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FUNDAMENTALS OF ENGINEERING NUMERICAL ANALYSIS
SECOND EDITION
Sincetheoriginalpublicationofthisbook,availablecomputerpowerhasincreased
greatly. Today, scientific computing is playing an ever more prominent role as a
tool in scientific discovery and engineering analysis. In this second edition, the
keyadditionisanintroductiontothefiniteelementmethod.Thisisawidelyused
technique for solving partial differential equations (PDEs) in complex domains.
Thistextintroducesnumericalmethodsandshowshowtodevelop,analyze,anduse
them.CompleteMATLABprogramsforalltheworkedexamplesarenowavailable
at www.cambridge.org/Moin, and more than 30 exercises have been added. This
thorough and practical book is intended as a first course in numerical analysis,
primarily for new graduate students in engineering and physical science. Along
withmasteringthefundamentalsofnumericalmethods,studentswilllearntowrite
theirowncomputerprogramsusingstandardnumericalmethods.
Parviz Moin is the Franklin P. and Caroline M. Johnson Professor of Mechanical
Engineering at Stanford University. He is the founder of the Center for Turbu-
lence Research and the Stanford Institute for Computational and Mathematical
Engineering.Hepioneeredtheuseofhigh-fidelitynumericalsimulationsandmas-
sivelyparallelcomputersforthestudyofturbulencephysics.ProfessorMoinisa
Fellow of the American Physical Society, American Institute of Aeronautics and
Astronautics, and the American Academy of Arts and Sciences. He is a Member
oftheNationalAcademyofEngineering.
FUNDAMENTALS OF ENGINEERING
NUMERICAL ANALYSIS
SECOND EDITION
PARVIZ MOIN
StanfordUniversity
CAMBRIDGEUNIVERSITYPRESS
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permissionofCambridgeUniversityPress.
Firstpublished2010
PrintedintheUnitedStatesofAmerica
AcatalogrecordforthispublicationisavailablefromtheBritishLibrary.
LibraryofCongressCataloginginPublicationdata
Moin,Parviz.
Fundamentalsofengineeringnumericalanalysis/ParvizMoin.–2nded.
p. cm.
Includesbibliographicalreferencesandindex.
ISBN978-0-521-88432-7(hardback)
1.Engineeringmathematics. 2.Numericalanalysis.
I.Title. II.Title:Engineeringnumericalanalysis.
TA335.M65 2010
620.001(cid:2)518–dc22 2010009012
ISBN978-0-521-88432-7Hardback
ISBN978-0-521-71123-4Paperback
Additionalresourcesforthispublicationatwww.cambridge.org/Moin
CambridgeUniversityPresshasnoresponsibilityforthepersistenceor
accuracyofURLsforexternalorthird-partyInternetWebsitesreferredtoin
thispublicationanddoesnotguaranteethatanycontentonsuchWebsitesis,
orwillremain,accurateorappropriate.
