260 2160 t ( \(16h restart for t =, 1pe10.2\))6 1860(1 format)1 720 2 900 2280 t (return)1260 2400 w (c get and print the error.)5 1560 1 720 2520 t (2 continue)1 780 1 900 2640 t (write\(iwrite, 3\)t)1 1020 1 1200 2760 t ( at t=,1pe10.2\))2 900(3 format\(6h)1 840 2 900 2880 t (ius=1)1200 3000 w (do 5 inu = 1, nu)5 960 1 1200 3120 t (iyr=iy)1380 3240 w (ixr=ix)1380 3360 w (do 4 ir=1,nr)2 720 1 1380 3480 t (ir1=ir-1)1560 3600 w (nx=is\(nxp+ir1\))1560 3720 w (ny=is\(nyp+ir1\))1560 3840 w (kx=is\(kxp+ir1\))1560 3960 w (ky=is\(kyp+ir1\))1560 4080 w (call gerr\(kx, ixr, nx, ky, iyr, ny, u\(ius\), inu, t, ir\))10 3300 1 1560 4200 t (ixr=ixr+nx)1560 4320 w (iyr=iyr+ny)1560 4440 w (ius=ius+\(nx-kx\)*\(ny-ky\))1560 4560 w (4 continue)1 1020 1 900 4680 t (5 continue)1 660 1 900 4800 t (return)1080 4920 w (end)1080 5040 w 10 R f (where the procedure)2 813 1 720 5316 t 10 CW f (GERR)1558 5316 w 10 R f (is used to print out the results)6 1178 1 1823 5316 t 10 CW f (subroutine gerr\(kx, ix, nx, ky, iy, ny, u, inu, t, ir\))10 3240 1 1080 5556 t (c to print the solution at each time-step)7 2460 1 720 5676 t (integer kx, ix, nx, ky, iy, ny)6 1800 1 1080 5796 t (integer inu, ir)2 900 1 1080 5916 t (double precision u\(1\), t)3 1440 1 1080 6036 t (common /cstak/ ds)2 1020 1 1080 6156 t (double precision ds\(500\))2 1440 1 1080 6276 t (integer ifa, ita\(2\), ixa\(2\), nta\(2\), nxa\(2\), idlumd)6 3060 1 1080 6396 t (integer ixs, iys, nxs, nys, istkgt, i)6 2220 1 1080 6516 t ( ma\(2\), is\(1000\), i1mach)3 1440(integer ka\(2\),)1 900 2 1080 6636 t (real rs\(1000\))1 780 1 1080 6756 t (logical ls\(1000\))1 960 1 1080 6876 t (complex cs\(500\))1 900 1 1080 6996 t ( ws\(500\))1 540(double precision)1 960 2 1080 7116 t (integer temp)1 720 1 1080 7236 t cleartomark showpage saveobj restore %%EndPage: 7 34 %%Page: 8 35 /saveobj save def mark 35 pagesetup 10 R f (\261 B-8 \261)2 300 1 2730 480 t 10 CW f (equivalence \(ds\(1\), cs\(1\), ws\(1\), rs\(1\), is\(1\), ls\(1\)\))6 3240 1 1080 840 t (c u\(nx-kx,ny-ky\).)1 1020 1 720 960 t (c the port library stack and its aliases.)7 2460 1 720 1080 t (call enter\(1\))1 780 1 1080 1200 t (c x search grid.)3 960 1 720 1320 t (c find the solution at 2 * 2 points / mesh rectangle.)11 3180 1 720 1440 t (ixs = idlumd\(ws\(ix\), nx, 2, nxs\))5 1920 1 1080 1560 t (c y search grid.)3 960 1 720 1680 t (iys = idlumd\(ws\(iy\), ny, 2, nys\))5 1920 1 1080 1800 t (c u search grid values.)4 1380 1 720 1920 t (ka\(1\) = kx)2 600 1 1080 2040 t (ka\(2\) = ky)2 600 1 1080 2160 t (ita\(1\) = ix)2 660 1 1080 2280 t (ita\(2\) = iy)2 660 1 1080 2400 t (nta\(1\) = nx)2 660 1 1080 2520 t (nta\(2\) = ny)2 660 1 1080 2640 t (ixa\(1\) = ixs)2 720 1 1080 2760 t (ixa\(2\) = iys)2 720 1 1080 2880 t (nxa\(1\) = nxs)2 720 1 1080 3000 t (nxa\(2\) = nys)2 720 1 1080 3120 t (ma\(1\) = 0)2 540 1 1080 3240 t (ma\(2\) = 0)2 540 1 1080 3360 t (c get solution.)2 900 1 720 3480 t (c approximate solution values.)3 1800 1 720 3600 t (ifa = istkgt\(nxs*nys, 4\))3 1440 1 1080 3720 t (c evaluate them.)2 960 1 720 3840 t (call dtsd1\(2, ka, ws, ita, nta, u, ws, ixa, nxa, ma, ws\(ifa\)\))11 3660 1 1080 3960 t (temp = i1mach\(2\))2 960 1 1080 4080 t (write\(temp,9001\)ir,inu,\(ws\(i\),i=iFA,IFa+nxs*nys-1\))1080 4200 w ( for rect",i3," u\(.,",i2,"\)=",)3 1800(9001 format\(")1 840 2 720 4320 t (1\(\(1p5e10.2/20x,1p4d10.2\)\)\))1020 4440 w (call leave)1 600 1 1080 4560 t (return)1080 4680 w (end)1080 4800 w 10 R f (The output of the above program unit is)7 1590 1 970 5076 t 10 CW f ( 1.00E+00)1 600(at t=)1 300 2 780 5436 t ( -1.31E-35 -5.00E-01 -2.50E-01)3 1800( 1.58E-19 5.25E-35)2 1200( u\(., 1\)=)2 540( 1)1 180(for rect)1 480 5 780 5556 t ( 3.49E-19)1 600(6.85E-19 -1.00E+00 -5.00E-01)2 1680 2 2040 5676 t ( 6.85E-19 2.50E-01)2 1200( u\(., 1\)= -1.31E-35 -3.00E-19 -1.37E-19)5 2340( 2)1 180(for rect)1 480 4 780 5796 t (5.00E-01 3.49E-19 5.00E-01 1.00E+00)3 2280 1 2040 5916 t ( 5.00E-01 7.50E-01)2 1200( -3.15E-19)1 600( 1.89E-34)1 600( u\(., 1\)= -1.37E-19)3 1140( 3)1 180(for rect)1 480 6 780 6036 t (1.00E+00 1.00E+00 1.50E+00 2.00E+00)3 2280 1 2040 6156 t ( -2.50E-01)1 600( 0.00E+00)1 600( -5.00E-01 -1.00E+00)2 1200( 0.00E+00)1 600( u\(., 1\)=)2 540( 4)1 180(for rect)1 480 7 780 6276 t (-5.00E-01 -1.31E-35 -3.00E-19 -1.37E-19)3 2340 1 1980 6396 t ( OF THE STACK ALLOWED.)4 1320( 700000)1 540( /)1 120(USED 5078)1 780 4 780 6516 t 10 R f ( 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 6792 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 6912 t ( procedure)1 423(tion. The)1 386 2 720 7032 t 10 CW f (GERR)1554 7032 w 10 R f (below checks the error rather than just evaluate the solution)9 2381 1 1819 7032 t 10 CW f (subroutine gerr\(kx, ix, nx, ky, iy, ny, u, inu, t, ir\))10 3240 1 1080 7272 t cleartomark showpage saveobj restore %%EndPage: 8 35 %%Page: 9 36 /saveobj save def mark 36 pagesetup 10 R f (\261 B-9 \261)2 300 1 2730 480 t 10 CW f (integer kx, ix, nx, ky, iy, ny, inu, ir)8 2340 1 1080 840 t (double precision u\(1\), t)3 1440 1 1080 960 t (common /cstak/ ds)2 1020 1 1080 1080 t (double precision ds\(500\))2 1440 1 1080 1200 t (integer ifa, ita\(2\), ixa\(2\), nta\(2\), nxa\(2\), idlumd)6 3060 1 1080 1320 t (integer ixs, iys, nxs, nys, istkgt, i)6 2220 1 1080 1440 t (integer iewe, ka\(2\), ma\(2\), is\(1000\), i1mach)5 2640 1 1080 1560 t (real rs\(1000\))1 780 1 1080 1680 t (logical ls\(1000\))1 960 1 1080 1800 t (complex cs\(500\))1 900 1 1080 1920 t (double precision dabs, erru, dmax1, ws\(500\))5 2580 1 1080 2040 t (integer temp, temp1, temp2)3 1560 1 1080 2160 t (equivalence \(ds\(1\), cs\(1\), ws\(1\), rs\(1\), is\(1\), ls\(1\)\))6 3240 1 1080 2280 t (c to get and print the error at each time-step.)9 2820 1 720 2400 t (c for variable inu for rectangle ir)6 2100 1 720 2520 t (c u\(nx-kx,ny-ky\).)1 1020 1 720 2640 t (c the port library stack and its aliases.)7 2460 1 720 2760 t (call enter\(1\))1 780 1 1080 2880 t (c find the error in the solution at 2*kx * 2*ky points / mesh rectangle.)14 4320 1 720 3000 t (c x search grid.)3 960 1 720 3120 t (ixs = idlumd\(ws\(ix\), nx, 2*kx, nxs\))5 2100 1 1080 3240 t (c y search grid.)3 960 1 720 3360 t (iys = idlumd\(ws\(iy\), ny, 2*ky, nys\))5 2100 1 1080 3480 t (c u search grid values.)4 1380 1 720 3600 t (iewe = istkgt\(nxs*nys, 4\))3 1500 1 1080 3720 t (c the exact solution.)3 1260 1 720 3840 t (call ewe2\(t, ws\(ixs\), nxs, ws\(iys\), nys, ws\(iewe\), inu, ir\))8 3540 1 1080 3960 t (ka\(1\) = kx)2 600 1 1080 4080 t (ka\(2\) = ky)2 600 1 1080 4200 t (ita\(1\) = ix)2 660 1 1080 4320 t (ita\(2\) = iy)2 660 1 1080 4440 t (nta\(1\) = nx)2 660 1 1080 4560 t (nta\(2\) = ny)2 660 1 1080 4680 t (ixa\(1\) = ixs)2 720 1 1080 4800 t (ixa\(2\) = iys)2 720 1 1080 4920 t (nxa\(1\) = nxs)2 720 1 1080 5040 t (nxa\(2\) = nys)2 720 1 1080 5160 t (ma\(1\) = 0)2 540 1 1080 5280 t (c get solution.)2 900 1 720 5400 t (ma\(2\) = 0)2 540 1 1080 5520 t (c approximate solution values.)3 1800 1 720 5640 t (ifa = istkgt\(nxs*nys, 4\))3 1440 1 1080 5760 t (c evaluate them.)2 960 1 720 5880 t (call dtsd1\(2, ka, ws, ita, nta, u, ws, ixa, nxa, ma, ws\(ifa\)\))11 3660 1 1080 6000 t (c error in solution values.)4 1620 1 720 6120 t (erru = 0)2 480 1 1080 6240 t (temp = nxs*nys)2 840 1 1080 6360 t ( i = 1, temp)4 720(do 1)1 300 2 1080 6480 t (temp2 = iewe+i)2 840 1 1260 6600 t (temp1 = ifa+i)2 780 1 1260 6720 t (erru = dmax1\(erru, dabs\(ws\(temp2-1\)-ws\(temp1-1\)\)\))3 2940 1 1260 6840 t (1 continue)1 840 1 900 6960 t (temp = i1mach\(2\))2 960 1 1080 7080 t ( ir, inu, erru)3 840( 2\))1 240(write \(temp,)1 720 3 1080 7200 t cleartomark showpage saveobj restore %%EndPage: 9 36 %%Page: 10 37 /saveobj save def mark 37 pagesetup 10 R f (\261 B-10 \261)2 350 1 2705 480 t 10 CW f ( i2,)1 300( for rect,i3,14h error in u\(.,,)5 1860(2 format\(9h)1 720 3 900 840 t ( =, 1pe10.2\))2 720(1 3h\))1 660 2 1020 960 t (call leave)1 600 1 1080 1080 t (return)1080 1200 w (end)1080 1320 w 10 R f (and the procedure)2 714 1 720 1560 t 10 CW f (EWE2)1459 1560 w 10 R f (below evaluates the exact solution at any position in time and space.)11 2730 1 1724 1560 t 10 CW f (subroutine ewe2\(t, x, nx, y, ny, u, inu, ir\))8 2640 1 1080 1800 t (integer inu, ir, nx, ny)4 1380 1 1080 1920 t (double precision t, x\(nx\), y\(ny\), u\(nx, ny\))6 2580 1 1080 2040 t (integer i, j)2 720 1 1080 2160 t (c the exact solution.)3 1260 1 720 2280 t ( i = 1, nx)4 600(do 2)1 300 2 1260 2400 t ( j = 1, ny)4 600(do 1)1 300 2 1440 2520 t (u\(i, j\) = t*x\(i\)*y\(j\))3 1260 1 1620 2640 t (1 continue)1 1200 1 900 2760 t (2 continue)1 1020 1 900 2880 t (return)1080 3000 w (end)1080 3120 w 10 R f (The above program unit gives)4 1198 1 720 3360 t 10 CW f ( 1.00E+00)1 600(at t=)1 300 2 780 3720 t ( 6.94E-18)1 600( error in u\(., 1\) =)5 1140( 1)1 180(for rect)1 480 4 780 3840 t ( 6.94E-18)1 600( error in u\(., 1\) =)5 1140( 2)1 180(for rect)1 480 4 780 3960 t ( 2.78E-17)1 600( error in