Spectral Methods for Time-Dependent Problems

281 Pages·2007·1.307 MB·English

by  Jan S. Hesthaven, Sigal Gottlieb, David Gottlieb

CAMBRIDGEMONOGRAPHSON
APPLIEDANDCOMPUTATIONAL
MATHEMATICS
SeriesEditors
M.ABLOWITZ,S.DAVIS,S.HOWISON,A.ISERLES,A.MAJDA,
J.OCKENDON,P.OLVER
21 Spectral Methods for Time-Dependent
Problems
TheCambridgeMonographsonAppliedandComputationalMathematicsreflectsthe
crucialroleofmathematicalandcomputationaltechniquesincontemporaryscience.The
seriespublishesexpositionsonallaspectsofapplicableandnumericalmathematics,with
anemphasisonnewdevelopmentsinthisfast-movingareaofresearch.
State-of-the-artmethodsandalgorithmsaswellasmodernmathematicaldescriptions
ofphysicalandmechanicalideasarepresentedinamannersuitedtograduateresearch
studentsandprofessionalsalike.Soundpedagogicalpresentationisaprerequisite.Itis
intendedthatbooksintheserieswillservetoinformanewgenerationofresearchers.
Alsointhisseries:
1. APracticalGuidetoPseudospectralMethods,BengtFornberg
2. DynamicalSystemsandNumericalAnalysis,A.M.StuartandA.R.Humphries
3. LevelSetMethodsandFastMarchingMethods,J.A.Sethian
4. TheNumericalSolutionofIntegralEquationsoftheSecondKind,Kendall
E.Atkinson
5. OrthogonalRationalFunctions,AdhemarBultheel,PabloGonza´lez-Vera,Erik
Hendiksen,andOlavNja˚stad
6. TheTheoryofComposites,GraemeW.Milton
7. GeometryandTopologyforMeshGeneration,HerbertEdelsbrunner
8. Schwarz-ChristoffelMapping,TofinA.DriscollandLloydN.Trefethen
9. High-OrderMethodsforIncompressibleFluidFlow,M.O.Deville,P.F.Fischer,
andE.H.Mund
10. PracticalExtrapolationMethods,AvramSidi
11. GeneralizedRiemannProblemsinComputationalFluidDynamics,Matania
Ben-ArtziandJosephFalcovitz
12. RadialBasisFunctions:TheoryandImplementations,MartinD.Buhmann
13. IterativeKrylovMethodsforLargeLinearSystems,HenkA.vanderVorst
14. SimulatingHamiltonianDynamics,BenLeimkuhlerandSebastianReich
15. CollocationMethodsforVolterraIntegralandRelatedFunctionalEquations,
HermannBrunner
16. Topologyforcomputing,AfraJ.Zomordia
17. ScatteredDataApproximation,HolgerWendland
19. MatrixPreconditioningTechniquesandApplications,KeChen
22. TheMathematicalFoundationsofMixing,RobSturman,JulioM.Ottinoand
StephenWiggins
Spectral Methods for Time-Dependent
Problems
JAN S. HESTHAVEN
BrownUniversity
SIGAL GOTTLIEB
UniversityofMassachusetts,Dartmouth
DAVID GOTTLIEB
BrownUniversity
Forourchildrenandgrandchildren
Contents
Introduction 1
1 Fromlocaltoglobalapproximation 5
1.1 Comparisonsoffinitedifferenceschemes 9
1.2 TheFourierspectralmethod:firstglance 16
2 Trigonometricpolynomialapproximation 19
2.1 Trigonometricpolynomialexpansions 19
2.2 Discretetrigonometricpolynomials 24
2.3 Approximationtheoryforsmoothfunctions 34
3 Fourierspectralmethods 43
3.1 Fourier–Galerkinmethods 43
3.2 Fourier–collocationmethods 48
3.3 StabilityoftheFourier–Galerkinmethod 52
3.4 StabilityoftheFourier–collocationmethodforhyperbolic
problemsI 54
