Numerical solution of hyperbolic partial differential equations

619 Pages·2009·7.984 MB·English

by  John A. Trangenstein

NUMERICAL SOLUTION OF
HYPERBOLIC PARTIAL DIFFERENTIAL EQUATIONS
This is a new type of graduate textbook, with both print and interactive electronic com-
ponents(onCD).Itisacomprehensivepresentationofmodernshock-capturingmethods,
includingbothfinitevolumeandfiniteelementmethods,coveringthetheoryofhyperbolic
conservationlawsandthetheoryofthenumericalmethods.
Classical techniques for judging the qualitative performance of the schemes, such as
modifiedequationanalysisandFourieranalysis,areusedtomotivatethedevelopmentof
classicalhigher-ordermethods(theLax–Wendroffprocess)andtoproveresultssuchasthe
LaxEquivalenceTheorem.
The range of applications (shallow water, compressible gas dynamics, magnetohydro-
dynamics,finitedeformationinsolids,plasticity,polymerfloodingandwater/gasinjection
inoilrecovery)isbroadenoughtoengagemostengineeringdisciplinesandmanyareasof
appliedmathematics.
The solution of the Riemann problems for these applications is developed, so that the
reader can use the theory to develop test problems for the methods, especially to mea-
sure errors for comparisions of accuracy and efficiency. The numerical methods involve
a variety of important approaches, such as MUSCL and PPM, TVD, wave propagation,
Lax–Friedrichs (aka central schemes), ENO and discontinuous Galerkin; all of these are
discussedinoneandmultiplespatialdimensions.Sincemanyofthesemethodsdependon
Riemannsolvers,thereisextensivediscussionofthebasicdesignprinciplesofapproximate
Riemannsolvers,andseveralcomputationallyusefultechniques.Thefinalchaptercontains
adiscussionofadaptivemeshrefinementviastructuredgrids.
TheaccompanyingCDcontainsahyperlinkedversionofthetextwhichprovidesaccess
tocomputercodesforallofthetextfigures.Throughthiselectronictextstudentscan:
(cid:1)
Seethecodesandrunthem,choosingtheirowninputparametersinteractively
(cid:1)
Viewtheonlinenumericalresultsasmovies
(cid:1)
Gainanappreciationforboththedynamicsoftheproblemapplication,andthegrowth
ofnumericalerrors
(cid:1)
Downloadandmodifythecodeforusewithotherapplications
(cid:1)
Study the code to learn how to structure their programs for modularity and ease of
debugging.
JohnA.TrangensteinisProfessorofMathematicsatDukeUniversity,NorthCarolina

