Numerical solution of differential equations : introduction to finite difference and finite difference and finite element methods

303 Pages·2018·3.574 MB·English

by  Li, Zhilin, Tang, Tao, Zhonghua, Qiao

“FM” — 2017/10/11 — 12:40 — page 1 — #1
NUMERICAL SOLUTION OF
DIFFERENTIAL EQUATIONS
This introduction to finite difference and finite element methods is aimed at
advancedundergraduateandgraduatestudentswhoneedtosolvedifferential
equations. The prerequisites are few (basic calculus, linear algebra, and ordi-
naryandpartialdifferentialequations)andsothebookwillbeaccessibleand
usefultoreadersfromarangeofdisciplinesacrossscienceandengineering.
Part I begins with finite difference methods. Finite element methods are
thenintroducedinPartII.Ineachpart,theauthorsbeginwithacomprehen-
sive discussion of one-dimensional problems, before proceeding to consider
two or higher dimensions. An emphasis is placed on numerical algorithms,
relatedmathematicaltheory,andessentialdetailsintheimplementation,while
someusefulpackagesarealsointroduced.Theauthorsalsoprovidewell-tested
Matlab® codes,allavailableonline.
ZHILINLIisatenuredfullprofessorattheCenterforScientificComputation&
DepartmentofMathematicsatNorthCarolinaStateUniversity.Hisresearch
area is in applied mathematics in general, particularly in numerical analysis
for partial differential equations, moving interface/free boundary problems,
irregular domain problems, computational mathematical biology, and sci-
entific computing and simulations for interdisciplinary applications. Li has
authoredonemonograph,TheImmersedInterfaceMethod,andalsoeditedsev-
eralbooksandproceedings.
ZHONGHUAQIAO is an associate professor in the Department of Applied
MathematicsattheHongKongPolytechnicUniversity.
TAOTANG is a professor in the Department of Mathematics at Southern
UniversityofScienceandTechnology,China.
13:40:38, subject to the Cambridge Core
“FM” — 2017/10/11 — 12:40 — page 2 — #2
13:40:38, subject to the Cambridge Core
“FM” — 2017/10/11 — 12:40 — page 3 — #3
NUMERICAL SOLUTION OF
DIFFERENTIAL EQUATIONS
Introduction to Finite Difference and
Finite Element Methods
ZHILIN LI
NorthCarolinaStateUniversity,USA
ZHONGHUA QIAO
HongKongPolytechnicUniversity,China
TAO TANG
SouthernUniversityofScienceandTechnology,China
13:40:38, subject to the Cambridge Core
“FM” — 2017/10/11 — 12:40 — page 4 — #4
UniversityPrintingHouse,CambridgeCB28BS,UnitedKingdom
OneLibertyPlaza,20thFloor,NewYork,NY10006,USA
477WilliamstownRoad,PortMelbourne,VIC3207,Australia
314–321,3rdFloor,Plot3,SplendorForum,JasolaDistrictCentre,NewDelhi–110025,India
79AnsonRoad,#06-04/06,Singapore079906
CambridgeUniversityPressispartoftheUniversityofCambridge.
ItfurtherstheUniversity’smissionbydisseminatingknowledgeinthepursuitof
education,learning,andresearchatthehighestinternationallevelsofexcellence.
www.cambridge.org
Informationonthistitle:www.cambridge.org/9781107163225
DOI:10.1017/9781316678725
©ZhilinLi,ZhonghuaQiao,andTaoTang2018
Thispublicationisincopyright.Subjecttostatutoryexception
andtotheprovisionsofrelevantcollectivelicensingagreements,
noreproductionofanypartmaytakeplacewithoutthewritten
permissionofCambridgeUniversityPress.
Firstpublished2018
PrintedintheUnitedKingdombyClays,StIvesplc
AcataloguerecordforthispublicationisavailablefromtheBritishLibrary.
ISBN978-1-107-16322-5Hardback
ISBN978-1-316-61510-2Paperback
CambridgeUniversityPresshasnoresponsibilityforthepersistenceoraccuracy
ofURLsforexternalorthird-partyinternetwebsitesreferredtointhispublication
anddoesnotguaranteethatanycontentonsuchwebsitesis,orwillremain,
accurateorappropriate.