3.5 StabilityoftheFourier–collocationmethodforhyperbolic
problemsII 58
3.6 Stabilityforparabolicequations 62
3.7 Stabilityfornonlinearequations 64
3.8 Furtherreading 65
4 Orthogonalpolynomials 66
4.1 ThegeneralSturm–Liouvilleproblem 67
4.2 Jacobipolynomials 69
5 Polynomialexpansions 79
5.1 Thecontinuousexpansion 79
5.2 Gaussquadratureforultrasphericalpolynomials 83
vii
viii Contents
5.3 Discreteinnerproductsandnorms 88
5.4 Thediscreteexpansion 89
6 Polynomialapproximationtheoryforsmoothfunctions 109
6.1 Thecontinuousexpansion 109
6.2 Thediscreteexpansion 114
7 Polynomialspectralmethods 117
7.1 Galerkinmethods 117
7.2 Taumethods 123
7.3 Collocationmethods 129
7.4 Penaltymethodboundaryconditions 133
8 Stabilityofpolynomialspectralmethods 135
8.1 TheGalerkinapproach 135
8.2 Thecollocationapproach 142
8.3 Stabilityofpenaltymethods 145
8.4 Stabilitytheoryfornonlinearequations 150
8.5 Furtherreading 152
9 Spectralmethodsfornonsmoothproblems 153
9.1 TheGibbsphenomenon 154
9.2 Filters 160
9.3 TheresolutionoftheGibbsphenomenon 174
9.4 Linearequationswithdiscontinuoussolutions 182
9.5 Furtherreading 186
10 Discretestabilityandtimeintegration 187
10.1 Stabilityoflinearoperators 188
10.2 Standardtimeintegrationschemes 192
10.3 Strongstabilitypreservingmethods 197
10.4 Furtherreading 202
11 Computationalaspects 204
11.1 Fastcomputationofinterpolationanddifferentiation 204
11.2 ComputationofGaussianquadraturepointsandweights 210
11.3 Finiteprecisioneffects 214
11.4 Ontheuseofmappings 225
12 Spectralmethodsongeneralgrids 235
12.1 Representingsolutionsandoperatorsongeneralgrids 236
12.2 Penaltymethods 238
Contents ix
12.3 DiscontinuousGalerkinmethods 246
12.4 Referencesandfurtherreading 248
AppendixA Elementsofconvergencetheory 249
AppendixB Azooofpolynomials 252
B.1 Legendrepolynomials 252
B.2 Chebyshevpolynomials 255
Bibliography 260
Index 272
Introduction
Thepurposeofthisbookistocollect,inonevolume,alltheingredientsnec-
essaryfortheunderstandingofspectralmethodsfortime-dependentproblems,
and, in particular, hyperbolic partial differential equations. It is intended as
a graduate-level text, covering not only the basic concepts in spectral meth-
ods,butsomeofthemoderndevelopmentsaswell.Therearealreadyseveral
excellentbooksonspectralmethodsbyauthorswhoarewell-knownandactive
researchersinthisfield.Thisbookisdistinguishedbytheexclusivetreatment
oftime-dependentproblems,andsothederivationofspectralmethodsisinflu-
encedprimarilybytheresearchonfinite-differenceschemes,andlesssobythe
finite-element methodology. Furthermore, this book is unique in its focus on
thestabilityanalysisofspectralmethods,bothforthesemi-discreteandfully
discretecases.Inthebookweaddressadvancedtopicssuchasspectralmeth-
odsfordiscontinuousproblemsandspectralmethodsonarbitrarygrids,which
arenecessaryfortheimplementationofpseudo-spectralmethodsoncomplex
multi-dimensionaldomains.