Contents
PrefacetotheSecondEdition pageix
PrefacetotheFirstEdition xi
1 INTERPOLATION 1
1.1 LagrangePolynomialInterpolation 1
1.2 CubicSplineInterpolation 4
Exercises 8
FurtherReading 12
2 NUMERICAL DIFFERENTIATION – FINITE DIFFERENCES 13
2.1 ConstructionofDifferenceFormulasUsingTaylorSeries 13
2.2 AGeneralTechniqueforConstructionofFiniteDifference
Schemes 15
2.3 AnAlternativeMeasurefortheAccuracyofFinite
Differences 17
2.4 Pade´ Approximations 20
2.5 Non-UniformGrids 23
Exercises 25
FurtherReading 29
3 NUMERICAL INTEGRATION 30
3.1 TrapezoidalandSimpson’sRules 30
3.2 ErrorAnalysis 31
3.3 TrapezoidalRulewithEnd-Correction 34
3.4 RombergIntegrationandRichardsonExtrapolation 35
3.5 AdaptiveQuadrature 37
3.6 GaussQuadrature 40
Exercises 44
FurtherReading 47
v
vi CONTENTS
4 NUMERICAL SOLUTION OF ORDINARY DIFFERENTIAL EQUATIONS 48
4.1 InitialValueProblems 48
4.2 NumericalStability 50
4.3 StabilityAnalysisfortheEulerMethod 52
4.4 ImplicitorBackwardEuler 55
4.5 NumericalAccuracyRevisited 56
4.6 TrapezoidalMethod 58
4.7 LinearizationforImplicitMethods 62
4.8 Runge–KuttaMethods 64
4.9 Multi-StepMethods 70
4.10 SystemofFirst-OrderOrdinaryDifferentialEquations 74
4.11 BoundaryValueProblems 78
4.11.1 ShootingMethod 79
4.11.2 DirectMethods 82
Exercises 84
FurtherReading 100
5 NUMERICAL SOLUTION OF PARTIAL DIFFERENTIAL EQUATIONS 101
5.1 Semi-Discretization 102
5.2 vonNeumannStabilityAnalysis 109
5.3 ModifiedWavenumberAnalysis 111
5.4 ImplicitTimeAdvancement 116
5.5 AccuracyviaModifiedEquation 119
5.6 DuFort–FrankelMethod:AnInconsistentScheme 121
5.7 Multi-Dimensions 124
5.8 ImplicitMethodsinHigherDimensions 126
5.9 ApproximateFactorization 128
5.9.1 StabilityoftheFactoredScheme 133
5.9.2 AlternatingDirectionImplicitMethods 134
5.9.3 MixedandFractionalStepMethods 136
5.10 EllipticPartialDifferentialEquations 137
5.10.1 IterativeSolutionMethods 140
5.10.2 ThePointJacobiMethod 141
5.10.3 Gauss–SeidelMethod 143
5.10.4 SuccessiveOverRelaxationScheme 144
5.10.5 MultigridAcceleration 147
Exercises 154
FurtherReading 166
6 DISCRETE TRANSFORM METHODS 167
6.1 FourierSeries 167
6.1.1 DiscreteFourierSeries 168
6.1.2 FastFourierTransform 169
6.1.3 FourierTransformofaRealFunction 170
6.1.4 DiscreteFourierSeriesinHigherDimensions 172
CONTENTS vii
6.1.5 DiscreteFourierTransformofaProductofTwo
Functions 173
6.1.6 DiscreteSineandCosineTransforms 175
6.2 ApplicationsofDiscreteFourierSeries 176
6.2.1 DirectSolutionofFiniteDifferencedEllipticEquations 176
6.2.2 DifferentiationofaPeriodicFunctionUsingFourier
SpectralMethod 180
6.2.3 NumericalSolutionofLinear,ConstantCoefficient
DifferentialEquationswithPeriodicBoundary
Conditions 182
6.3 MatrixOperatorforFourierSpectralNumerical
Differentiation 185
6.4 DiscreteChebyshevTransformandApplications 188
6.4.1 NumericalDifferentiationUsingChebyshevPolynomials 192
6.4.2 QuadratureUsingChebyshevPolynomials 195
6.4.3 MatrixFormofChebyshevCollocationDerivative 196
6.5 MethodofWeightedResiduals 200
6.6 TheFiniteElementMethod 201
6.6.1 ApplicationoftheFiniteElementMethodtoaBoundary
ValueProblem 202
6.6.2 ComparisonwithFiniteDifferenceMethod 207
6.6.3 ComparisonwithaPade´ Scheme 209
6.6.4 ATime-DependentProblem 210
6.7 ApplicationtoComplexDomains 213
6.7.1 ConstructingtheBasisFunctions 215
Exercises 221
FurtherReading 226
A A REVIEW OF LINEAR ALGEBRA 227
A.1 Vectors,MatricesandElementaryOperations 227
A.2 SystemofLinearAlgebraicEquations 230
A.2.1 EffectsofRound-offError 230
A.3 OperationsCounts 231
A.4 EigenvaluesandEigenvectors 232
Index 235
ToLinda
