Spline Collocation Methods for Partial Differential Equations. With Applications in R

550 Pages·2017·2.951 MB·english

by  William E. Schiesser

Spline Collocation Methods
for Partial Differential Equations
WithApplicationsinR
WilliamE.Schiesser
LehighUniversity
Bethlehem,PA,USA
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Thiseditionfirstpublished2017
©2017JohnWiley&SonsInc.
LibraryofCongressCataloging-in-PublicationData
Names:Schiesser,W.E.,author.
Title:Splinecollocationmethodsforpartialdifferentialequations:with
applicationsinR/WilliamE.Schiesser.
Description:Hoboken,NJ:JohnWiley&Sons,2017.|Includes
bibliographicalreferencesandindex.
Identifiers:LCCN2017000481(print)|LCCN2017007223(ebook)|ISBN
9781119301035(cloth)|ISBN9781119301059(pdf)|ISBN9781119301042
(epub)
Subjects:LCSH:Differentialequations,Partial–Mathematicalmodels.|
Splinetheory.Classification:LCCQA377.S3552017(print)|LCCQA377(ebook)|DDC
515/.353–dc23LCrecordavailableathttps://lccn.loc.gov/2017000481
Setin10/12ptWarnockProbySPiGlobal,Chennai,India
PrintedintheUnitedStatesofAmerica
(cid:2)
Contents
Preface xiii
AbouttheCompanionWebsite xv
1 Introduction 1
1.1 UniformGrids 2
1.2 VariableGrids 18
1.3 StagewiseDifferentiation 24
AppendixA1–OnlineDocumentationforsplinefun 27
Reference 30
2 One-DimensionalPDEs 31
2.1 ConstantCoefficient 31
2.1.1 DirichletBCs 32
2.1.1.1 MainProgram 33
2.1.1.2 ODERoutine 40
2.1.2 NeumannBCs 43
2.1.2.1 MainProgram 44
2.1.2.2 ODERoutine 46
2.1.3 RobinBCs 49
2.1.3.1 MainProgram 50
2.1.3.2 ODERoutine 55
2.1.4 NonlinearBCs 60
2.1.4.1 MainProgram 61
2.1.4.2 ODERoutine 63
2.2 VariableCoefficient 64
2.2.1 MainProgram 67
2.2.2 ODERoutine 71
2.3 Inhomogeneous,Simultaneous,Nonlinear 76
2.3.1 MainProgram 78
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2.3.2 ODEroutine 85
2.3.3 SubordinateRoutines 88
2.4 FirstOrderinSpaceandTime 94
2.4.1 MainProgram 96
2.4.2 ODERoutine 101
2.4.3 SubordinateRoutines 105
2.5 SecondOrderinTime 107
2.5.1 MainProgram 109
2.5.2 ODERoutine 114
2.5.3 SubordinateRoutine 117
2.6 FourthOrderinSpace 120
2.6.1 FirstOrderinTime 120
2.6.1.1 MainProgram 121
2.6.1.2 ODERoutine 125
2.6.2 SecondOrderinTime 138
2.6.2.1 MainProgram 140
2.6.2.2 ODERoutine 143
References 155
3 MultidimensionalPDEs 157
3.1 2DinSpace 157
(cid:2) 3.1.1 MainProgram 158 (cid:2)
3.1.2 ODERoutine 163
3.2 3DinSpace 170
3.2.1 MainProgram,Case1 170
3.2.2 ODERoutine 174
3.2.3 MainProgram,Case2 183
3.2.4 ODERoutine 187
3.3 SummaryandConclusions 193
4 Navier–Stokes,Burgers’Equations 197
4.1 PDEModel 197
4.2 MainProgram 198
4.3 ODERoutine 203
4.4 SubordinateRoutine 205
4.5 ModelOutput 206
4.6 SummaryandConclusions 208
Reference 209
5 Korteweg–deVriesEquation 211
5.1 PDEModel 211
5.2 MainProgram 212
5.3 ODERoutine 225
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5.4 SubordinateRoutines 228
5.5 ModelOutput 234
5.6 SummaryandConclusions 238
References 239
6 MaxwellEquations 241
6.1 PDEModel 241
6.2 MainProgram 243
6.3 ODERoutine 248
6.4 ModelOutput 252
6.5 SummaryandConclusions 252
AppendixA6.1.DerivationoftheAnalyticalSolution 257
Reference 259
7 Poisson–Nernst–PlanckEquations 261
7.1 PDEModel 261
7.2 MainProgram 265
7.3 ODERoutine 271