u\(., 1\) =)5 1140( 3)1 180(for rect)1 480 4 780 4080 t ( 6.94E-18)1 600( error in u\(., 1\) =)5 1140( 4)1 180(for rect)1 480 4 780 4200 t ( OF THE STACK ALLOWED.)4 1320( 700000)1 540( /)1 120(USED 5078)1 780 4 780 4320 t 10 R f (and we can see that \(A.1\)-\(A.2\) has indeed been solved to within rounding error.)13 3220 1 720 4680 t ( for Example 1 of TTGR[15] and that of Example)9 2015(The only differences between the user written code)7 2055 2 970 4836 t ( domain rather than a rectangle, are several statements in)9 2338(1 here, which solves the problem on a T-shaped)8 1982 2 720 4956 t ( defines the mesh and the subroutine HANDLU is different from HANDLE. The)12 3320(the main program which)3 1000 2 720 5076 t (subroutines AF and BC have not been changed.)7 1903 1 720 5196 t 10 B f (Example 2 - A Coupled System of pdes.)7 1678 1 720 5436 t 10 R f (The solution of a coupled system of)6 1432 1 970 5592 t 10 B f (pde)2427 5592 w 10 R f ( The)1 205('s is now illustrated.)3 811 2 2583 5592 t 10 B f (pde)3624 5592 w 10 R f (is)3805 5592 w 10 I f (a a)1 50 1 1220 5772 t 7 R f (1)1275 5791 w (\( 1 \))2 91 1 1275 5732 t 10 S f (= =)1 55 1 1431 5772 t 10 I f (u u)1 50 1 1535 5772 t 7 R f (1)1596 5792 w 7 I f (x x)1 31 1 1636 5792 t 10 R f (,)1683 5772 w 10 I f (a a)1 50 1 1815 5772 t 7 R f (2)1870 5791 w (\( 1 \))2 91 1 1870 5732 t 10 S f (= =)1 55 1 2026 5772 t 10 I f (u u)1 50 1 2130 5772 t 7 R f (2)2191 5792 w 7 I f (x x)1 31 1 2231 5792 t 10 R f (,)2278 5772 w 10 I f (a a)1 50 1 1228 5952 t 7 R f (1)1283 5971 w (\( 2 \))2 91 1 1283 5912 t 10 S f (= =)1 55 1 1439 5952 t 10 I f (u u)1 50 1 1543 5952 t 7 R f (1)1604 5972 w 7 I f (y y)1 31 1 1644 5972 t 10 R f (,)1691 5952 w 10 I f (a a)1 50 1 1823 5952 t 7 R f (2)1878 5971 w (\( 2 \))2 91 1 1878 5912 t 10 S f (= =)1 55 1 2034 5952 t 10 I f (u u)1 50 1 2138 5952 t 7 R f (2)2199 5972 w 7 I f (y y)1 31 1 2239 5972 t 10 R f (, \(A.3\))1 2754 1 2286 5952 t 10 I f (f f)1 28 1 1236 6132 t 7 R f (1)1275 6152 w 10 S f (= =)1 55 1 1367 6132 t 10 I f (u u)1 50 1 1471 6132 t 7 R f (1)1532 6152 w 7 I f (t t)1 20 1 1572 6152 t 10 S f (+ +)1 55 1 1640 6132 t 10 I f (u u)1 50 1 1735 6132 t 7 R f (1)1796 6152 w 10 I f (u u)1 50 1 1871 6132 t 7 R f (2)1932 6152 w 10 S f (- -)1 55 1 2015 6132 t 10 I f (g g)1 50 1 2110 6132 t 7 R f (1)2171 6152 w 10 R f (,)2222 6132 w 10 I f (f f)1 28 1 2362 6132 t 7 R f (2)2401 6152 w 10 S f (= =)1 55 1 2493 6132 t 10 I f (u u)1 50 1 2597 6132 t 7 R f (2)2658 6152 w 7 I f (t t)1 20 1 2698 6152 t 10 S f (+ +)1 55 1 2766 6132 t 10 I f (u u)1 50 1 2861 6132 t 7 R f (1)2922 6152 w 10 I f (u u)1 50 1 2997 6132 t 7 R f (2)3058 6152 w 10 S f (- -)1 55 1 3141 6132 t 10 I f (g g)1 50 1 3236 6132 t 7 R f (2)3297 6152 w 10 R f (with)720 6312 w 10 B f (bc)923 6312 w 10 R f (s)1023 6312 w 10 I f (b b)1 50 1 1220 6492 t 7 R f (1)1281 6512 w 10 S f (= =)1 55 1 1373 6492 t 10 I f (u u)1 50 1 1477 6492 t 7 R f (1)1538 6512 w 10 R f (\()1589 6492 w 10 I f (t t)1 28 1 1630 6492 t 10 R f (,)1666 6492 w 10 I f (x x)1 44 1 1699 6492 t 10 R f (,)1751 6492 w 10 I f (y y)1 44 1 1784 6492 t 10 R f (\))1836 6492 w 10 S f (- -)1 55 1 1917 6492 t 10 I f (e e)1 44 1 2012 6492 t 7 I f (t t)1 20 1 2067 6452 t 7 R f (\()2092 6452 w 7 I f (x x)1 31 1 2120 6452 t 7 S f (- -)1 39 1 2167 6452 t 7 I f (y y)1 31 1 2217 6452 t 7 R f (\))2253 6452 w 10 R f (and)2498 6492 w 10 I f (b b)1 50 1 2848 6492 t 7 R f (2)2909 6512 w 10 S f (= =)1 55 1 3001 6492 t 10 I f (u u)1 50 1 3105 6492 t 7 R f (2)3166 6512 w 10 R f (\()3217 6492 w 10 I f (t t)1 28 1 3258 6492 t 10 R f (,)3294 6492 w 10 I f (x x)1 44 1 3327 6492 t 10 R f (,)3379 6492 w 10 I f (y y)1 44 1 3412 6492 t 10 R f (\))3464 6492 w 10 S f (- -)1 55 1 3545 6492 t 10 I f (e e)1 44 1 3640 6492 t 7 S f (- -)1 39 1 3695 6452 t 7 I f (t t)1 20 1 3745 6452 t 7 R f (\()3770 6452 w 7 I f (x x)1 31 1 3798 6452 t 7 S f (- -)1 39 1 3845 6452 t 7 I f (y y)1 31 1 3895 6452 t 7 R f (\))3931 6452 w 10 R f (\(A.4\))4827 6492 w (on the L-shaped domain)3 968 1 720 6672 t cleartomark showpage saveobj restore %%EndPage: 10 37 %%Page: 11 38 /saveobj save def mark 38 pagesetup 10 R f (\261 B-11 \261)2 350 1 2705 480 t 2160 840 2160 1560 Dl 2880 840 2160 840 Dl 2880 1560 2880 840 Dl 2160 1560 2880 1560 Dl (0,1)2035 1580 w (0,2 1,2)1 970 1 2035 860 t 2160 1560 2160 2280 Dl 2880 1560 2160 1560 Dl 2880 2280 2880 1560 Dl 2160 2280 2880 2280 Dl (0,0)2035 2300 w (1,0)2818 2360 w 2880 1560 2880 2280 Dl 3600 1560 2880 1560 Dl 3600 2280 3600 1560 Dl 2880 2280 3600 2280 Dl (0,2)3625 2300 w (1,2)3600 1580 w (1,1)2916 1544 w (The solution is)2 595 1 1420 2496 t 10 I f (u u)1 50 1 2040 2496 t 7 R f (1)2101 2516 w 10 S f (\272)2168 2496 w 10 I f (e e)1 44 1 2264 2496 t 7 I f (t t)1 20 1 2319 2456 t 7 R f (\()2344 2456 w 7 I f (x x)1 31 1 2372 2456 t 7 S f (- -)1 39 1 2419 2456 t 7 I f (y y)1 31 1 2469 2456 t 7 R f (\))2505 2456 w 10 R f (and)2561 2496 w 10 I f (u u)1 50 1 2730 2496 t 7 R f (2)2791 2516 w 10 S f (\272)2875 2496 w 10 I f (e e)1 44 1 2971 2496 t 7 S f (- -)1 39 1 3026 2456 t 7 I f (t t)1 20 1 3076 2456 t 7 R f (\()3101 2456 w 7 I f (x x)1 31 1 3129 2456 t 7 S f (- -)1 39 1 3176 2456 t 7 I f (y y)1 31 1 3226 2456 t 7 R f (\))3262 2456 w 10 R f (.)3293 2496 w ( program solves \(A.3\)-\(A.4\) using)4 1380(The following)1 572 2 1670 2652 t 10 CW f (TTGU)3651 2652 w 10 R f (, with a cubic B-spline, over)5 1149 1 3891 2652 t (a spatial mesh consisting of 3 equally spaced, distinct points each side of each rectangle)14 3620 1 1420 2772 t ( out to 10)3 384(with the time-evolution carried)3 1240 2 1420 2892 t 7 S f (- -)1 39 1 3055 2852 t 7 R f (2)3105 2852 w 10 R f ( error at each time-step)4 923( The)1 206(absolute accuracy.)1 737 3 3174 2892 t ( main program is)3 680( The)1 205(is printed out to confirm the accuracy of the computed solution.)10 2544 3 1420 3012 t 10 CW f ( program)1 480(c main)1 420 2 1420 3252 t (common /cstak/ ds)2 1020 1 1780 3372 t (double precision ds\(350000\))2 1620 1 1780 3492 t (external handlu, bc, af)3 1380 1 1780 3612 t (integer ndx, ndy, istkgt, is\(1000\), iu)5 2280 1 1780 3732 t (integer nu, nr, iyb\(3\), ixb\(3\), kx, ky)6 2280 1 1780 3852 t (integer nxr\(3\), nyr\(3\), kxr\(3\), kyr\(3\))4 2280 1 1780 3972 t (integer idumb)1 780 1 1780 4092 t (real errpar\(2\), rs\(1000\))2 1440 1 1780 4212 t (logical ls\(1000\))1 960 1 1780 4332 t (complex cs\(500\))1 900 1 1780 4452 t (double precision tstart, dt)3 1620 1 1780 4572 t (double precision ws\(500\), tstop)3 1860 1 1780 4692 t (equivalence \(ds\(1\), cs\(1\), ws\(1\), rs\(1\), is\(1\), ls\(1\)\))6 3240 1 1780 4812 t (c to solve two coupled, nonlinear heat equations.)7 2940 1 1420 4932 t ( sub t = div . \( u1x, u1y \) - u1*u2 + g1)13 2400(c u1)1 360 2 1420 5052 t ( sub t = div . \( u2x, u2y \) - u1*u2 + g2)13 2400(c u2)1 360 2 1420 5172 t (c the port library stack and its aliases.)7 2460 1 1420 5292 t (c initialize the port library stack length.)6 2580 1 1420 5412 t (call istkin\(350000, 4\))2 1320 1 1780 5532 t (call enter\(1\))1 780 1 1780 5652 t (nu = 2)2 360 1 1780 5772 t (kx = 4)2 360 1 1780 5892 t (ky = 4)2 360 1 1780 6012 t (ndx = 3)2 420 1 1780 6132 t (ndy = 3)2 420 1 1780 6252 t (nr = 3)2 360 1 1780 6372 t (tstart = 0)2 600 1 1780 6492 t (dt = 1e-2)2 540 1 1780 6612 t (tstop =1.d0)1 660 1 1780 6732 t (errpar\(1\) = 1e-2)2 960 1 1780 6852 t (errpar\(2\) = 1e-4)2 960 1 1780 6972 t (c uniform grid.)2 900 1 1420 7092 t (ixb\(1\) = idumb\(0.0d0, 1.0d0, ndx, kx, nxr\(1\)\))6 2700 1 1780 7212 t cleartomark showpage saveobj restore %%EndPage: 11 38 %%Page: 12 39 /saveobj save def mark 39 pagesetup 10 R f (\261 B-12 \261)2 350 1 2705 480 t 10 CW f (ixb\(2\) = idumb\(0.0d0, 1.0d0, ndx, kx, nxr\(2\)\))6 2700 1 1780 840 t (ixb\(3\) = idumb\(1.0d0, 2.0d0, ndx, kx, nxr\(3\)\))6 2700 1 1780 960 t (iyb\(1\) = idumb\(0.0d0, 1.0d0, ndy, ky, nyr\(1\)\))6 2700 1 1780 1080 t (iyb\(2\) = idumb\(1.0d0, 2.0d0, ndy, ky, nyr\(2\)\))6 2700 1 1780 1200 t (iyb\(3\) = idumb\(0.0d0, 1.0d0, ndy, ky, nyr\(3\)\))6 2700 1 1780 1320 t (c uniform grid.)2 900 1 1420 1440 t (c space for the solution.)4 1500 1 1420 1560 t (nnu=0)1780 1680 w (do 1 i=1,nr)2 660 1 1780 1800 t (nnu=nnu+nu*\(\(nxr\(i\)-kx\)*\(nyr\(i\)-ky\)\))1960 1920 w (1 continue)1 780 1 1480 2040 t (iu = istkgt\(nnu, 4\))3 1140 1 1780 2160 t (do 2 i=1,nr)2 660 1 1780 2280 t (kxr\(i\)=kx)1960 2400 w (kyr\(i\)=ky)1960 2520 w (2 continue)1 780 1 1480 2640 t (call setd\(nnu, 1.d0,ws\(iu\)\))2 1620 1 1780 2760 t (call dttgu\(ws\(iu\),nu,nr,kxr,ws,nxr,ixb,kyr,ws,nyr,iyb,tstart,)1 3660 1 1780 2880 t ( dt, af, bc, errpar, handlu\))5 1680(1 tstop,)1 600 2 1720 3000 t (call leave)1 600 1 1780 3120 t (call wrapup)1 660 1 1780 3240 t (stop)1780 3360 w (end)1780 3480 w 10 R f (The only change in the subroutine)5 1407 1 1420 3756 t 10 CW f (AF)2861 3756 w 10 R f ( is the code for specifying the)6 1241(of the last example)3 784 2 3015 3756 t 10 B f (pde)1420 3876 w 10 CW f (integer