High Order Difference Methods for Time Dependent PDE

345 Pages·2008·3.69 MB·English

by  Bertil Gustafsson (auth.)

S 38
pringer Series in
C
omputational
M
athematics
EditorialBoard
R.Bank
R.L.Graham
J.Stoer
R.Varga
H.Yserentant
Bertil Gustafsson
High Order
Difference Methods
for Time
Dependent PDE
With94Figuresand12Tables
123
BertilGustafsson
Ledungsvägen28
75440Uppsala
Sweden
[email protected]
ISBN 978-3-540-74992-9 e-ISBN 978-3-540-74993-6
DOI 10.1007/978-3-540-74993-6
SpringerSeriesinComputationalMathematicsISSN0179-3632
LibraryofCongressControlNumber:2007940500
MathematicsSubjectClassification(2000):65M06
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Preface
Many books have been written on finite difference methods (FDM), but there are
goodreasonstowritestillanotherone.Themainreasonisthatevenifhigherorder
methods have been known for a long time, the analysis of stability, accuracy and
effectivenessismissingtoalargeextent.Forexample,thedefinitionoftheformal
high order accuracy is based on the assumption that the true solution is smooth,
or expresseddifferently,that the grid is fine enoughsuch that all variationsin the
solution are well resolved. In many applications, this assumption is not fulfilled,
and thenit is interestingto knowif a high ordermethodis still effective.Another
problem that needs thorough analysis is the construction of boundary conditions
suchthatbothaccuracyandstabilityis upheld.Andfinally,therehasbeenquitea
strongdevelopmentduringthelastyears,inparticularwhenitcomestoverygeneral
andstabledifferenceoperatorsforapplicationoninitial–boundaryvalueproblems.
Thecontentofthebookisnotpurelytheoretical,neitherisitasetofrecipesfor
varioustypesofapplications.Theideaistogiveanoverviewofthebasictheoryand
constructionprinciplesfordifferencemethodswithoutgoingintoalldetails.Forex-
ample,certaintheoremsarepresented,buttheproofsareinmostcasesleftout.The
explanationand applicationof the theory is illustrated by using simple modelex-
amples.Amongengineers,oneisoftentalkingabout“toyproblems”withacertain
scepticism,undertheassumptionthattheresultshavenosignificanceforrealworld
problems.When lookingatthe scientific productionoverthe years,there is a cer-
taintruthtothisclaim.Amethodmaybeworkingverywell,andbetterthanother
wellknownmethods,foracertainmodelprobleminonespacedimension,butitis
notevenclearhowthe methodcanbe generalizedto severalspace dimensions.In
thisbookwetrytoavoidfallingintothistrap.Thegeneralizationshouldofcourse
always bepossible in the sense thatan algorithmbased on the same principlecan
be constructed for the full problem, and hopefully the essential properties of the
numericalmethodhavebeencaughtbytheanalysisofthesimplerproblem.Some-
times,thetheoreticalconsiderationscarryoverwithoutdifficulty,butsometimeswe
have to rely upon intuition and numerical experiments to be confident about the
performanceonthelargeproblem.Ontheotherhand,therearemanycaseswhere
themodelproblemtellsitall.Forexample,forahyperbolicsystemofPDEinone
v
vi Preface
spacedimension,onecantransformthesystemtoadiagonaloneandlimittheanal-
ysistoascalarPDEwithconstantcoefficients.Byapplyingthetransformationback,
andgeneralizingtovariablecoefficients,theresultsholdfortheoriginalproblemas
well.
Whendiscussingstabilityandaccuracy,themodelexamplesinthisbookareof-
ten of low order accuracy,since they havea simpler structure. The goalis simply
toillustratehowthetheoryworks,anditiseasiertoseethebasicmechanismsfor
simplerschemes.Oncetheapplicationofthetheoryiswellunderstood,itshouldbe
clearhowtoapplyitformorecomplicatedandpowerfulhighordermethods.How-
ever,therearetwo mainapplicationareaswherewe chooseto gointomoredetail
about the full implementation of the suggested methods. One application is wave
propagationwithrelevanceforacousticsandelectromagnetics.Inthiscasethemost
interestingproblemsaresolarge,thatthereisnorealisticlowordercomputational
alternative. The other one is incompressible flow governed by the Navier-Stokes
equations. Even if low order methods have been used for many applications, the
realchallengeisturbulentflow,wherethereisnorealisticalternativetohighorder
methodsifthesmallestscalesaretoberepresentedwell.Inbothcases,wepresent
afairlydetaileddescriptionofthemethods.Onegoodreasonforthisistoillustrate
howthe analysisandconstructiondescribedin earlierchaptersiscarriedoutfora
moretechnicallycomplicatedproblem.
Thereisapervadingthemein thebook,andthatiscompactnessofthecompu-
tationalstencils. Acomparisonbetweentwoapproximationsof thesame orderal-
wayscomesoutinfavorofthemorecompactone,i.e.,asfewgridpointsaspossible
shouldbeinvolved,bothinspaceandtime.Thesmallererrorconstantsoftenmake
quiteadramaticchange.Pade´ approximationsandstaggeredgridsareexamplesof
this,aswellastheboxschemedescribedin8.
Theoutlineofthebookisasfollows.
InChapter1thereisananalysisoftheeffectivenessofhigherordermethods.It
isbasedonFourieranalysis,andthenecessarynumberofpointsperwavelengthis