13:40:38, subject to the Cambridge Core
“FM” — 2017/10/11 — 12:40 — page v — #5
Table of Contents
Preface pageix
1 Introduction 1
1.1 BoundaryValueProblemsofDifferentialEquations 1
1.2 FurtherReading 5
PARTI FINITEDIFFERENCEMETHODS 7
2 FiniteDifferenceMethodsfor1DBoundaryValueProblems 9
2.1 ASimpleExampleofaFiniteDifferenceMethod 9
2.2 FundamentalsofFiniteDifferenceMethods 14
2.3 DerivingFDFormulasUsingtheMethodofUndetermined
Coefficients 19
2.4 Consistency, Stability, Convergence, and Error Estimates of
FDMethods 21
2.5 FDMethodsfor1DSelf-adjointBVPs 27
2.6 FDMethodsforGeneral1DBVPs 29
2.7 TheGhostPointMethodforBoundaryConditionsInvolving
Derivatives 30
2.8 AnExampleofaNonlinearBVP 34
2.9 TheGridRefinementAnalysisTechnique 37
2.10 *1DIIMforDiscontinuousCoefficients 39
Exercises 44
3 FiniteDifferenceMethodsfor2DEllipticPDEs 47
3.1 BoundaryandCompatibilityConditions 49
3.2 TheCentralFiniteDifferenceMethodforPoissonEquations 51
3.3 TheMaximumPrincipleandErrorAnalysis 55
13:40:46, subject to the Cambridge Core
“FM” — 2017/10/11 — 12:40 — page vi — #6
vi TableofContents
3.4 FiniteDifferenceMethodsforGeneralSecond-orderElliptic
PDEs 60
3.5 SolvingtheResultingLinearSystemofAlgebraicEquations 61
3.6 AFourth-orderCompactFDSchemeforPoissonEquations 67
3.7 A Finite Difference Method for Poisson Equations in Polar
Coordinates 69
3.8 Programmingof2DFiniteDifferenceMethods 72
Exercises 75
4 FDMethodsforParabolicPDEs 78
4.1 TheEulerMethods 80
4.2 TheMethodofLines 85
4.3 TheCrank–Nicolsonscheme 87
4.4 StabilityAnalysisforTime-dependentProblems 89
4.5 FDMethodsandAnalysisfor2DParabolicEquations 97
4.6 TheADIMethod 99
4.7 An Implicit–explicit Method for Diffusion and Advection
Equations 104
4.8 SolvingEllipticPDEsusingNumericalMethodsforParabolic
PDEs 105
Exercises 105
5 FiniteDifferenceMethodsforHyperbolicPDEs 108
5.1 CharacteristicsandBoundaryConditions 109
5.2 FiniteDifferenceSchemes 110
5.3 TheModifiedPDEandNumericalDiffusion/Dispersion 115
5.4 TheLax–WendroffSchemeandOtherFDmethods 117
5.5 NumericalBoundaryConditions 120
5.6 Finite Difference Methods for Second-order Linear
HyperbolicPDEs 121
5.7 Some Commonly Used FD Methods for Linear System of
HyperbolicPDEs 127
5.8 FiniteDifferenceMethodsforConservationLaws 127
Exercises 131
PARTII FINITEELEMENTMETHODS 133
6 FiniteElementMethodsfor1DBoundaryValueProblems 135
6.1 TheGalerkinFEMethodforthe1DModel 135
6.2 DifferentMathematicalFormulationsforthe1DModel 138
6.3 KeyComponentsoftheFEMethodforthe1DModel 143
13:40:46, subject to the Cambridge Core
“FM” — 2017/10/11 — 12:40 — page vii — #7
TableofContents vii
6.4 Matlab Programming of the FE Method for the 1D Model
Problem 152
Exercises 156
7 TheoreticalFoundationsoftheFiniteElementMethod 158
7.1 FunctionalSpaces 158
7.2 SpacesforIntegralForms,L2(Ω)andLp(Ω) 160
7.3 SobolevSpacesandWeakDerivatives 164
7.4 FEAnalysisfor1DBVPs 168
7.5 ErrorAnalysisoftheFEMethod 173
Exercises 178
8 IssuesoftheFEMethodinOneSpaceDimension 181
8.1 BoundaryConditions 181
8.2 TheFEMethodforSturm–LiouvilleProblems 185