p, q)2 720 1 1780 4116 t (double precision dexp)2 1260 1 1780 4236 t ( q = 1, ny)4 600(do 2)1 300 2 1780 4356 t ( p = 1, nx)4 600(do 1)1 300 2 1960 4476 t (a\(p, q, 1, 1\) = ux\(p, q, 1\))7 1620 1 2140 4596 t (aux\(p, q, 1, 1, 1\) = 1)6 1320 1 2140 4716 t (a\(p, q, 1, 2\) = uy\(p, q, 1\))7 1620 1 2140 4836 t (auy\(p, q, 1, 1, 2\) = 1)6 1320 1 2140 4956 t (f\(p, q, 1\) = ut\(p, q, 1\)+u\(p, q, 1\)*u\(p, q, 2\))10 2760 1 2140 5076 t (fu\(p, q, 1, 1\) = u\(p, q, 2\))7 1620 1 2140 5196 t (fu\(p, q, 1, 2\) = u\(p, q, 1\))7 1620 1 2140 5316 t (fut\(p, q, 1, 1\) = 1)5 1140 1 2140 5436 t (a\(p, q, 2, 1\) = ux\(p, q, 2\))7 1620 1 2140 5556 t (aux\(p, q, 2, 2, 1\) = 1)6 1320 1 2140 5676 t (a\(p, q, 2, 2\) = uy\(p, q, 2\))7 1620 1 2140 5796 t (auy\(p, q, 2, 2, 2\) = 1)6 1320 1 2140 5916 t (f\(p, q, 2\) = ut\(p, q, 2\)+u\(p, q, 1\)*u\(p, q, 2\))10 2760 1 2140 6036 t (fu\(p, q, 2, 1\) = u\(p, q, 2\))7 1620 1 2140 6156 t (fu\(p, q, 2, 2\) = u\(p, q, 1\))7 1620 1 2140 6276 t (fut\(p, q, 2, 2\) = 1)5 1140 1 2140 6396 t (f\(p, q, 1\) = f\(p, q, 1\)-\(dexp\(t*\(x\(p\)-y\(q\)\)\)*\(x\(p\)-y\(q\)-2d0*)6 3600 1 2140 6516 t (1 t*t\)+1d0\))1 1140 1 1720 6636 t (f\(p, q, 2\) = f\(p, q, 2\)-\(dexp\(t*\(y\(q\)-x\(p\)\)\)*\(y\(q\)-x\(p\)-2d0*)6 3600 1 2140 6756 t (1 t*t\)+1d0\))1 1140 1 1720 6876 t (1 continue)1 1020 1 1600 6996 t (2 continue)1 840 1 1600 7116 t cleartomark showpage saveobj restore %%EndPage: 12 39 %%Page: 13 40 /saveobj save def mark 40 pagesetup 10 R f (\261 B-13 \261)2 350 1 2705 480 t (The only change in the subroutine)5 1392 1 1420 840 t 10 CW f (BC)2843 840 w 10 R f ( for specifying)2 596(of the previous example is the code)6 1450 2 2994 840 t (the)1420 960 w 10 B f (bc)1567 960 w 10 R f (s)1667 960 w 10 CW f (double precision dexp)2 1260 1 1780 1200 t ( j = 1, ny)4 600(do 2)1 300 2 1780 1320 t ( i = 1, nx)4 600(do 1)1 300 2 1960 1440 t (bu\(i, j, 1, 1\) = 1)5 1080 1 2140 1560 t (b\(i, j, 1\) = u\(i, j, 1\)-dexp\(t*\(x\(i\)-y\(j\)\)\))6 2580 1 2140 1680 t (bu\(i, j, 2, 2\) = 1)5 1080 1 2140 1800 t (b\(i, j, 2\) = u\(i, j, 2\)-dexp\(t*\(y\(j\)-x\(i\)\)\))6 2580 1 2140 1920 t (1 continue)1 1020 1 1600 2040 t (2 continue)1 840 1 1600 2160 t 10 R f (The)1420 2436 w 10 CW f (HANDLU)1606 2436 w 10 R f ( the computed solution and is the)6 1364(subroutine simply checks the accuracy of)5 1679 2 1997 2436 t ( following body for the sub-)5 1134( The)1 207( example 1 which checks the solution.)6 1535(same as the one in)4 744 4 1420 2556 t (routine)1420 2676 w 10 CW f (EWE2)1728 2676 w 10 R f (computes the exact solution.)3 1138 1 1993 2676 t 10 CW f (double precision dble, dexp)3 1620 1 1780 2916 t (c the exact solution.)3 1260 1 1420 3036 t ( i = 1, nx)4 600(do 2)1 300 2 1960 3156 t ( j = 1, ny)4 600(do 1)1 300 2 2140 3276 t (u\(i, j\) = dexp\(dble\(float\(\(-1\)**\(inu+1\)\)\)*t*\(x\(i\)-y\(j\)\)\))3 3360 1 2320 3396 t (1 continue)1 1200 1 1600 3516 t (2 continue)1 1020 1 1600 3636 t 10 R f (The output of the above program unit is)7 1590 1 1420 3876 t 10 CW f ( 1.00E-02)1 600(at t=)1 300 2 1480 4236 t ( 1.39E-05)1 600( error in u\(., 1\) =)5 1140( 1)1 180(for rect)1 480 4 1480 4356 t ( 5.48E-05)1 600( error in u\(., 1\) =)5 1140( 2)1 180(for rect)1 480 4 1480 4476 t ( 5.58E-05)1 600( error in u\(., 1\) =)5 1140( 3)1 180(for rect)1 480 4 1480 4596 t ( 1.39E-05)1 600( error in u\(., 2\) =)5 1140( 1)1 180(for rect)1 480 4 1480 4716 t ( 5.58E-05)1 600( error in u\(., 2\) =)5 1140( 2)1 180(for rect)1 480 4 1480 4836 t ( 5.48E-05)1 600( error in u\(., 2\) =)5 1140( 3)1 180(for rect)1 480 4 1480 4956 t ( 9.57E-02)1 600(at t=)1 300 2 1480 5076 t ( 5.49E-04)1 600( error in u\(., 1\) =)5 1140( 1)1 180(for rect)1 480 4 1480 5196 t ( 1.27E-03)1 600( error in u\(., 1\) =)5 1140( 2)1 180(for rect)1 480 4 1480 5316 t ( 1.54E-03)1 600( error in u\(., 1\) =)5 1140( 3)1 180(for rect)1 480 4 1480 5436 t ( 5.49E-04)1 600( error in u\(., 2\) =)5 1140( 1)1 180(for rect)1 480 4 1480 5556 t ( 1.54E-03)1 600( error in u\(., 2\) =)5 1140( 2)1 180(for rect)1 480 4 1480 5676 t ( 1.27E-03)1 600( error in u\(., 2\) =)5 1140( 3)1 180(for rect)1 480 4 1480 5796 t ( 2.80E-01)1 600(at t=)1 300 2 1480 5916 t ( 2.00E-03)1 600( error in u\(., 1\) =)5 1140( 1)1 180(for rect)1 480 4 1480 6036 t ( 2.73E-03)1 600( error in u\(., 1\) =)5 1140( 2)1 180(for rect)1 480 4 1480 6156 t ( 5.02E-03)1 600( error in u\(., 1\) =)5 1140( 3)1 180(for rect)1 480 4 1480 6276 t ( 2.00E-03)1 600( error in u\(., 2\) =)5 1140( 1)1 180(for rect)1 480 4 1480 6396 t ( 5.02E-03)1 600( error in u\(., 2\) =)5 1140( 2)1 180(for rect)1 480 4 1480 6516 t ( 2.73E-03)1 600( error in u\(., 2\) =)5 1140( 3)1 180(for rect)1 480 4 1480 6636 t ( 5.49E-01)1 600(at t=)1 300 2 1480 6756 t ( 4.16E-03)1 600( error in u\(., 1\) =)5 1140( 1)1 180(for rect)1 480 4 1480 6876 t ( 3.04E-03)1 600( error in u\(., 1\) =)5 1140( 2)1 180(for rect)1 480 4 1480 6996 t ( 1.09E-02)1 600( error in u\(., 1\) =)5 1140( 3)1 180(for rect)1 480 4 1480 7116 t ( 4.16E-03)1 600( error in u\(., 2\) =)5 1140( 1)1 180(for rect)1 480 4 1480 7236 t cleartomark showpage saveobj restore %%EndPage: 13 40 %%Page: 14 41 /saveobj save def mark 41 pagesetup 10 R f (\261 B-14 \261)2 350 1 2705 480 t 10 CW f ( 1.09E-02)1 600( error in u\(., 2\) =)5 1140( 2)1 180(for rect)1 480 4 1480 840 t ( 3.04E-03)1 600( error in u\(., 2\) =)5 1140( 3)1 180(for rect)1 480 4 1480 960 t ( 8.92E-01)1 600(at t=)1 300 2 1480 1080 t ( 7.70E-04)1 600( error in u\(., 1\) =)5 1140( 1)1 180(for rect)1 480 4 1480 1200 t ( 2.39E-04)1 600( error in u\(., 1\) =)5 1140( 2)1 180(for rect)1 480 4 1480 1320 t ( 1.52E-03)1 600( error in u\(., 1\) =)5 1140( 3)1 180(for rect)1 480 4 1480 1440 t ( 7.70E-04)1 600( error in u\(., 2\) =)5 1140( 1)1 180(for rect)1 480 4 1480 1560 t ( 1.52E-03)1 600( error in u\(., 2\) =)5 1140( 2)1 180(for rect)1 480 4 1480 1680 t ( 2.39E-04)1 600( error in u\(., 2\) =)5 1140( 3)1 180(for rect)1 480 4 1480 1800 t ( 1.00E+00)1 600(at t=)1 300 2 1480 1920 t ( 1.98E-03)1 600( error in u\(., 1\) =)5 1140( 1)1 180(for rect)1 480 4 1480 2040 t ( 5.24E-04)1 600( error in u\(., 1\) =)5 1140( 2)1 180(for rect)1 480 4 1480 2160 t ( 7.31E-03)1 600( error in u\(., 1\) =)5 1140( 3)1 180(for rect)1 480 4 1480 2280 t ( 1.98E-03)1 600( error in u\(., 2\) =)5 1140( 1)1 180(for rect)1 480 4 1480 2400 t ( 7.31E-03)1 600( error in u\(., 2\) =)5 1140( 2)1 180(for rect)1 480 4 1480 2520 t ( 5.24E-04)1 600( error in u\(., 2\) =)5 1140( 3)1 180(for rect)1 480 4 1480 2640 t ( OF THE STACK ALLOWED.)4 1320( 700000)1 540( /)1 120(USED 57108)1 780 4 1480 2760 t 10 R f ( it is)2 185( Thus,)1 281( this problem is not exactly representable as a spline.)9 2162(Note that the solution of)4 992 4 1420 3156 t (the first non-trivial example use of)5 1384 1 1420 3276 t 10 CW f (TTGU)2829 3276 w 10 R f (.)3069 3276 w 10 B f (Example 3 - Interfaces.)3 987 1 1420 3516 t 10 R f (Consider the)1 508 1 1670 3672 t 10 B f (pde)2203 3672 w 10 I f (f f)1 28 1 1220 4172 t 10 S f (= =)1 55 1 1313 4172 t 10 B f (u)1417 4172 w 7 I f (t t)1 20 1 1484 4192 t 10 S f (- -)1 55 1 1552 4172 t 10 I f (g g)1 50 1 1647 4172 t (a a)1 50 1 1220 4012 t 7 R f (\( 2 \))2 91 1 1281 3972 t 10 S f ( k)1 104(= =)1 55 2 1437 4012 t 10 R f (\()1628 4012 w 10 I f (x x)1 44 1 1669 4012 t 10 R f (,)1721 4012 w 10 I f (y y)1 44 1 1754 4012 t 10 R f (\))1806 4012 w 10 B f (u)1879 4012 w 7 I f (y y)1 31 1 1946 4032 t 10 I f (a a)1 50 1 1220 3842 t 7 R f (\( 1 \))2 91 1 1281 3802 t 10 S f ( k)1 104(= =)1 55 2 1437 3842 t 10 R f (\()1628 3842 w 10 I f (x x)1 44 1 1669 3842 t 10 R f (,)1721 3842 w 10 I f (y y)1 44 1 1754 3842 t 10 R f (\))1806 3842 w 10 B f (u)1879 3842 w 7 I f (x x)1 31 1 1946 3862 t 10 R f (\(A.5\))4827 4012 w (where)1420 4352 w 10 S f (k \272)1 151 1 1220 4717 t (\354)1420 4530 w (\357)1420 4630 w (\355)1420 4730 w (\357)1420 4830 w (\356)1420 4930 w 10 R f (1)1535 4867 w 10 I f (/ /)1 28 1 1593 4867 t 10 R f (3 2)1 240 1 1629 4867 t 10 S f (< <)1 55 1 1885 4867 t 10 I f (y y)1 44 1 1956 4867 t 10 S f (\243)2008 4867 w 10 R f (3)2071 4867 w (1)1535 4727 w 10 I f (/ /)1 28 1 1593 4727 t 10 R f (2 1)1 240 1 1629 4727 t 10 S f (< <)1 55 1 1885 4727 t 10 I f (y y)1 44 1 1956 4727 t 10 S f (\243)2008 4727 w 10 R f (2)2071 4727 w (1 0)1 273 1 1535 4587 t 10 S f (\243)1816 4587 w 10 I f (y y)1 44 1 1879 4587 t 10 S f (\243)1931 4587 w 10 R f (1)1994 4587 w (,)2129 4717 w (with)1420 5102 w 10 B f (bc)1623 5102 w 10 R f (s on the bottom and top)5 942 1 1723 5102 t 10 B f (b)1220 5282 w 10 S f (= =)1 55 1 1325 5282 t 10 I f (u u)1 50 1 1429 5282 t 7 I f (y y)1 31 1 1490 5302 t 10 R f (\(A.6a\))4783 5282 w (and those on the sides)4 877 1 1420 5462 t 10 B f (b)1220 5642 w 10 S f (= =)1 55 1 1325 5642 t 10 B f (u)1429 5642 w 10 S f (- -)1 55 1 1534 5642 t 10 I f (s s)1 39 1 1638 5642 t 10 R f (\()1685 5642 w 10 I f (t t)1 28 1 1726 5642 t 10 R f (,)1762 5642 w 10 I f (x x)1 44 1 1795 5642 t 10 R f (,)1847 5642 w 10 I f (y y)1 44 1 1880 