Numerical Analysis Using R: Solutions to ODEs and PDEs

646 Pages·2016·17.47 MB·English

by  Graham W. Griffiths

Numerical Analysis Using R
Thisbookpresentsthelatestnumericalsolutionstoinitialvalueproblemsand
boundary value problems described by ODEs and PDEs.The author offers
practicalmethodsthatcanbeadaptedtosolvewiderangesofproblemsand
illustratesthemintheincreasinglypopularopensourcecomputerlanguageR,
allowingintegrationwithmorestatisticallybasedmethods.
The book begins with standard techniques, followed by an overview of
“high-resolution”flux limiters and WENO to solve problems with solutions
exhibiting high-gradient phenomena. Meshless methods using radial basis
functionsarethendiscussedinthecontextofscattereddatainterpolationand
the solution of PDEs on irregular grids.Three detailed case studies demon-
strate how numerical methods can be used to tackle very different complex
problems.
With its focus on practical solutions to real-world problems,this book is
useful to students and practitioners in all areas of science and engineering,
especially those using R.R Code is available for download from the book’s
homepage.
Graham W.Griffiths is a visiting professor in the Schoolof Engineeringand
MathematicalSciences,CityUniversityLondon.Hisprimaryinterestsarein
numerical methods and climate modeling, on which he has previously pub-
lishedfourbooks.GriffithswasafounderofSpecialAnalysisandSimulation
TechnologyLtd.andlaterbecamevicepresidentofoperationsandtechnology
withAspenTech.HeisaCharteredEngineerandaFellowoftheInstituteof
Measurement and Control and was granted Freedom of the City of London
in1995.

NUMERICAL
ANALYSIS USING R
Solutions to ODEs and PDEs
Graham W. Griffiths
CityUniversity,UnitedKingdom
32AvenueoftheAmericas,NewYork,NY10013
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©GrahamW.Griffiths2016
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Firstpublished2016
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AcatalogrecordforthispublicationisavailablefromtheBritishLibrary.
LibraryofCongressCataloginginPublicationData
Names:Griffiths,GrahamW.
Title:NumericalanalysisusingR:solutionstoODEsandPDEs/GrahamW.Griffiths,
CityUniversity,UnitedKingdom.
Description:NewYork,NY:CambridgeUniversityPress,2016.|Includesbibliographical
referencesandindex.
Identifiers:LCCN2015046150|ISBN9781107115613(hardback:alk.paper)
Subjects:LCSH:Initialvalueproblems–Dataprocessing.|Boundaryvalueproblems–
Dataprocessing.|Differentialequations–Dataprocessing.|Differentialequations,
Partial–Dataprocessing.|Numericalanalysis.|R(Computerprogramlanguage)
Classification:LCCQA378.G762016|DDC518.0285/5133–dc23
LCrecordavailableathttp://lccn.loc.gov/2015046150
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Contents
Preface pagexv
1 ODEIntegrationMethods . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1 Introduction 1
1.2 EulerMethods 11
1.2.1 ForwardEuler 11
1.2.2 BackwardEuler 12
1.3 Runge–KuttaMethods 12
1.3.1 RKCoefficients 15
1.3.2 VariableStepSizeMethods 19
1.3.3 SHK:Sommeijer,VanDerHouwen,andKokMethod 36
1.4 LinearMultistepMethods(LMMs) 37
1.4.1 General 37
1.4.2 BackwardDifferentiationFormulas(BDFs) 38
1.4.3 NumericalDifferentiationFormulas(NDFs) 44
1.4.4 Convergence 46
1.4.5 AdamsMethods 60
1.5 TruncationErrorandOrderofIntegration 61
1.5.1 LMMTruncationError 62
1.5.2 VerificationofIntegrationOrder 66
1.6 Stiffness 69
1.7 HowtoChooseaNumericalIntegrator 69
1.A InstallationoftheRPackageRyacas 70
1.B InstallationoftheRPackagerSymPy 71
References 72
2 StabilityAnalysisofODEIntegrators . . . . . . . . . . . . . . . . . . . . 74
2.1 General 74
2.1.1 DahlquistBarrierTheorems 75
2.2 DahlquistTestProblem 75
2.3 EulerMethods 76
vii
viii Contents
2.3.1 ForwardEuler 76
2.3.2 BackwardEuler 76
2.4 Runge–KuttaMethods 76
2.4.1 RK-1:First-OrderRunge–Kutta 76
2.4.2 RK-2:Second-OrderRunge–Kutta 79
2.4.3 RK-4:Fourth-OrderRunge–Kutta 80
2.4.4 RKF-54:FehlbergRunge–Kutta 83
2.4.5 SHK:Sommeijer,vanderHouwen,andKok 85
2.5 LinearMultistepMethods(LMMs) 87
2.5.1 General 87
2.5.2 BackwardDifferentiationFormulas(BDFs) 89
2.5.3 NumericalDifferentiationFormulas(NDFs) 95
2.5.4 AdamsMethods 97
References 101
3 NumericalSolutionofPDEs . . . . . . . . . . . . . . . . . . . . . . . . 102
3.1 SomePDEBasics 102
3.2 InitialandBoundaryConditions 103
3.3 TypesofPDESolutions 105
3.4 PDESubscriptNotation 105
3.5 AGeneralPDESystem 106
3.6 ClassificationofPDEs 107
3.7 Discretization 109
3.7.1 GeneralFiniteDifferenceTerminology 109
3.7.2 TheMesh 111
3.7.3 NonuniformGridSpacing 112
3.7.4 TheCourant–Friedrichs–LewyNumber 112
3.7.5 TheStencil 112
3.7.6 Upwinding 113
3.8 MethodofLines(MOL) 114
3.8.1 Introduction 114
3.8.2 FiniteDifferenceMatrices 115
3.8.3 MOL1D:CartesianCoordinates 123
3.8.4 MOL2D:CartesianCoordinates 141
3.8.5 MOL2D:PolarCoordinates 175
3.9 FullyDiscreteMethods 194
3.9.1 Introduction 194
3.9.2 OverviewofSomeCommonSchemes 194
3.9.3 ResultsfromSimulatingaHyperbolicEquation 197
3.10 FiniteVolumeMethod 207
3.10.1 General 207
3.10.2 Applicationtoa1DConservativeSystem 208
3.10.3 ApplicationtoaGeneralConservationLaw 210
3.11 InterpretationofResults 210
3.11.1 Verification 210
3.11.2 Validation 211
3.11.3 TruncationError 211