InChapter1,wedemonstratethebenefitsofhighordermethodsusingphase
erroranalysis.Typicalfinitedifferencemethodsusealocalstenciltocompute
thederivativeatagivenpoint;higherordermethodsarethenobtainedbyusinga
widerstencil,i.e.,morepoints.TheFourierspectralmethodisobtainedbyusing
allthepointsinthedomain.InChapter2,wediscussthetrigonometricpoly-
nomialapproximationstosmoothfunctions,andtheassociatedapproximation
theoryforboththecontinuousandthediscretecase.InChapter3,wepresent
Fourierspectralmethods,usingboththeGalerkinandcollocationapproaches,
anddiscusstheirstabilityforbothhyperbolicandparabolicequations.Wealso
presentwaysofstabilizingthesemethods,throughsuperviscosityorfiltering.
Chapter4featuresadiscussionoffamiliesoforthogonalpolynomialswhich
areeigensolutionsofaSturm–Liouvilleproblem.WefocusontheLegendreand
Chebyshevpolynomials,whicharesuitableforrepresentingfunctionsonfinite
1
2 Introduction
domains.Inthischapter,wepresentthepropertiesofJacobipolynomials,and
theirassociatedrecursionrelations.Manyusefulformulascanbefoundinthis
chapter.InChapter5,wediscussthecontinuousanddiscretepolynomialexpan-
sionsbasedonJacobipolynomials;inparticular,theLegendreandChebyshev
polynomials.WepresenttheGauss-typequadratureformulas,andthedifferent
pointsonwhicheachisaccurate.Finally,wediscusstheconnectionsbetween
Lagrangeinterpolationandelectrostatics.Chapter6presentstheapproximation
theory for polynomial expansions of smooth functions using the ultraspheri-
cal polynomials. Both the continuous and discrete expansions are discussed.
ThisdiscussionsetsthestageforChapter7,inwhichweintroducepolynomial
spectralmethods,usefulforproblemswithnon-periodicboundaryconditions.
We present the Galerkin, tau, and collocation approaches and give examples
oftheformulationofChebyshevandLegendrespectralmethodsforavariety
ofproblems.Wealsointroducethepenaltymethodapproachfordealingwith
boundary conditions. In Chapter 8 we analyze the stability properties of the
methodsdiscussedinChapter7.
Inthefinalchapters,weintroducesomemoreadvancedtopics.InChapter9
wediscussthespectralapproximationsofnon-smoothproblems.Weaddress
theGibbsphenomenonanditseffectontheconvergencerateoftheseapproxi-
mations,andpresentmethodswhichcan,partiallyorcompletely,overcomethe
Gibbsphenomenon.Wepresentavarietyoffilters,bothforFourierandpoly-
nomial methods, and an approximation theory for filters. Finally, we discuss
theresolutionoftheGibbsphenomenonusingspectralreprojectionmethods.
In Chapter 10, we turn to the issues of time discretization and fully discrete
stability. We discuss the eigenvalue spectrum of each of the spectral spatial
discretizations, which provides a necessary, but not sufficient, condition for
stability.Weproceedtothefullydiscreteanalysisofthestabilityoftheforward
EulertimediscretizationfortheLegendrecollocationmethod.Wethenpresent
some of the standard time integration methods, especially the Runge–Kutta
methods. At the end of the chapter, we introduce the class of strong stability
preservingmethodsandpresentsomeoftheoptimalschemes.InChapter11,we
turntothecomputationalissueswhicharisewhenusingspectralmethods,such
astheuseofthefastFouriertransformforinterpolationanddifferentiation,the
efficientcomputationoftheGaussquadraturepointsandweights,andtheeffect
of round-off errors on spectral methods. Finally, we address the use of map-
pingsfortreatmentofnon-standardintervalsandforimprovingaccuracyinthe
computationofhigherorderderivatives.InChapter12,wetalkabouttheimple-
mentation of spectral methods on general grids. We discuss how the penalty
methodformulationenablestheuseofspectralmethodsongeneralgridsinone
dimension,andincomplexdomainsinmultipledimensions,andillustratethis