5642 t 10 R f (\) \(A.6b\))1 3108 1 1932 5642 t (The following program solves \(A.5\)-\(A.6\) using)5 2008 1 1420 5858 t 10 CW f (TTGU)3469 5858 w 10 R f (, with a linear B-spline \()5 1051 1 3709 5858 t 10 I f (k k)1 44 1 4760 5858 t 10 S f (= =)1 55 1 4853 5858 t 10 R f (2\))4957 5858 w ( of 3 rectangles on top of each other each of unit height and)13 2492(over a domain that consists)4 1128 2 1420 5978 t ( rectangle consists of 3 equally spaced points on each)9 2195( spatial meshes on each)4 961(width. The)1 464 3 1420 6098 t ( time evolution is carried out to roughly 10)8 1796( The)1 215(side of each rectangle.)3 921 3 1420 6218 t 7 S f (- -)1 39 1 4363 6178 t 7 R f (2)4413 6178 w 10 R f (relative accu-)1 549 1 4491 6218 t ( printed out to confirm the accuracy of the numerical)9 2172( error at each time-step is)5 1041(racy. The)1 407 3 1420 6338 t (solution.)1420 6458 w ( TTGR one was)3 656( For)1 196( of TTGR[15].)2 599(This problem is also example 3 in the documentation)8 2169 4 1420 6614 t ( handle the interface conditions, multiple knots were)7 2272(limited to one rectangle and to)5 1348 2 1420 6734 t (inserted. For TTGU there is no reason to insert multiple knots.)10 2498 1 1420 6854 t (The main program is)3 835 1 1670 7046 t 10 CW f ( program)1 480(c main)1 420 2 1420 7286 t cleartomark showpage saveobj restore %%EndPage: 14 41 %%Page: 15 42 /saveobj save def mark 42 pagesetup 10 R f (\261 B-15 \261)2 350 1 2705 480 t 10 CW f (common /cstak/ ds)2 1020 1 1780 840 t (double precision ds\(350000\))2 1620 1 1780 960 t (external handlu, bc, af)3 1380 1 1780 1080 t (integer ndx, ndy, istkgt, is\(1000\), iu)5 2280 1 1780 1200 t (integer nu, nr, iyb\(3\), ixb\(3\), kx, ky)6 2280 1 1780 1320 t (integer nxr\(3\), nyr\(3\), kxr\(3\), kyr\(3\))4 2280 1 1780 1440 t (integer idumb)1 780 1 1780 1560 t (real errpar\(2\), rs\(1000\))2 1440 1 1780 1680 t (logical ls\(1000\))1 960 1 1780 1800 t (complex cs\(500\))1 900 1 1780 1920 t (double precision tstart, dt, ws\(500\))4 2160 1 1780 2040 t (double precision tstop)2 1320 1 1780 2160 t (equivalence \(ds\(1\), cs\(1\), ws\(1\), rs\(1\), is\(1\), ls\(1\)\))6 3240 1 1780 2280 t (c to solve the layered heat equation, with kappa = 1, 1/2, 1/3,)12 3780 1 1420 2400 t ( . \( kappa\(x,y\) * grad u \) = ut + g)11 2100(c div)1 420 2 1420 2520 t (c the port library stack and its aliases.)7 2460 1 1420 2640 t (c initialize the port library stack length.)6 2580 1 1420 2760 t (call istkin\(350000, 4\))2 1320 1 1780 2880 t (call enter\(1\))1 780 1 1780 3000 t (nu = 1)2 360 1 1780 3120 t (nr = 3)2 360 1 1780 3240 t (kx = 2)2 360 1 1780 3360 t (ky = 2)2 360 1 1780 3480 t (ndx = 3)2 420 1 1780 3600 t (ndy = 3)2 420 1 1780 3720 t (tstart = 0)2 600 1 1780 3840 t (tstop = 1)2 540 1 1780 3960 t (dt = 1)2 360 1 1780 4080 t (errpar\(1\) = 1e-2)2 960 1 1780 4200 t (errpar\(2\) = 1e-4)2 960 1 1780 4320 t (c uniform grid.)2 900 1 1420 4440 t (ixb\(1\) = idumb\(0.0d0, 1.0d0, ndx, kx, nxr\(1\)\))6 2700 1 1780 4560 t (ixb\(2\) = idumb\(0.0d0, 1.0d0, ndx, kx, nxr\(2\)\))6 2700 1 1780 4680 t (ixb\(3\) = idumb\(0.0d0, 1.0d0, ndx, kx, nxr\(3\)\))6 2700 1 1780 4800 t (iyb\(1\) = idumb\(0.0d0, 1.0d0, ndy, ky, nyr\(1\)\))6 2700 1 1780 4920 t (iyb\(2\) = idumb\(1.0d0, 2.0d0, ndy, ky, nyr\(2\)\))6 2700 1 1780 5040 t (iyb\(3\) = idumb\(2.0d0, 3.0d0, ndy, ky, nyr\(3\)\))6 2700 1 1780 5160 t (c space for the solution.)4 1500 1 1420 5280 t (nnu=0)1780 5400 w (do 1 i=1,nr)2 660 1 1780 5520 t (nnu=nnu+nu*\(\(nxr\(i\)-kx\)*\(nyr\(i\)-ky\)\))1960 5640 w (1 continue)1 780 1 1480 5760 t (iu = istkgt\(nnu, 4\))3 1140 1 1780 5880 t (do 2 i=1,nr)2 660 1 1780 6000 t (kxr\(i\)=kx)1960 6120 w (kyr\(i\)=ky)1960 6240 w (2 continue)1 780 1 1480 6360 t (call setd\(nnu, 0.d0,ws\(iu\)\))2 1620 1 1780 6480 t (call dttgu\(ws\(iu\),nu,nr,kxr,ws,nxr,ixb,kyr,ws,nyr,iyb,tstart,)1 3660 1 1780 6600 t ( dt, af, bc, errpar, handlu\))5 1680(1 tstop,)1 600 2 1720 6720 t (call leave)1 600 1 1780 6840 t (call wrapup)1 660 1 1780 6960 t (stop)1780 7080 w (end)1780 7200 w cleartomark showpage saveobj restore %%EndPage: 15 42 %%Page: 16 43 /saveobj save def mark 43 pagesetup 10 R f (\261 B-16 \261)2 350 1 2705 480 t (The body of the)3 674 1 1420 840 t 10 CW f (AF)2132 840 w 10 R f (and)2290 840 w 10 CW f (BC)2472 840 w 10 R f (subroutines for specifying the)3 1229 1 2630 840 t 10 B f (pde)3897 840 w 10 R f (\(A.5\) and the)2 557 1 4092 840 t 10 B f (bc)4688 840 w 10 R f (\(A.7\))4827 840 w ( exactly the same as those given for example 3 in the documentation of)13 2980(respectively are)1 640 2 1420 960 t (TTGR[15].)1420 1080 w (There is no change in the)5 1031 1 1420 1236 t 10 CW f (HANDLU)2481 1236 w 10 R f (or)2872 1236 w 10 CW f (GERR)2986 1236 w 10 R f ( only change)2 522( The)1 211(subroutines of example 1.)3 1050 3 3257 1236 t (in the body of the subroutine)5 1187 1 1420 1356 t 10 CW f (EWE2)2639 1356 w 10 R f (, for computing)2 633 1 2879 1356 t 10 I f (u u)1 50 1 3544 1356 t 10 R f ( the code for com-)4 755(, of example 2 is)4 691 2 3594 1356 t (puting)1420 1476 w 10 I f (u u)1 50 1 1701 1476 t 10 R f (,)1751 1476 w 10 CW f (u\(i, j\) = dble\(float\(ir\)\)*t*y\(j\)-dble\(float\(ir-1\)\)*t)3 3120 1 2260 1716 t (if\(ir.eq.3\) u\(i,j\)=u\(i,j\)-t)1 1620 1 2260 1836 t 10 R f (The output from this program is)5 1280 1 1420 2112 t 10 CW f ( 1.00E+00)1 600(at t=)1 300 2 1480 2472 t ( 1.39E-17)1 600( error in u\(., 1\) =)5 1140( 1)1 180(for rect)1 480 4 1480 2592 t ( 5.55E-17)1 600( error in u\(., 1\) =)5 1140( 2)1 180(for rect)1 480 4 1480 2712 t ( 1.11E-16)1 600( error in u\(., 1\) =)5 1140( 3)1 180(for rect)1 480 4 1480 2832 t ( OF THE STACK ALLOWED.)4 1320( 700000)1 540( /)1 120(USED 4006)1 780 4 1480 2952 t 10 R f (and we see that \(A.5\)-\(A.6\) has indeed been solved to rounding error.)11 2776 1 1420 3312 t 10 B f (Example 4 - Determining the initial conditions)6 1973 1 1420 3552 t 10 R f ( domain of this example is exactly the same as that of Example)12 2543(The PDE and spatial)3 827 2 1670 3708 t ( 0)1 81( only difference is that in example 2 we considered)9 2100(2. The)1 287 3 1420 3828 t 10 S f (\243 \243)1 55 1 3904 3828 t 10 I f (t t)1 28 1 3975 3828 t 10 S f (\243 \243)1 55 1 4027 3828 t 10 R f ( which meant the)3 703(1. 0,)1 158 2 4098 3828 t 10 I f (u u)1 50 1 4990 3828 t 10 R f ( 0)1 58( and that in this example we wish that 1.)9 1628(initially was a constant)3 922 3 1420 3948 t 10 S f (\243 \243)1 55 1 4044 3948 t 10 I f (t t)1 28 1 4115 3948 t 10 S f (\243 \243)1 55 1 4167 3948 t 10 R f ( so that)2 293(1. 01)1 183 2 4238 3948 t 10 I f (u u)1 50 1 4741 3948 t 10 R f (is not)1 222 1 4818 3948 t ( us determine the initial B-spline coefficients we call the sub-)10 2482( help)1 201( To)1 165(a constant initially.)2 772 4 1420 4068 t ( actually entails a one line)5 1045( This)1 229( document.)1 445(routine DICON documented in section 5 of this)7 1901 4 1420 4188 t ( main program of example 2 besides the changes to TSTART and TSTOP.)12 3049( the)1 154(change in)1 417 3 1420 4308 t ( initial B-spline coefficients are computed, we call GERR)8 2369(In the program below after the)5 1251 2 1420 4428 t (to determine the error in the pde before calling DTTGU to solve the pde.)13 2905 1 1420 4548 t 10 CW f ( program)1 480(c main)1 420 2 1420 4908 t (common /cstak/ ds)2 1020 1 1780 5028 t (double precision ds\(350000\))2 1620 1 1780 5148 t (external handlu, bc, af, ic)4 1620 1 1780 5268 t (integer ndx, ndy, istkgt, is\(1000\), iu)5 2280 1 1780 5388 t (integer nu, nr, iyb\(3\), ixb\(3\), kx, ky)6 2280 1 1780 5508 t (integer nxr\(3\), nyr\(3\), kxr\(3\), kyr\(3\))4 2280 1 1780 5628 t (integer idumb)1 780 1 1780 5748 t (real errpar\(2\), rs\(1000\))2 1440 1 1780 5868 t (logical ls\(1000\))1 960 1 1780 5988 t (complex cs\(500\))1 900 1 1780 6108 t (double precision tstart, dt)3 1620 1 1780 6228 t (double precision ws\(500\), tstop)3 1860 1 1780 6348 t (equivalence \(ds\(1\), cs\(1\), ws\(1\), rs\(1\), is\(1\), ls\(1\)\))6 3240 1 1780 6468 t (c to solve two coupled, nonlinear heat equations.)7 2940 1 1420 6588 t ( sub t = div . \( u1x, u1y \) - u1*u2 + g1)13 2400(c u1)1 360 2 1420 6708 t ( sub t = div . \( u2x, u2y \) - u1*u2 + g2)13 2400(c u2)1 360 2 1420 6828 t (c the port library stack and its aliases.)7 2460 1 1420 6948 t (c initialize the port library stack length.)6 2580 1 1420 7068 t (call istkin\(350000, 4\))2 1320 1 1780 7188 t (call enter\(1\))1 780 1 1780 7308 t cleartomark showpage saveobj restore %%EndPage: 16 43 %%Page: 17 44 /saveobj save def mark 44 pagesetup 10 R f (\261 B-17 \261)2 350 1 2705 480 t 10 CW f (nu = 2)2 360 1 1780 840 t (kx = 4)2 360 1 1780 960 t (ky = 4)2 360 1 1780 1080 t (ndx = 3)2 420 1 1780 1200 t (ndy = 3)2 420 1 1780 1320 t (nr = 3)2 360 1 1780 1440 t (tstart = 1.0d0)2 840 1 1780 1560 t (dt = 1e-2)2 540 1 1780 1680 t (tstop =1.01d0)1 780 1 1780 1800 t (errpar\(1\) = 1e-2)2 960 1 1780 1920 t (errpar\(2\) = 1e-4)2 960 1 1780 