7.4 ModelOutput 276
7.5 SummaryandConclusions 284
References 286
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8 Fokker–PlanckEquation 287
8.1 PDEModel 287
8.2 MainProgram 288
8.3 ODERoutine 293
8.4 ModelOutput 295
8.5 SummaryandConclusions 301
References 303
9 Fisher–KolmogorovEquation 305
9.1 PDEModel 305
9.2 MainProgram 306
9.3 ODERoutine 311
9.4 SubordinateRoutine 313
9.5 ModelOutput 314
9.6 SummaryandConclusions 316
Reference 316
10 Klein–GordonEquation 317
10.1 PDEModel,LinearCase 317
10.2 MainProgram 318
10.3 ODERoutine 323
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10.4 ModelOutput 326
10.5 PDEModel,NonlinearCase 328
10.6 MainProgram 330
10.7 ODERoutine 335
10.8 SubordinateRoutines 338
10.9 ModelOutput 339
10.10 SummaryandConclusions 342
Reference 342
11 BoussinesqEquation 343
11.1 PDEModel 343
11.2 MainProgram 344
11.3 ODERoutine 350
11.4 SubordinateRoutines 354
11.5 ModelOutput 355
11.6 SummaryandConclusions 358
References 358
12 Cahn–HilliardEquation 359
12.1 PDEModel 359
12.2 MainProgram 360
(cid:2) 12.3 ODERoutine 366 (cid:2)
12.4 ModelOutput 369
12.5 SummaryandConclusions 379
References 379
13 Camassa–HolmEquation 381
13.1 PDEModel 381
13.2 MainProgram 382
13.3 ODERoutine 388
13.4 ModelOutput 391
13.5 SummaryandConclusions 394
13.6 AppendixA13.1:SecondExampleofaPDEwithaMixedPartial
Derivative 395
13.7 MainProgram 395
13.8 ODERoutine 398
13.9 ModelOutput 400
Reference 403
14 Burgers–HuxleyEquation 405
14.1 PDEModel 405
14.2 MainProgram 406
14.3 ODERoutine 411
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14.4 SubordinateRoutine 416
14.5 ModelOutput 417
14.6 SummaryandConclusions 422
References 422
15 Gierer–MeinhardtEquations 423
15.1 PDEModel 423
15.2 MainProgram 424
15.3 ODERoutine 429
15.4 ModelOutput 432
15.5 SummaryandConclusions 437
Reference 440
16 Keller–SegelEquations 441
16.1 PDEModel 441
16.2 MainProgram 443
16.3 ODERoutine 449
16.4 SubordinateRoutines 453
16.5 ModelOutput 453
16.6 SummaryandConclusions 458
AppendixA16.1.DiffusionModels 458
(cid:2) References 459 (cid:2)
17 Fitzhugh–NagumoEquations 461
17.1 PDEModel 461
17.2 MainProgram 462
17.3 ODERoutine 467
17.4 ModelOutput 470
17.5 SummaryandConclusions 475
Reference 475
18 Euler–Poisson–DarbouxEquation 477
18.1 PDEModel 477
18.2 MainProgram 478
18.3 ODERoutine 483
18.4 ModelOutput 488
18.5 SummaryandConclusions 493
References 493
19 Kuramoto–SivashinskyEquation 495
19.1 PDEModel 495
19.2 MainProgram 496
19.3 ODERoutine 503
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19.4 SubordinateRoutines 506
19.5 ModelOutput 508
19.6 SummaryandConclusions 513
References 514
20 Einstein–MaxwellEquations 515
20.1 PDEModel 515
20.2 MainProgram 516
20.3 ODERoutine 521
20.4 ModelOutput 526
20.5 SummaryandConclusions 533
Reference 536
A DifferentialOperatorsinThreeOrthogonalCoordinate
Systems 537
References 539
Index 541
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Preface
Thisbookisanintroductiontotheuseofsplinecollocation(SC)forthenumer-
icalanalysisofpartialdifferentialequations(PDEs).SCisanalternativetofinite
differences(FDs),finiteVolumes(FVs),finiteelements(FEs),spectralmethods,
weightedresiduals,andleastsquares.