8.3 High-orderElements 189
8.4 A1DMatlabFEPackage 195
8.5 TheFEMethodforFourth-orderBVPsin1D 208
8.6 TheLax–MilgramLemmaandtheExistenceofFESolutions 214
8.7 *1DIFEMforDiscontinuousCoefficients 221
Exercises 223
9 TheFiniteElementMethodfor2DEllipticPDEs 228
9.1 TheSecondGreen’sTheoremandIntegrationbyPartsin2D 228
9.2 WeakFormofSecond-orderSelf-adjointEllipticPDEs 231
9.3 TriangulationandBasisFunctions 233
9.4 Transforms,ShapeFunctions,andQuadratureFormulas 246
9.5 SomeImplementationDetails 248
9.6 SimplificationoftheFEMethodforPoissonEquations 251
9.7 SomeFESpacesinH1(Ω)andH2(Ω) 257
9.8 TheFEMethodforParabolicProblems 272
Exercises 275
Appendix:NumericalSolutionsofInitialValueProblems 279
A.1 SystemofFirst-orderODEsofIVPs 279
A.2 Well-posednessofanIVP 280
A.3 SomeFiniteDifferenceMethodsforSolvingIVPs 281
A.4 SolvingIVPsUsingMatlabODESuite 284
Exercises 288
References 289
Index 291
13:40:46, subject to the Cambridge Core
“FM” — 2017/10/11 — 12:40 — page viii — #8
13:40:46, subject to the Cambridge Core
“FM” — 2017/10/11 — 12:40 — page ix — #9
Preface
Thepurposeofthisbookistoprovideanintroductiontofinitedifferenceand
finite element methods for solving ordinary and partial differential equations
ofboundaryvalueproblems.Thebookisdesignedforbeginninggraduatestu-
dents,upperlevelundergraduatestudents,andstudentsfrominterdisciplinary
areasincludingengineersandotherswhoneedtoobtainsuchnumericalsolu-
tions. The prerequisite is a basic knowledge of calculus, linear algebra, and
ordinary differential equations. Some knowledge of numerical analysis and
partialdifferentialequationswouldalsobehelpfulbutnotessential.
Theemphasisisontheunderstandingoffinitedifferenceandfiniteelement
methodsandessentialdetailsintheirimplementationwithreasonablymathe-
maticaltheory.PartIconsidersfinitedifferencemethods,andPartIIisabout
finite element methods. In each part, we start with a comprehensive discus-
sionofone-dimensionalproblemsbeforeproceedingtoconsidertwoorhigher
dimensions. We also list some useful references for those who wish to know
moreinrelatedareas.
Thematerialsofthistextbookingeneralcanbecoveredinanacademicyear.
Thetwopartsofthebookareessentiallyindependent.Thusitispossibletouse
onlyonepartforaclass.
Thisisatextbookbasedonmaterialsthattheauthorshaveused,andsome
are from Dr. Randall J. LeVeque’s notes, in teaching graduate courses on the
numericalsolutionofdifferentialequations.Mostsamplecomputerprogram-
ming is written in Matlab®. Some advantages of Matlab are its simplicity, a
widerangeoflibrarysubroutines,doubleprecisionaccuracy,andmanyexisting
andemergingtool-boxes.
Awebsitewww4.ncsu.edu/~zhilin/FD_FEM_Bookhasbeensetup,topostor
linkcomputercodesaccompanyingthistextbook.
We would like to thank Dr. Roger Hoskin, Lilian Wang, Peiqi Huang, and
HongruChenforproofreadingthebook,orforprovidingMatlabcode.
ix
08:57:00, subject to the Cambridge Core terms of use,
01
“FM” — 2017/10/11 — 12:40 — page x — #10
08:57:00, subject to the Cambridge Core terms of use,
01