2040 t (c uniform grid.)2 900 1 1420 2160 t (ixb\(1\) = idumb\(0.0d0, 1.0d0, ndx, kx, nxr\(1\)\))6 2700 1 1780 2280 t (ixb\(2\) = idumb\(0.0d0, 1.0d0, ndx, kx, nxr\(2\)\))6 2700 1 1780 2400 t (ixb\(3\) = idumb\(1.0d0, 2.0d0, ndx, kx, nxr\(3\)\))6 2700 1 1780 2520 t (iyb\(1\) = idumb\(0.0d0, 1.0d0, ndy, ky, nyr\(1\)\))6 2700 1 1780 2640 t (iyb\(2\) = idumb\(1.0d0, 2.0d0, ndy, ky, nyr\(2\)\))6 2700 1 1780 2760 t (iyb\(3\) = idumb\(0.0d0, 1.0d0, ndy, ky, nyr\(3\)\))6 2700 1 1780 2880 t (c uniform grid.)2 900 1 1420 3000 t (c space for the solution.)4 1500 1 1420 3120 t (nnu=0)1780 3240 w (do 1 i=1,nr)2 660 1 1780 3360 t (nnu=nnu+nu*\(\(nxr\(i\)-kx\)*\(nyr\(i\)-ky\)\))1960 3480 w (1 continue)1 780 1 1480 3600 t (iu = istkgt\(nnu, 4\))3 1140 1 1780 3720 t (do 2 i=1,nr)2 660 1 1780 3840 t (kxr\(i\)=kx)1960 3960 w (kyr\(i\)=ky)1960 4080 w (2 continue)1 780 1 1480 4200 t (call dicon\(ws\(iu\),nu,nr,kxr,ws,nxr,ixb,kyr,ws,nyr,iyb,ic\))1 3420 1 1780 4320 t (iu1=iu)1780 4440 w (iwrite=i1mach\(2\))1780 4560 w (write\(iwrite,3\))1780 4680 w ( initially\))1 660(3 format\(10h)1 960 2 1420 4800 t (do 5 inu=1,nu)2 780 1 1780 4920 t (do 4 i=1,nr)2 660 1 1960 5040 t (call gerr\(kxr\(i\),ixb\(i\),nxr\(i\),kyr\(i\),iyb\(i\),nyr\(i\),)1 3120 1 2140 5160 t (1 ws\(iu1\),inu,1.0d0,i\))1 1620 1 1720 5280 t (iu1=iu1+\(nxr\(i\)-kxr\(i\)\)*\(nyr\(i\)-kyr\(i\)\))2140 5400 w (4 continue)1 1020 1 1420 5520 t (5 continue)1 840 1 1420 5640 t (call dttgu\(ws\(iu\),nu,nr,kxr,ws,nxr,ixb,kyr,ws,nyr,iyb,tstart,)1 3660 1 1780 5760 t ( dt, af, bc, errpar, handlu\))5 1680(1 tstop,)1 600 2 1720 5880 t (call leave)1 600 1 1780 6000 t (call wrapup)1 660 1 1780 6120 t (stop)1780 6240 w (end)1780 6360 w 10 R f ( determine the ini-)3 756( To)1 168(The subroutines AF and BC are exactly those given in example 2.)11 2696 3 1420 6636 t ( initial conditions at)3 807(tial conditions we also have to supply a subroutine IC to compute the)12 2813 2 1420 6756 t (specific points. This subroutine is called once per rectangle.)8 2390 1 1420 6876 t 10 CW f (subroutine ic\(nu,ir,xq,nxq,yq,nyq,ui\))1 2220 1 1780 7116 t (integer nu, ir, nxq, nyq)4 1440 1 1780 7236 t cleartomark showpage saveobj restore %%EndPage: 17 44 %%Page: 18 45 /saveobj save def mark 45 pagesetup 10 R f (\261 B-18 \261)2 350 1 2705 480 t 10 CW f (double precision xq\(nxq\), yq\(nyq\), ui\(nxq, nyq,nu\))5 3000 1 1780 840 t (double precision dble, dexp)3 1620 1 1780 960 t (integer p)1 540 1 1780 1080 t (do 30 p=1,nu)2 720 1 1780 1200 t (do 20 j=1,nyq)2 780 1 1900 1320 t (do 10 i=1, nxq)3 840 1 2080 1440 t (ui\(i, j, p\) = dexp\(dble\(float\(\(-1\)**\(p+1\)\)\)*\(xq\(i\)-yq\(j\)\)\))4 3480 1 2200 1560 t (10 continue)1 1140 1 1420 1680 t (20 continue)1 960 1 1420 1800 t (30 continue)1 840 1 1420 1920 t (return)1780 2040 w (end)1780 2160 w 10 R f ( subroutines HANDLU, GERR, and EWE2 of example 2 were used, the follow-)12 3233(When the)1 387 2 1420 2436 t (ing output was produced:)3 1013 1 1420 2556 t 10 CW f (initially)1480 2916 w ( 3.70E-04)1 600( error in u\(., 1\) =)5 1140( 1)1 180(for rect)1 480 4 1480 3036 t ( 1.39E-04)1 600( error in u\(., 1\) =)5 1140( 2)1 180(for rect)1 480 4 1480 3156 t ( 9.90E-04)1 600( error in u\(., 1\) =)5 1140( 3)1 180(for rect)1 480 4 1480 3276 t ( 3.70E-04)1 600( error in u\(., 2\) =)5 1140( 1)1 180(for rect)1 480 4 1480 3396 t ( 9.90E-04)1 600( error in u\(., 2\) =)5 1140( 2)1 180(for rect)1 480 4 1480 3516 t ( 1.39E-04)1 600( error in u\(., 2\) =)5 1140( 3)1 180(for rect)1 480 4 1480 3636 t ( 1.01E+00)1 600(at t=)1 300 2 1480 3756 t ( 1.98E-04)1 600( error in u\(., 1\) =)5 1140( 1)1 180(for rect)1 480 4 1480 3876 t ( 7.21E-05)1 600( error in u\(., 1\) =)5 1140( 2)1 180(for rect)1 480 4 1480 3996 t ( 6.54E-04)1 600( error in u\(., 1\) =)5 1140( 3)1 180(for rect)1 480 4 1480 4116 t ( 1.98E-04)1 600( error in u\(., 2\) =)5 1140( 1)1 180(for rect)1 480 4 1480 4236 t ( 6.54E-04)1 600( error in u\(., 2\) =)5 1140( 2)1 180(for rect)1 480 4 1480 4356 t ( 7.21E-05)1 600( error in u\(., 2\) =)5 1140( 3)1 180(for rect)1 480 4 1480 4476 t ( OF THE STACK ALLOWED.)4 1320( 700000)1 540( /)1 120(USED 56508)1 780 4 1480 4596 t 10 R f (Notice that initially the B spline coefficients do not give the exact initial conditions.)13 3354 1 1420 4872 t 10 B f (Example 5 - A Static Problem.)5 1293 1 1420 5112 t 10 R f (Consider the)1 508 1 1670 5268 t 10 B f (pde)2203 5268 w 10 I f (f f)1 28 1 1220 5768 t 10 S f (= =)1 55 1 1313 5768 t 10 R f (0)1417 5768 w 10 I f (a a)1 50 1 1220 5608 t 7 R f (\( 2 \))2 91 1 1281 5568 t 10 S f (= =)1 55 1 1437 5608 t 10 B f (u)1565 5608 w 7 I f (y y)1 31 1 1632 5628 t 10 I f (a a)1 50 1 1220 5438 t 7 R f (\( 1 \))2 91 1 1281 5398 t 10 S f (= =)1 55 1 1437 5438 t 10 B f (u)1565 5438 w 7 I f (x x)1 31 1 1632 5458 t 10 R f (\(A.9\))4827 5608 w (on the domain)2 572 1 1420 5964 t cleartomark showpage saveobj restore %%EndPage: 18 45 %%Page: 19 46 /saveobj save def mark 46 pagesetup 10 R f (\261 B-19 \261)2 350 1 2705 480 t 1800 840 1800 1560 Dl 2520 840 1800 840 Dl 2520 1560 2520 840 Dl 1800 1560 2520 1560 Dl (2,0)1675 860 w (1,0)1675 1580 w (2,1)2520 860 w 1800 1560 1800 2280 Dl 2520 1560 1800 1560 Dl 2520 2280 2520 1560 Dl 1800 2280 2520 2280 Dl (0,0)1675 2300 w (0,1)2458 2360 w 2520 1560 2520 2280 Dl 3240 1560 2520 1560 Dl 3240 2280 3240 1560 Dl 2520 2280 3240 2280 Dl (0,2)3178 2360 w 3240 1560 3240 2280 Dl 3960 1560 3240 1560 Dl 3960 2280 3960 1560 Dl 3240 2280 3960 2280 Dl (0,3)3960 2300 w (1,3)3960 1580 w 3240 840 3240 1560 Dl 3960 840 3240 840 Dl 3960 1560 3960 840 Dl 3240 1560 3960 1560 Dl (2,3 2,2)1 -720 1 3960 860 t (1,1 1,2)1 648 1 2556 1544 t (with)1420 2460 w 10 B f (bc)1623 2460 w 10 R f (s on the bottom)3 620 1 1723 2460 t 10 B f (b)1220 2640 w 10 S f (= =)1 55 1 1325 2640 t 10 I f (u u)1 50 1 1429 2640 t 7 I f (y y)1 31 1 1490 2660 t 10 R f (\(A.10a\))4733 2640 w (and on the other sides)4 871 1 1420 2820 t 10 B f (b)1220 3000 w 10 S f (= =)1 55 1 1325 3000 t 10 B f (u)1429 3000 w 10 S f (- -)1 55 1 1534 3000 t 10 I f (s s)1 39 1 1638 3000 t 10 R f (\()1685 3000 w 10 I f (t t)1 28 1 1726 3000 t 10 R f (,)1762 3000 w 10 I f (x x)1 44 1 1795 3000 t 10 R f (,)1847 3000 w 10 I f (y y)1 44 1 1880 3000 t 10 R f (\) \(A.10b\))1 3108 1 1932 3000 t (where)1420 3180 w 10 I f (s s)1 39 1 1688 3180 t 10 R f (is chosen so that the solution is)6 1245 1 1752 3180 t 10 I f (u u)1 50 1 3022 3180 t 10 S f (= =)1 55 1 3121 3180 t 10 I f ( l)1 0( al)1 28( ea)1 50(R Re)1 105 4 3225 3180 t 10 R f (\()3416 3180 w 10 I f ( og g)2 50( lo)1 50( l)1 60(z z)1 39 4 3457 3180 t 10 R f (\()3664 3180 w 10 I f (z z)1 39 1 3705 3180 t 10 R f (\) \).)1 99 1 3752 3180 t (The following program solves the)4 1414 1 1670 3336 t 10 B f (pde)3124 3336 w 10 R f (-)3280 3336 w 10 B f (bc)3313 3336 w 10 R f (combination \(A.9\)-\(A.10\), using cubic)3 1587 1 3453 3336 t ( consisting of 3 non-uniformly spaced, distinct points whenever)8 2681(B-splines over a mesh)3 939 2 1420 3456 t (0)1420 3576 w 10 S f (\243 \243)1 55 1 1486 3576 t 10 I f (x x)1 44 1 1557 3576 t 10 S f (\243 \243)1 55 1 1625 3576 t 10 R f (1 and whenever 0)3 748 1 1696 3576 t 10 S f (\243 \243)1 55 1 2460 3576 t 10 I f (y y)1 44 1 2531 3576 t 10 S f (\243 \243)1 55 1 2599 3576 t 10 R f (1 and a uniform mesh elsewhere with the time-evolution)8 2370 1 2670 3576 t (carried out to 10)3 669 1 1420 3736 t 7 S f (- -)1 39 1 2100 3696 t 7 R f (2)2150 3696 w 10 R f ( non-uniform mesh used is)4 1078( The)1 208(absolute accuracy.)1 739 3 2221 3736 t 10 I f (x x)1 44 1 4274 3736 t 7 I f (i i)1 20 1 4329 3756 t 10 S f (\272)4398 3736 w 10 R f (\()4494 3736 w 10 I f (n n)1 50 1 4560 3806 t 10 S f (- -)1 55 1 4634 3806 t 10 R f (1)4705 3806 w 10 I f (i i)1 28 1 4571 3676 t 10 S f (- -)1 55 1 4623 3676 t 10 R f (1)4694 3676 w 10 S1 f (_ ____)1 225 1 4545 3706 t 10 R f (\))4788 3736 w 7 I f (k k)1 31 1 4832 3696 t 10 R f (, for)1 169 1 4871 3736 t 10 I f (i i)1 28 1 1420 3906 t 10 S f (= =)1 55 1 1497 3906 t 10 R f (1 ,)1 83 1 1601 3906 t (. . .)2 125 1 1717 3881 t (,)1875 3906 w 10 I f (n n)1 50 1 1908 3906 t 10 R f ( non-uniform mesh is used in the)6 1340(. The)1 234 2 1958 3906 t 10 I f (x x)1 44 1 3561 3906 t 10 R f (direction in rectangles 1 and 2 and)6 1405 1 3635 3906 t (in the)1 234 1 1420 4026 t 10 I f (y y)1 44 1 1688 4026 t 10 R f ( mesh is used in the)5 826(direction in rectangles 1, 3, and 4. A uniform)8 1872 2 1766 4026 t 10 I f (x x)1 44 1 4497 4026 t 10 R f (direction in)1 466 1 4574 4026 t ( the)1 154(rectangles 3, 4, and 5 and in)6 1156 2 1420 4146 t 10 I f (y y)1 44 1 2762 4146 t 10 R f ( the mesh would)3 679( Thus)1 257(direction in rectangles 2 and 5.)5 1266 3 2838 4146 t (look like)1 353 1 1420 4266 t 1800 5148 1800 5868 Dl 2520 5148 1800 5148 Dl 2520 5868 2520 5148 Dl 1800 5868 2520 5868 Dl 1800 4428 1800 5148 Dl 2520 4428 1800 4428 Dl 2520 5148 2520 4428 Dl 1800 5148 2520 5148 Dl 2520 5148 2520 5868 Dl 3240 5148 2520 5148 Dl 3240 5868 3240 