ThefeaturesandadvantagesofSCaredemonstratedthroughaseriesofPDE
examples,includingtheuseofSClibraryroutinesthatarepartoftheRpro-
grammingsystem[1,2].ApplicationsofthePDEsarementionedonlybriefly.
Rather,theemphasisisontheSCmethodology,particularlyintermsofPDE
examplesdevelopedanddiscussedindetail.
(cid:2) ThepaperscitedasasourceofthePDEmodelsgenerallyconsistof(i)astate- (cid:2)
mentoftheequationsfollowedby(ii)areportednumericalsolution.Generally,
littleornoinformationisgivenabouthowthesolutionswerecomputed(the
algorithms),andinallcases,thecomputercodethatwasusedtocalculatethe
solutionsisnotprovided.
Inotherwords,whatismissingis(i)adetaileddiscussionofthenumerical
methods used to produce the reported solutions and (ii) the computer rou-
tinesusedtocalculatethereportedsolutions.Forthereadertocompletethese
twostepstoverifythereportedsolutionswithreasonableeffortisessentially
impossible.
Aprincipalobjectiveofthisbookisthereforetoprovidethereaderwithaset
ofdocumentedRroutinesthatarediscussedindetail,andtheycanbedown-
loadedandexecutedwithouthavingtofirstmasterthedetailsoftherelevant
numericalanalysisandthencodeasetofroutines.
Theexampleapplicationsareintendedasintroductoryandopenended.They
are based mainly on legacy PDEs. Since each of the legacy equations is well
known,asearchontheequationnamewillgiveanextensivesetofreferences,
generallyincludingmanyapplicationstudies.Ratherthanfocusontheapplica-
tions,whichwouldrequireextendeddiscussion,theemphasisineachchapter
isonthefollowing:
1. AstatementofthePDEsystem,includinginitialConditions(ICs),boundary
conditions(BCs),andparameters
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2. The algorithms for the calculation of numerical solutions, with particular
emphasisonSC
3. A set of R routines for the calculationof numericalsolutions, including a
detailedexplanationofeachsectionofthecode
4. Discussionofthenumericalsolution
5. Summary and conclusions about extensions of the computer-based
analysis
In summary, the presentation is not as formal mathematics, for example,
theorems and proofs. Rather, the presentation is by examples of SC analysis
oflegacyPDEs,includingthedetailsforcomputingnumericalsolutions,par-
ticularlywithdocumentedsourcecode.Theauthorwouldwelcomecomments,
especiallypertainingtothisformatandexperienceswiththeuseoftheRrou-
tines.Commentsandquestionscanbedirectedto([email protected]).
References
1 Hiebeler, D.E. (2015), R and Matlab, CRC/Taylor and Francis, Boca Raton,
FL.
2 Soetaert, K., J. Cash, and F. Mazzia (2012), Solving Differential Equations in
R, Springer-Verlag, Heidelberg, Germany.
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WilliamE.Schiesser
Bethlehem,PA,USA
March1,2017
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