NUMERICAL SOLUTION OF
HYPERBOLIC PARTIAL
DIFFERENTIAL EQUATIONS
JOHN A. TRANGENSTEIN
DepartmentofMathematics,DukeUniversity
Durham,NC27708-0320
cambridge university press
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Contents
Preface pageix
1 IntroductiontoPartialDifferentialEquations 1
2 ScalarHyperbolicConservationLaws 6
2.1 LinearAdvection 6
2.1.1 ConservationLawonanUnboundedDomain 6
2.1.2 IntegralFormoftheConservationLaw 8
2.1.3 Advection–DiffusionEquation 9
2.1.4 AdvectionEquationonaHalf-Line 10
2.1.5 AdvectionEquationonaFiniteInterval 11
2.2 LinearFiniteDifferenceMethods 12
2.2.1 BasicsofDiscretization 12
2.2.2 ExplicitUpwindDifferences 14
2.2.3 ProgramsforExplicitUpwindDifferences 16
2.2.3.1 FirstUpwindDifferenceProgram 16
2.2.3.2 SecondUpwindDifferenceProgram 17
2.2.3.3 ThirdUpwindDifferenceProgram 18
2.2.3.4 FourthUpwindDifferenceProgram 20
2.2.3.5 FifthUpwindDifferenceProgram 21
2.2.4 ExplicitDownwindDifferences 23
2.2.5 ImplicitDownwindDifferences 24
2.2.6 ImplicitUpwindDifferences 25
2.2.7 ExplicitCenteredDifferences 26
2.3 ModifiedEquationAnalysis 30
2.3.1 ModifiedEquationAnalysisforExplicitUpwind
Differences 30
vii
viii Contents
2.3.2 ModifiedEquationAnalysisforExplicitDownwind
Differences 31
2.3.3 ModifiedEquationAnalysisforExplicitCentered
Differences 32
2.3.4 ModifiedEquationAnalysisLiterature 33
2.4 Consistency,StabilityandConvergence 35
2.5 FourierAnalysisofFiniteDifferenceSchemes 38
2.5.1 ConstantCoefficientEquationsandWaves 39
2.5.2 DimensionlessGroups 40
2.5.3 LinearFiniteDifferencesandAdvection 41
2.5.4 FourierAnalysisofIndividualSchemes 44
2.6 L2 StabilityforLinearSchemes 53
2.7 LaxEquivalenceTheorem 55
2.8 MeasuringAccuracyandEfficiency 69
3 NonlinearScalarLaws 81
3.1 NonlinearHyperbolicConservationLaws 81
3.1.1 NonlinearEquationsonUnboundedDomains 81
3.1.2 Characteristics 82
3.1.3 DevelopmentofSingularities 84
3.1.4 PropagationofDiscontinuities 85
3.1.5 TravelingWaveProfiles 89
3.1.6 EntropyFunctions 92
3.1.7 OleinikChordCondition 95
3.1.8 RiemannProblems 97
3.1.9 GalileanCoordinateTransformations 99
3.2 CaseStudies 102
3.2.1 TrafficFlow 102
3.2.2 MiscibleDisplacementModel 103
3.2.3 Buckley–LeverettModel 105
3.3 First-OrderFiniteDifferenceMethods 111
3.3.1 ExplicitUpwindDifferences 111
3.3.2 Lax–FriedrichsScheme 112
3.3.3 TimestepSelection 117
3.3.4 Rusanov’sScheme 118
3.3.5 Godunov’sScheme 120
3.3.6 ComparisonofLax–Friedrichs,GodunovandRusanov 124
3.4 NonreflectingBoundaryConditions 125
3.5 Lax–WendroffProcess 129
3.6 OtherSecondOrderSchemes 132
Contents ix
4 NonlinearHyperbolicSystems 135
4.1 TheoryofHyperbolicSystems 135
4.1.1 HyperbolicityandCharacteristics 135
4.1.2 LinearSystems 139
4.1.3 FramesofReference 140
4.1.3.1 UsefulIdentities 141
4.1.3.2 ChangeofFrameofReferencefor
ConservationLaws 143
4.1.3.3 ChangeofFrameofReferencefor
PropagatingDiscontinuities 145
4.1.4 Rankine–HugoniotJumpCondition 146
4.1.5 LaxAdmissibilityConditions 150
4.1.6 AsymptoticBehaviorofHugoniotLoci 152
4.1.7 CenteredRarefactions 156
4.1.8 RiemannProblems 159
4.1.9 RiemannProblemforLinearSystems 159
4.1.10 RiemannProblemforShallowWater 162
4.1.11 EntropyFunctions 164
4.2 UpwindSchemes 176
4.2.1 Lax–FriedrichsScheme 176
4.2.2 RusanovScheme 179
4.2.3 GodunovScheme 179
4.3 CaseStudy:Maxwell’sEquations 183
4.3.1 ConservationLaws 183
4.3.2 CharacteristicAnalysis 184
4.4 CaseStudy:GasDynamics 186
4.4.1 ConservationLaws 187
4.4.2 Thermodynamics 187
4.4.3 CharacteristicAnalysis 188
4.4.4 EntropyFunction 190
4.4.5 CenteredRarefactionCurves 192
4.4.6 JumpConditions 194
4.4.7 RiemannProblem 200
4.4.8 ReflectingWalls 205
4.5 CaseStudy:Magnetohydrodynamics(MHD) 208
4.5.1 ConservationLaws 208
4.5.2 CharacteristicAnalysis 209
4.5.3 EntropyFunction 218
4.5.4 CenteredRarefactionCurves 218
4.5.5 JumpConditions 220
x Contents
4.6 CaseStudy:FiniteDeformationinElasticSolids 221
4.6.1 EulerianFormulationofEquationsofMotionforSolids 221
4.6.2 LagrangianFormulationofEquationsofMotionforSolids 222
4.6.3 ConstitutiveLaws 223
4.6.4 ConservationFormoftheEquationsofMotionforSolids 225
4.6.5 JumpConditionsforIsothermalSolids 226
4.6.6 CharacteristicAnalysisforSolids 227
4.7 CaseStudy:LinearElasticity 233
4.8 CaseStudy:VibratingString 235
4.8.1 ConservationLaws 235
4.8.2 CharacteristicAnalysis 237
4.8.3 JumpConditions 238
4.8.4 LaxAdmissibilityConditions 240
4.8.5 EntropyFunction 240
4.8.6 WaveFamiliesforConcaveTension 241
4.8.7 WaveFamilyIntersections 245
4.8.8 RiemannProblemSolution 249
4.9 CaseStudy:Plasticity 255
4.9.1 LagrangianEquationsofMotion 255
4.9.2 ConstitutiveLaws 256
4.9.3 CenteredRarefactions 258
4.9.4 HugoniotLoci 259
4.9.5 EntropyFunction 261
4.9.6 RiemannProblem 261
4.10 CaseStudy:PolymerModel 267
4.10.1 ConstitutiveLaws 268
4.10.2 CharacteristicAnalysis 269
4.10.3 JumpConditions 270
4.10.4 RiemannProblemSolution 271
4.11 CaseStudy:Three-PhaseBuckley–LeverettFlow 274
4.11.1 ConstitutiveModels 274
4.11.2 CharacteristicAnalysis 276
4.11.3 UmbilicPoint 277
4.11.4 EllipticRegions 277
4.12 CaseStudy:Schaeffer–Schechter–ShearerSystem 278
4.13 ApproximateRiemannSolvers 284
4.13.1 DesignofApproximateRiemannSolvers 284
4.13.2 ArtificialDiffusion 291
4.13.3 RusanovSolver 293
4.13.4 WeakWaveRiemannSolver 294