5148 Dl 2520 5868 3240 5868 Dl 3240 5148 3240 5868 Dl 3960 5148 3240 5148 Dl 3960 5868 3960 5148 Dl 3240 5868 3960 5868 Dl 3240 4428 3240 5148 Dl 3960 4428 3240 4428 Dl 3960 5148 3960 4428 Dl 3240 5148 3960 5148 Dl 1845 4428 1845 5868 Dl 3960 5823 1800 5823 Dl 2520 4788 1800 4788 Dl 3960 4788 3240 4788 Dl 2880 5148 2880 5868 Dl 3600 4428 3600 5868 Dl ( place, it demon-)3 707( the first)2 355( In)1 144(This unusual grid has been used for several reasons.)8 2164 4 1670 6084 t ( rectangles, but currently, on contiguous)5 1606(strates that one can use different grids on different)8 2014 2 1420 6204 t (rectangles the grids must match. Thus in the)7 1849 1 1420 6324 t 10 I f (x x)1 44 1 3306 6324 t 10 R f (direction one could not have a different)6 1652 1 3388 6324 t ( \()1 41( a log)2 222( since the solution has)4 887( Secondly,)1 448(mesh in rectangles 1 and 2.)5 1092 5 1420 6444 t 10 I f (z z)1 39 1 4118 6444 t 10 R f (\) singularity at)2 583 1 4165 6444 t 10 I f (z z)1 39 1 4773 6444 t 10 S f (= =)1 55 1 4861 6444 t 10 R f (0,)4965 6444 w ( 1 is sufficient to give)5 914(the non uniform grading of the mesh in rectangle)8 2022 2 1420 6564 t 10 I f (O O)1 72 1 4390 6564 t 10 R f (\()4470 6564 w 10 I f (n n)1 50 1 4511 6564 t 7 S f (- -)1 39 1 4572 6524 t 7 I f (k k)1 31 1 4622 6524 t 10 R f (\) conver-)1 371 1 4669 6564 t (gence where)1 520 1 1420 6684 t 10 I f (k k)1 44 1 1985 6684 t 10 R f ( the grading, the convergence)4 1253( Without)1 397( order of the spline used.)5 1082(is the)1 234 4 2074 6684 t (would only be)2 592 1 1420 6804 t 10 I f (O O)1 72 1 2047 6804 t 10 R f (\()2127 6804 w 10 I f (n n)1 50 1 2168 6804 t 7 S f (- -)1 39 1 2229 6764 t 7 R f (1)2279 6764 w 10 R f ( require many more points to get comparable accu-)8 2122(\) and it would)3 588 2 2330 6804 t ( data will indicate, the singularity has little effect on rectangle 5, so)12 2709( as our)2 272(racy. However,)1 639 3 1420 6924 t (that the need for a non uniform mesh is not as great.)11 2080 1 1420 7044 t ( program below the error at each time-step is printed out to confirm the)13 2907(In the main)2 463 2 1670 7200 t cleartomark showpage saveobj restore %%EndPage: 19 46 %%Page: 20 47 /saveobj save def mark 47 pagesetup 10 R f (\261 B-20 \261)2 350 1 2705 480 t (accuracy of the computed solution.)4 1400 1 1420 840 t 10 CW f ( program)1 480(c main)1 420 2 1420 1080 t (common /cstak/ ds)2 1020 1 1780 1200 t (double precision ds\(350000\))2 1620 1 1780 1320 t (external handlu, bc, af)3 1380 1 1780 1440 t (integer ndx, ndy, istkgt, is\(1000\), iu, ix, temp, temp1)8 3300 1 1780 1560 t (integer nu, nr, iyb\(5\), ixb\(5\), kx, ky)6 2280 1 1780 1680 t (integer nxr\(5\), nyr\(5\), kxr\(5\), kyr\(5\))4 2280 1 1780 1800 t (integer idumb)1 780 1 1780 1920 t (real errpar\(2\), rs\(1000\))2 1440 1 1780 2040 t (logical ls\(1000\))1 960 1 1780 2160 t (complex cs\(500\))1 900 1 1780 2280 t (double precision tstart, dt, rx)4 1860 1 1780 2400 t (double precision ws\(500\), tstop)3 1860 1 1780 2520 t (equivalence \(ds\(1\), cs\(1\), ws\(1\), rs\(1\), is\(1\), ls\(1\)\))6 3240 1 1780 2640 t (c to solve laplaces equation with real \( z*log\(z\) \) as solution.)11 3840 1 1420 2760 t (c the port library stack and its aliases.)7 2460 1 1420 2880 t (c initialize the port library stack length.)6 2580 1 1420 3000 t (call istkin\(350000, 4\))2 1320 1 1780 3120 t (call enter\(1\))1 780 1 1780 3240 t (nu = 1)2 360 1 1780 3360 t (kx = 4)2 360 1 1780 3480 t (ky = 4)2 360 1 1780 3600 t (ndx = 3)2 420 1 1780 3720 t (ndy = 3)2 420 1 1780 3840 t (nr = 5)2 360 1 1780 3960 t (tstart = 0)2 600 1 1780 4080 t (dt = 1.d0)2 540 1 1780 4200 t (tstop =1.d0)1 660 1 1780 4320 t (errpar\(1\) = 1e-2)2 960 1 1780 4440 t (errpar\(2\) = 1e-4)2 960 1 1780 4560 t (nx = ndx+2*\(kx-1\))2 1020 1 1780 4680 t (rx=1.0d0)1780 4800 w (c space for x mesh for rectangle 1)7 2040 1 1420 4920 t (ix = istkgt\(nx, 4\))3 1080 1 1780 5040 t (c 0 and rx mult = kx.)6 1260 1 1420 5160 t (ixb\(1\)=ix)1780 5280 w ( i = 1, kx)4 600(do 1)1 300 2 1780 5400 t (temp = ix+i)2 660 1 1960 5520 t (ws\(temp-1\) = 0)2 840 1 1960 5640 t (temp = ix+nx-i)2 840 1 1960 5760 t (ws\(temp\) = rx)2 780 1 1960 5880 t (1 continue)1 840 1 1600 6000 t (temp = ndx-1)2 720 1 1780 6120 t ( i = 1, temp)4 720(do 2)1 300 2 1780 6240 t (temp1 = ix+kx-2+i)2 1020 1 1960 6360 t (ws\(temp1\) = rx*\(dble\(float\(i-1\)\)/\(dble\(float\(ndx\)\)-1d0\)\)**kx)2 3600 1 1960 6480 t (2 continue)1 840 1 1600 6600 t (c rectangle 2 has same grid in x direction as rectangle 1)11 3420 1 1420 6720 t (ixb\(2\)=istkgt\(nx, 4\))1 1200 1 1780 6840 t (call dcopy\(nx, ws\(ix\), 1, ws\(ixb\(2\)\), 1\))5 2400 1 1780 6960 t (c uniform grid for rectangles 3,4, and 5 in x direction)10 3300 1 1420 7080 t (ixb\(3\) = idumb\(1.0d0, 2.0d0, ndx, kx, nxr\(3\)\))6 2700 1 1780 7200 t cleartomark showpage saveobj restore %%EndPage: 20 47 %%Page: 21 48 /saveobj save def mark 48 pagesetup 10 R f (\261 B-21 \261)2 350 1 2705 480 t 10 CW f (ixb\(4\) = idumb\(2.0d0, 3.0d0, ndx, kx, nxr\(4\)\))6 2700 1 1780 840 t (ixb\(5\) = idumb\(2.0d0, 3.0d0, ndx, kx, nxr\(5\)\))6 2700 1 1780 960 t (ny = ndy+2*\(ky-1\))2 1020 1 1780 1080 t (c rectangles 1,3, and 4 use the same grid in the y direction as)13 3780 1 1420 1200 t (c is used for the x direction in rectangle 1)9 2640 1 1420 1320 t (c space for y mesh.)4 1140 1 1420 1440 t (iyb\(1\) = istkgt\(ny, 4\))3 1320 1 1780 1560 t (call dcopy\( nx, ws\(ix\), 1, ws\(iyb\(1\)\), 1\))6 2460 1 1780 1680 t (iyb\(3\) =istkgt\(ny, 4\))2 1260 1 1780 1800 t (call dcopy\( nx, ws\(ix\), 1, ws\(iyb\(3\)\), 1\))6 2460 1 1780 1920 t (iyb\(4\) =istkgt\(ny, 4\))2 1260 1 1780 2040 t (call dcopy\( nx, ws\(ix\), 1, ws\(iyb\(4\)\), 1\))6 2460 1 1780 2160 t (c rectangles 2 and 5 use uniform mesh in y direction)10 3120 1 1420 2280 t (iyb\(2\) = idumb\(1.0d0, 2.0d0, ndy, ky, nyr\(2\)\))6 2700 1 1780 2400 t (iyb\(5\) = idumb\(1.0d0, 2.0d0, ndy, ky, nyr\(5\)\))6 2700 1 1780 2520 t (c space for the solution.)4 1500 1 1420 2640 t (nnu=0)1780 2760 w (do 3 i=1,nr)2 660 1 1780 2880 t (nxr\(i\)=nx)1960 3000 w (nyr\(i\)=ny)1960 3120 w (nnu=nnu+nu*\(\(nxr\(i\)-kx\)*\(nyr\(i\)-ky\)\))1960 3240 w (3 continue)1 780 1 1480 3360 t (iu = istkgt\(nnu, 4\))3 1140 1 1780 3480 t (do 4 i=1,nr)2 660 1 1780 3600 t (kxr\(i\)=kx)1960 3720 w (kyr\(i\)=ky)1960 3840 w (4 continue)1 780 1 1480 3960 t (call setd\(nnu, 0.0d0,ws\(iu\)\))2 1680 1 1780 4080 t (call istkck)1 660 1 1780 4200 t (write\(6,25\)nnu)1780 4320 w ( nnu",i5\))1 540(25 format\(")1 900 2 1420 4440 t (call dttgu\(ws\(iu\),nu,nr,kxr,ws,nxr,ixb,kyr,ws,nyr,iyb,tstart,)1 3660 1 1780 4560 t ( dt, af, bc, errpar, handlu\))5 1680(1 tstop,)1 600 2 1720 4680 t (call leave)1 600 1 1780 4800 t (call wrapup)1 660 1 1780 4920 t (stop)1780 5040 w (end)1780 5160 w 10 R f (The body of the)3 674 1 1670 5436 t 10 CW f (AF)2382 5436 w 10 R f (and)2540 5436 w 10 CW f (BC)2722 5436 w 10 R f ( the)1 161(subroutines for specifying)2 1069 2 2880 5436 t 10 B f (pde)4149 5436 w 10 R f (\(A.9\) and the)2 557 1 4344 5436 t 10 B f (bc)4940 5436 w 10 R f (\(A.10\) respectively are exactly the same as those given for example 5 in the documenta-)14 3620 1 1420 5556 t (tion of TTGR[15].)2 741 1 1420 5676 t (There is no change in the)5 1041 1 1420 5832 t 10 CW f (HANDLU)2493 5832 w 10 R f (or)2885 5832 w 10 CW f (GERR)3000 5832 w 10 R f ( of the)2 271( body)1 232( The)1 212(subroutines of example 1.)3 1053 4 3272 5832 t (subroutine)1420 5952 w 10 CW f (EWE2)1902 5952 w 10 R f (, for computing)2 619 1 2142 5952 t 10 I f (u u)1 50 1 2786 5952 t 10 R f (, is changed as follows:)4 935 1 2836 5952 t 10 CW f (double precision r, dcos, dlog, dsin, datan, theta)7 3000 1 1780 6192 t (double precision dsqrt)2 1320 1 1780 6312 t (c the exact solution.)3 1260 1 1420 6432 t ( i = 1, nx)4 600(do 6)1 300 2 1960 6552 t ( j = 1, ny)4 600(do 5)1 300 2 2140 6672 t (r = dsqrt\(x\(i\)**2+y\(j\)**2\))2 1560 1 2320 6792 t (if \(x\(i\) .le. 0d0\) goto 1)5 1500 1 2320 6912 t (theta = datan\(y\(j\)/x\(i\)\))2 1440 1 2500 7032 t (goto 2)1 420 1 2500 7152 t ( = 2d0*datan\(1d0\))2 1020(1 theta)1 1200 2 1600 7272 t cleartomark showpage saveobj restore %%EndPage: 21 48 %%Page: 22 49 /saveobj save def mark 49 pagesetup 10 R f (\261 B-22 \261)2 350 1 2705 480 t 10 CW f ( \(r .le. 0d0\) goto 3)5 1200(2 if)1 840 2 1600 840 t (u\(i, j\) = r*\(dcos\(theta\)*dlog\(r\)-theta*dsin\(theta\)\))3 3060 1 2500 960 t (goto 4)1 420 1 2500 1080 t ( j\) = 0)3 420(3 u\(i,)1 1140 2 1600 1200 t (4 continue)1 1200 1 1600 1320 t (5 continue)1 1200 1 1600 1440 t (6 continue)1 1020 1 1600 1560 t 10 R f (The output of this program is)5 1169 1 1420 1836 t 10 CW f ( 1.00E+00)1 600(at t=)1 300 2 1480 2076 t ( 1.48E-02)1 600( error in u\(., 1\) =)5 1140( 1)1 180(for rect)1 480 4 1480 2196 t ( 2.12E-03)1 600( error in u\(., 1\) =)5 1140( 2)1 180(for rect)1 480 4 1480 2316 t ( 3.20E-03)1 600( error in u\(., 1\) =)5 1140( 3)1 180(for rect)1 480 4 1480 2436 t ( 7.18E-05)1 600( error in u\(., 1\) =)5 1140( 4)1 180(for rect)1 480 4 1480 2556 t ( 1.64E-05)1 600( error in u\(., 1\) =)5 1140( 5)1 180(for rect)1 480 4 1480 2676 t ( OF THE STACK ALLOWED.)4 1320( 700000)1 540( /)1 120(USED 26612)1 780 4 1480 2796 t 10 R f ( more than it affects the)5 1015(One notices that the singularity affects rectangle 1 much)8 2355 2 1670 3072 t ( the above program was executed, the mesh was refined by chang-)11 2692( After)1 264(other rectangles.)1 664 3 1420 3192 t (ing NDX and NDY to 5 and the program was run again with the following result:)15 3254 1 1420 3312 t 10 CW f ( 3.21E-03)1 600( 1\) =)2 300( 1.00E+00,)1 660( error in u\(.,)3 840( 1)1 180(for rect)1 480 6 1480 3552 t ( 2.93E-04)1 600( 1\) =)2 300( 1.00E+00,)1 660( error in u\(.,)3 840( 2)1 180(for rect)1 480 6 1480 3672 t ( 2.52E-04)1 600( 1\) =)2 300( 1.00E+00,)1 660( error in u\(.,)3 840( 3)1 180(for rect)1 480 6 1480 3792 t ( 2.48E-05)1 600( 1\) =)2 300( 1.00E+00,)1 660( error in u\(.,)3 840( 4)1 180(for rect)1 480 6 1480 3912 t ( 7.02E-06)1 600( 1\) =)2 300( 1.00E+00,)1 660( error in u\(.,)3 840( 5)1 180(for rect)1 480 6 1480 4032 t ( OF THE STACK ALLOWED.)4 1320( 700000)1 540( /)1 120(USED 75804)1 780 4 1480 4152 t cleartomark showpage saveobj restore %%EndPage: 22 49 %%Page: 23 50 /saveobj save def mark 50 pagesetup 10 R f (\261 B-23 \261)2 350 1 2705 480 t (Appendix 2)1 469 1 2995 840 t 10 B f (Routine, Common and Error State Summary)5 1932 1 2264 1080 t 10 R f ( sequences for the routines of)5 1192(This appendix summarizes the calling)4 1527 2 1670 1476 t 10 CW f (TTGU)4418 1476 w 10 R f (, the con-)2 382 1 4658 1476 t ( is terse and meant to)5 851( It)1 112( public Common regions and the error states.)7 1805(tents of the relatively)3 852 4 1420 1596 t ( top level of)3 480( The)1 205(serve as a reference guide, not as a tutorial.)8 1722 3 1420 1716 t 10 CW f (TTGU)3852 1716 w 10 R f (is)4117 1716 w 10 CW f (Call TTGU\(U,Nu,Nr,kx,x,nx,ixb, ky,y,ny,iyb,)2 2580 1 1540 2076 t (tstart,tstop,dt,)2140 2196 w (AF,BC,)2140 2316 w (errpar,)2140 2436 w (HANDLU\))2140 2556 w 10 R f (The)1420 2832 w 10 CW f (AF)1600 2832 w 10 R f (procedure is of the form)4 964 1 1745 2832 t 10 CW f (Subroutine AF\(t,x,nx,y,ny,U,Ux,Uy,Ut,Utx,Uty,Nu,)1 2880 1 1540 3192 t (A,AU,AUx,AUy,AUt,AUtx,AUty,)2380 3312 w (f,fU,fUx,fUy,fUt,fUtx,fUty\))2380 3432 w 10 R f (the)1420 3792 w 10 CW f (BC)1567 3792 w 10 R f (procedure is of the form)4 964 1 1712 3792 t 10 CW f (Subroutine BC\(t,Lx,Rx,Ly,Ry,U,Ux,Uy,Ut,Utx,Uty,Nu,)1 3000 1 1540 4152 t (B,BU,BUx,BUy,BUt,BUtx,BUty\))2380 4272 w 10 R f (and the)1 291 1 1420 4632 t 10 CW f (HANDLU)1736 4632 w 10 R f (procedure has the form)3 922 1 2121 4632 t 10 CW f (Subroutine HANDLU\(t0,U0,V0,t,U,V,Nu,dt,tstop\))1 2700 1 1540 4992 t 10 R f (The "Return-End")1 728 1 1420 5268 t 10 CW f (HANDLU)2173 5268 w 10 R f (procedure is)1 490 1 2558 5268 t 10 CW f (TTGUH)3073 5268 w 10 R f (.)3373 5268 w (The "Return-End")1 728 1 1420 5424 t 10 CW f (BC)2173 5424 w 10 R f (procedure is)1 490 1 2318 5424 t 10 CW f (TTGUP)2833 5424 w 10 R f (, for when)2 407 1 3133 5424 t 10 I f (n n)1 50 1 3565 5424 t 7 I f (u u)1 35 1 3626 5444 t 10 S f (= =)1 55 1 3718 5424 t 10 R f (0.)3822 5424 w (The statistics printing procedure is)4 1382 1 1420 5580 t 10 CW f (TTGUX)2827 5580 w 10 R f (.)3127 5580 w (The basic knob twiddling routine for)5 1468 1 1670 5736 t 10 CW f (TTGU)3163 5736 w 10 R f (is)3428 5736 w 10 CW f (Call TTGUV\(j,f,r,i,l\))1 1260 1 1540 6096 t 10 R f (The following table summarizes the values that can be set by)10 2435 1 1420 6372 t 10 CW f (TTGUV)3880 6372 w cleartomark showpage saveobj restore %%EndPage: 23 50 %%Page: 24 51 /saveobj save def mark 51 pagesetup 10 R f (\261 B-24 \261)2 350 1 2705 480 t 10 S f (_ __________________________________)1 1725 1 2367 740 t 10 R f ( to)1 103( Set)1 279( Default)1 577(Name j)1 605 4 2478 860 t 10 S f (_ __________________________________)1 1725 1 2367 880 t (_ __________________________________)1 1725 1 2367 900 t 10 CW f (theta)2447 1080 w 10 R f (1 1)1 492 1 3044 1080 t 10 CW f (f)3896 1080 w (beta)2477 1200 w 10 R f (2 1)1 492 1 3044 1200 t 10 CW f (f)3896 1200 w (gamma)2447 1320 w 10 R f (3 1)1 492 1 3044 1320 t 10 CW f (f)3896 1320 w (delta)2447 1440 w 10 R f (4 0)1 492 1 3044 1440 t 10 CW f (f)3896 1440 w 10 S f (_ __________________________________)1 1725 1 2367 1460 t 10 CW f (hfract)2417 1580 w 10 R f (1001 1)1 567 1 2969 1580 t 10 CW f (r)3896 1580 w (egive)2447 1700 w 10 R f (1002 100)1 617 1 2969 1700 t 10 CW f (r)3896 1700 w 10 S f (_ __________________________________)1 1725 1 2367 1720 t 10 CW f (kj)2537 1840 w 10 R f (2001 0)1 567 1 2969 1840 t 10 CW f (i)3896 1840 w (minit)2447 1960 w 10 R f (2002 10)1 592 1 2969 1960 t 10 CW f (i)3896 1960 w (maxit)2447 2080 w 10 R f (2003 50)1 592 1 2969 2080 t 10 CW f (i)3896 2080 w (kmax)2477 2200 w 10 R f (2004 10)1 592 1 2969 2200 t 10 CW f (i)3896 2200 w (kinit)2447 2320 w 10 R f (2005 2)1 567 1 2969 2320 t 10 CW f (i)3896 2320 w (mmax)2477 2440 w 10 R f (2006 15)1 592 1 2969 2440 t 10 CW f (i)3896 2440 w (mxq)2507 2560 w 10 R f (2008)2969 2560 w 10 CW f (0 i)1 475 1 3481 2560 t (myq)2507 2680 w 10 R f (2009)2969 2680 w 10 CW f (0 i)1 475 1 3481 2680 t (la)2537 2800 w 10 R f (2010 1)1 567 1 2969 2800 t 10 CW f (i)3896 2800 w 10 S f (_ __________________________________)1 1725 1 2367 2820 t 10 CW f (xpoly)2447 2940 w 10 R f (3001)2969 2940 w 10 CW f (False l)1 595 1 3361 2940 t (erputs)2417 3060 w 10 R f (3002)2969 3060 w 10 CW f (False l)1 595 1 3361 3060 t 10 S f (_ __________________________________)1 1725 1 2367 3080 t 10 CW f (N)2520 3200 w 10 R f (\(i\) 4000+i)1 631 1 2580 3200 t 10 S f (- -)1 55 1 3483 3200 t 10 CW f (i)3896 3200 w 10 S f ( \347)1 -1725(_ __________________________________)1 1725 2 2367 3220 t (\347)2367 3140 w (\347)2367 3040 w (\347)2367 2940 w (\347)2367 2840 w (\347)2367 2740 w (\347)2367 2640 w (\347)2367 2540 w (\347)2367 2440 w (\347)2367 2340 w (\347)2367 2240 w (\347)2367 2140 w (\347)2367 2040 w (\347)2367 1940 w (\347)2367 1840 w (\347)2367 1740 w (\347)2367 1640 w (\347)2367 1540 w (\347)2367 1440 w (\347)2367 1340 w (\347)2367 1240 w (\347)2367 1140 w (\347)2367 1040 w (\347)2367 940 w (\347)2367 840 w (\347)2852 3220 w (\347)2852 3140 w (\347)2852 3040 w (\347)2852 2940 w (\347)2852 2840 w (\347)2852 2740 w (\347)2852 2640 w (\347)2852 2540 w (\347)2852 2440 w (\347)2852 2340 w (\347)2852 2240 w (\347)2852 2140 w (\347)2852 2040 w (\347)2852 1940 w (\347)2852 1840 w (\347)2852 1740 w (\347)2852 1640 w (\347)2852 1540 w (\347)2852 1440 w (\347)2852 1340 w (\347)2852 1240 w (\347)2852 1140 w (\347)2852 1040 w (\347)2852 940 w (\347)2852 840 w (\347)3286 3220 w (\347)3286 3140 w (\347)3286 3040 w (\347)3286 2940 w (\347)3286 2840 w (\347)3286 2740 w (\347)3286 2640 w (\347)3286 2540 w (\347)3286 2440 w (\347)3286 2340 w (\347)3286 2240 w (\347)3286 2140 w (\347)3286 2040 w (\347)3286 1940 w (\347)3286 1840 w (\347)3286 1740 w (\347)3286 1640 w (\347)3286 1540 w (\347)3286 1440 w (\347)3286 1340 w (\347)3286 1240 w (\347)3286 1140 w (\347)3286 1040 w (\347)3286 940 w (\347)3286 840 w (\347)3736 3220 w (\347)3736 3140 w (\347)3736 3040 w (\347)3736 2940 w (\347)3736 2840 w (\347)3736 2740 w (\347)3736 2640 w (\347)3736 2540 w (\347)3736 2440 w (\347)3736 2340 w (\347)3736 2240 w (\347)3736 2140 w (\347)3736 2040 w (\347)3736 1940 w (\347)3736 1840 w (\347)3736 1740 w (\347)3736 1640 w (\347)3736 1540 w (\347)3736 1440 w (\347)3736 1340 w (\347)3736 1240 w (\347)3736 1140 w (\347)3736 1040 w (\347)3736 940 w (\347)3736 840 w (\347)4092 3220 w (\347)4092 3140 w (\347)4092 3040 w (\347)4092 2940 w (\347)4092 2840 w (\347)4092 2740 w (\347)4092 2640 w (\347)4092 2540 w (\347)4092 2440 w (\347)4092 2340 w (\347)4092 2240 w (\347)4092 2140 w (\347)4092 2040 w (\347)4092 1940 w (\347)4092 1840 w (\347)4092 1740 w (\347)4092 1640 w (\347)4092 1540 w (\347)4092 1440 w (\347)4092 1340 w (\347)4092 1240 w (\347)4092 1140 w (\347)4092 1040 w (\347)4092 940 w (\347)4092 840 w 10 R f (where)1420 3436 w 10 CW f (mxq)1688 3436 w 10 S f (= =)1 55 1 1893 3436 t 10 R f (0 means that)2 505 1 1997 3436 t 10 CW f (kx)2527 3436 w 10 R f (quadrature points will be used in)5 1307 1 2672 3436 t 10 I f (x x)1 44 1 4004 3436 t 10 R f (, similarly for)2 547 1 4048 3436 t 10 CW f (myq)4620 3436 w 10 R f (.)4800 3436 w (The procedural knob twiddler is)4 1281 1 1670 3592 t 10 CW f (Call TTGUR\(U,Nu,Nr,kx,x,ixb,nx,ky,y,iyb,ny,)1 2580 1 1480 3952 t (tstart,tstop,dt,)2140 4072 w (AF,BC,)2140 4192 w (ERROR,errpar,)2140 4312 w (HANDLU\))2140 4432 w 10 R f (where the)1 390 1 1420 4792 t 10 CW f (ERROR)1835 4792 w 10 R f (procedure has the form)3 922 1 2160 4792 t 10 CW f (Logical Function ERROR\(U,Nu,t,dt,)2 1980 1 1540 5152 t (errpar,)2920 5272 w (erputs,)2920 5392 w (eU\))2920 5512 w 10 B f (Common Regions)1 758 1 1420 5992 t 10 R f (When the user cannot evaluate any of)6 1543 1 1670 6148 t 10 CW f (AF)3245 6148 w 10 R f (or)3397 6148 w 10 CW f (BC)3512 6148 w 10 R f ( to)1 111(that fact can be signaled)4 992 2 3664 6148 t 10 CW f (TTGU)4800 6148 w 10 R f (via)1420 6268 w 10 CW f (Common / TTGUF / Failed; Logical Failed)6 2340 1 1540 6628 t cleartomark showpage saveobj restore %%EndPage: 24 51 %%Page: 25 52 /saveobj save def mark 52 pagesetup 10 R f (\261 B-25 \261)2 350 1 2705 480 t 10 B f (Naming Conventions)1 898 1 1420 840 t 10 R f (The naming convention for)3 1090 1 1670 996 t 10 CW f (TTGU)2785 996 w 10 R f ( same as that for the Port Library: all hidden)9 1776(is the)1 214 2 3050 996 t ( If)1 117( subroutines have names beginning with a letter followed by a digit.)11 2725(\( not user callable \))4 778 3 1420 1116 t (users avoid such names, there will be no name conflicts.)9 2249 1 1420 1236 t 10 B f (Error States.)1 554 1 1420 1512 t 10 R f ( list of the error states [9] that may be encountered when)11 2407(This section provides a)3 963 2 1670 1668 t (using)1420 1788 w 10 CW f (TTGU)1667 1788 w 10 R f ( the user in find-)4 676( interpretation of these error messages is made to aid)9 2149(. Some)1 308 3 1907 1788 t (ing bugs \(if they exist\) in the user-supplied code)8 1995 1 1420 1908 t 10 CW f (AF)3448 1908 w 10 R f (,)3568 1908 w 10 CW f (BC)3626 1908 w 10 R f (or)3779 1908 w 10 CW f (HANDLU)3896 1908 w 10 R f ( each level of)3 561(. For)1 223 2 4256 1908 t (\(entry to\))1 378 1 1420 2028 t 10 CW f (TTGU)1827 2028 w 10 R f (, the error message and number for a given error state is the same, always)14 2973 1 2067 2028 t (reflecting the error from the bottom layer of)7 1803 1 1420 2148 t 10 CW f (TTGU)3255 2148 w 10 R f ( error states below, along)4 1038( list of)2 270(. The)1 237 3 3495 2148 t ( for the)2 306(with interpretation, is the complete set of error states)8 2185 2 1420 2268 t 10 CW f (TTGU)3945 2268 w 10 R f (package as obtained)2 821 1 4219 2268 t (from the lowest level of)4 954 1 1420 2388 t 10 CW f (TTGU)2399 2388 w 10 R f (.)2639 2388 w (There are many internal variables of)5 1453 1 1670 2544 t 10 CW f (TTGU)3149 2544 w 10 R f ( controlled by the subroutine)4 1157(that may be)2 468 2 3415 2544 t 10 CW f (TTGUV)1420 2664 w 10 R f ( there are many more ways to call)7 1391(. Thus,)1 306 2 1720 2664 t 10 CW f (TTGU)3448 2664 w 10 R f ( just the obvi-)3 568(with bad data than)3 753 2 3719 2664 t ( in the calling sequence for)5 1123(ous ones involving data)3 971 2 1420 2784 t 10 CW f (TTGU)3548 2784 w 10 R f ( list of error states given)5 1013(. The)1 239 2 3788 2784 t (below is complete for)3 904 1 1420 2904 t 10 CW f (TTGU)2361 2904 w 10 R f (/)2601 2904 w 10 CW f (TTGUV)2629 2904 w 10 R f ( not mentioned in)3 736(, and therefore involves variables)4 1375 2 2929 2904 t (the calling sequence for)3 989 1 1420 3024 t 10 CW f (TTGU)2482 3024 w 10 R f ( you are only using)4 818(. If)1 154 2 2722 3024 t 10 CW f (TTGU)3732 3024 w 10 R f (, and not)2 373 1 3972 3024 t 10 CW f (TTGUV)4383 3024 w 10 R f (as well,)1 319 1 4721 3024 t ( error states mentioning variables not in the)7 1790(then the)1 328 2 1420 3144 t 10 CW f (TTGU)3571 3144 w 10 R f ( if you)2 277( So)1 164(call can not occur.)3 755 3 3844 3144 t ( or unknown variable, simply)4 1186(bump into an error state below that mentions an undescribed)9 2434 2 1420 3264 t (ignore it, it cannot happen to you.)6 1349 1 1420 3384 t cleartomark showpage saveobj restore %%EndPage: 25 52 %%Page: 26 53 /saveobj save def mark 53 pagesetup 10 R f (\261 B-26 \261)2 350 1 2705 480 t ( .lt. 1.)2 231(1 Nu)1 372 2 720 840 t ( .lt. 2. in some rectangle)5 960(2 kx)1 350 2 720 996 t ( .lt. 2*kx. in some rectangle)5 1110(3 nx)1 350 2 720 1152 t ( .lt. 2. in some rectangle)5 960(4 ky)1 350 2 720 1308 t ( .lt. 2*ky. in some rectangle)5 1110(5 ny)1 350 2 720 1464 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(6 dt=0)1 434 4 720 1620 t 10 S f (\272)4458 1620 w 10 R f (tstart.)4538 1620 w ( and tstop-tstart must have the same sign.)7 1649( dt)1 128( has wrong sign on input.)5 1011(7 dt)1 328 4 720 1776 t ( .lt. kx-1.)2 364(8 mxq)1 428 2 720 1932 t ( .lt. ky-1.)2 364(9 myq)1 428 2 720 2088 t ( must be 1.)3 439(10 Abs\(LA\))1 610 2 720 2244 t (16 nr.lt.1)1 489 1 720 2400 t ( of meshes along interfaces)4 1086(17 Disagreement)1 804 2 720 2556 t ( from)1 219(1000 dt)1 328 2 720 2712 t 10 CW f (HANDLE)1292 2712 w 10 R f (has wrong sign. Recoverable.)3 1178 1 1677 2712 t ( raise dt in)3 419(1001 Cannot)1 539 2 720 2868 t 10 CW f (HANDLE)1703 2868 w 10 R f (when Failure is set. Recoverable.)4 1325 1 2088 2868 t ( returned by)2 496( .le. 0)2 234(1002 E\(i\))1 405 3 720 3024 t 10 CW f (ERROR)1887 3024 w 10 R f ( a relative error)3 632( Having)1 351( error request is too small.)5 1078( The)1 212(. Recoverable.)1 580 5 2187 3024 t (request on a variable going to 0 can cause this.)9 1863 1 970 3144 t ( have mgq=k when one of the)6 1187( Must)1 256( and Order=0. Recoverable.)3 1105(1003 mxq=kx-1)1 667 4 720 3300 t 10 CW f (pde)3960 3300 w 10 R f (s is of zero order.)4 695 1 4140 3300 t (1004)720 3456 w 10 CW f (pde)970 3456 w 10 R f ( is no)2 217( There)1 282(\(i\) is vacuous. Recoverable.)3 1111 3 1150 3456 t 10 I f (i i)1 28 1 2785 3456 t 7 I f ( h)1 0(t th)1 55 2 2824 3416 t 10 CW f (pde)2912 3456 w 10 R f (.)3092 3456 w ( The)1 205( BCs. Recoverable.)2 771(1005 Improper)1 621 3 720 3612 t 10 B f (bc)2342 3612 w 10 R f (s and)1 208 1 2442 3612 t 10 CW f (pde)2675 3612 w 10 R f (s do not match properly.)4 974 1 2855 3612 t (1006)720 3768 w 10 CW f (pde)970 3768 w 10 R f ( The)1 212(system not in minimal order form. Recoverable.)6 1962 2 1182 3768 t 10 CW f (pde)3388 3768 w 10 R f (can have derivatives removed from)4 1439 1 3601 3768 t (it.)970 3888 w ( few boundary conditions. Recoverable.)4 1591(1007 Too)1 411 2 720 4044 t ( many boundary conditions. Recoverable.)4 1664(1008 Too)1 411 2 720 4200 t ( are too many mixed)4 821( There)1 282( boundary conditions are overdetermined. Recoverable.)5 2214(1009 Mixed)1 511 4 720 4356 t 10 B f (bc)4573 4356 w 10 R f (s.)4673 4356 w ( mixed)1 275( The)1 205( Mixed BCs. Recoverable.)3 1057(1010 Singular)1 589 4 720 4512 t 10 B f (bc)2871 4512 w 10 R f (s were singular so frequently that dt went to 0.)9 1853 1 2971 4512 t ( are too many Dirichlet)4 926( There)1 282( boundary conditions are overdetermined. Recoverable.)5 2214(1011 Dirichlet)1 605 4 720 4668 t 10 B f (bc)4772 4668 w 10 R f (s.)4872 4668 w ( Dirichlet)1 380( The)1 205( Dirichlet BCs. Recoverable.)3 1151(1012 Singular)1 589 4 720 4824 t 10 B f (bc)3070 4824 w 10 R f (s were singular so frequently that dt went to 0.)9 1853 1 3170 4824 t ( Jacobian for the)3 668( The)1 207( Jacobian. Recoverable.)2 951(1013 Singular)1 589 4 720 4980 t 10 CW f (pde)3162 4980 w 10 R f ( so frequently that dt went to)6 1167(was singular)1 504 2 3369 4980 t (0.)970 5100 w ( of stack space for LU decomposition. Recoverable.)7 2070(1014 Out)1 400 2 720 5256 t ( very badly scaled,)3 787( problem may be)3 710( The)1 217( time-step has become too small.)5 1370( The)1 217( Recoverable.)1 560(1016 dt=0.)1 459 7 720 5412 t ( is too)2 265( cause)1 257( Another)1 388(that is units like light-years and micro-grams are being used simultaneously.)10 3160 4 970 5532 t (small an accuracy requirement, like errpar\(2\)=0 when the solution is exceedingly small.)11 3507 1 970 5652 t ( returned from)2 576(1017 dt=0)1 434 2 720 5808 t 10 CW f (HANDLE)1755 5808 w 10 R f ( lowered dt and it became too small.)7 1448( Handle)1 338(. Recoverable.)1 573 3 2115 5808 t (1018)720 5964 w 10 CW f (AF)970 5964 w 10 R f (,)1090 5964 w 10 CW f (BC)1140 5964 w 10 R f ( = True occurred in)4 770( Failed)1 300(failure. Recoverable.)1 833 3 1285 5964 t 10 CW f (AF)3213 5964 w 10 R f (or)3358 5964 w 10 CW f (BC)3466 5964 w 10 R f (so often that dt went to 0.)6 1019 1 3611 5964 t ( of Newton iterations was pre-)5 1301( number)1 347( The)1 222( many Newton iterations predicted. Recoverable.)5 2039(1019 Too)1 411 5 720 6120 t ( cause is a bad Jacobian, see next error state.)9 1775( Probable)1 405(dicted to be too large so often that dt went to 0.)11 1887 3 970 6240 t ( Newton iterations were needed so)5 1467( many)1 264( Too)1 228( many Newton iterations needed. Recoverable.)5 1950(1020 Too)1 411 5 720 6396 t ( possible cause)2 605( Another)1 380( is an incorrectly computed Jacobian.)5 1501( cause)1 250( Probable)1 409(often that dt went to 0.)5 925 6 970 6516 t ( another possible cause is a very badly conditioned Jaco-)9 2274( Yet)1 195( small.)1 268(is that Minit and/or Maxit are too)6 1333 4 970 6636 t ( possible causes: too stringent an error request or a mesh that is too non-uniform.)14 3232(bian. Further)1 541 2 970 6756 t cleartomark showpage saveobj restore %%EndPage: 26 53 %%Trailer done %%Pages: 53 %%DocumentFonts: Courier Times-Bold Times-Italic Times-Roman Times-Roman Symbol