estimatedfordifferenttypesofPDEanddifferentordersofaccuracy.
Chapter2containsasurveyofthetheoryforwell-posednessandstability,andthe
differenttoolsforanalysisaredescribed.ThesetoolsarebasedonFourieranalysis
forproblemswithperiodicsolutionsandtheenergymethodandLaplacetransform
methodforinitial–boundaryvalueproblems.Differentkindsofstabilitydefinitions
arenecessaryinthiscase,andwediscusstheimplicationsofeachone.
In Chapter 3 we discuss how the order of accuracyis connectedto the conver-
gencerate,i.e.,howfastthenumericalsolutionapproachesthetruesolutionwhen
thegridisrefined.Thisisastraightforwardanalysisforperiodicproblems,butless
obviouswhenboundariesareinvolved.
Chapters2and3 canbereadindependentlyoftherestofthe bookasanintro-
ductionandasurveyofthestabilitytheoryfordifferenceapproximationsingeneral.
Thenextthreechapterscontainasystematicpresentationofthedifferentwaysof
constructinghighorderapproximations.Therearethestandardcentereddifference
operators,butalsoPade´typeoperatorsaswellasschemesbasedonstaggeredgrids.
Whenconstructingthedifferenceschemes,oneprincipleistoapproximatethedif-
Preface vii
ferential equation in space first, and then approximate the resulting ODE system
in time.Butthereare otherwaysto obtaineffectivedifferenceschemeswherethe
spaceandtimediscretizationsarenotseparated,andthisisdescribedinChapter6.
InChapter7webringupthespecialproblemofconstructingproperapproxima-
tions of the boundaryconditions,and of the modificationsof the scheme near the
boundaries.AmajorpartofthischapterisdevotedtothesocalledSBPoperators,
whicharebasedonsummationbypartsleadingtostabilityinthesenseoftheenergy
method.
Chapter8isalittledifferentfromtheotherones,sincetheboxschemediscussed
thereis onlysecondorderaccurate.However,becauseofthe compactnature,itis
sometimesmoreaccuratethanhigherorderones,andsomerecentresultsregarding
thispropertyarepresented.Anotheradvantageisthatitcaneasilybegeneralizedto
nonuniformgrids.
The next two chapters are devoted to applications. The purpose is not to give
a survey of problems and numerical methods, but rather to illustrate the applica-
tionofcertainhighordermethodsinsomedetailforafewproblemsofhighinter-
est in the engineeringandscientific communities.In this way, we hope to givean
ideaofhowtohandlethetechnicaldetails.InChapter9wediscusswavepropaga-
tionproblemsdescribedbyfirstorderhyperbolicPDEwithapplicationinacoustics
andelectromagnetics.Hereweconcentrateonaclassofhighordermethodsbased
onstaggeredgrids,anddemonstratetheeffectiveness,evenifthesolutionsarenot
smooth.We also presenta new methodofembeddingthe boundarywith a certain
definitionofthecoefficientsinthePDEsuchthatthetrueboundaryconditionsare
wellrepresented.InChapter10wediscussincompressiblefluiddynamics.Theflow
is governedby the Navier–Stokesequations,but we give a specialpresentationof
theStokesequations,sincetheyplayaspecialroleintheNavier–Stokessolver.The
methodissemi-implicitandfourthorderaccurateinspace.Largelinearsystemsof
equationsmustbesolvedforeachtimestep, andwe presenttheiterativesolverin
somedetailaswell.
A big challenge, particularly in gas dynamics, is computation of solutions to
nonlinearproblemscontainingdiscontinuitiesorshocks.Thisrequiressomespecial
theoryandspecializedmethods.Techniquesthatworkwellfornonlinearproblems
withsmoothsolutions,liketheincompressibleNavier–StokesequationsinChapter
10, are not well suited. We discuss the theory and methods for these problemsin
Chapter11.Thisisanareawherelowordermethodsdominateevenmorethanfor
linearproblems,butwegivesomeemphasisonthosemethodsthatcanbegeneral-
izedtohighorder.
Each chapter has a summary section at the end. These sections contain a brief
summaryofthetheoryandresultsofthechapter,andalsosomehistoricalremarks
andcommentsonavailableliterature.
When electronic computers came in use in the forties, the field of Numerical
Analysisexpandedveryquickly.Alreadyfromthebeginning,thesolutionofordi-
naryand partialdifferentialequationswas in focus, andthe developmentof FDM
set full speed. Hardly no other methodswere considered, and it was not until the
late sixties, that finite element methods (FEM) started emerging for solving PDE
viii Preface
problems.Inthebeginning,FEMwereusedmostlyforsteadystateproblems,and
for approximation of the space part of time dependent problems. For approxima-
tionintime,finitedifferencemethodswereused,andthisisstillthecaseformany
applications.ThegreatadvantagewithFEMistheirflexibilitywhenitcomestoap-
proximationofirregulardomains.Thisflexibilityissharedbyfinitevolumemethods
(FVM),butconstructionofhighorderFVMisnotthateasy.
Thegeometricflexibilityof FEM is notsharedby spectraland pseudo-spectral
methods, which was the next class of numerical methods that emerged for PDE.
Their strength is the very high accuracy relative to the required work. However,
these methodshave at least as manyrestrictionsas FDM on the type of computa-
tionaldomain,inparticularthosethatarebasedonapproximationbytrigonometric
polynomials(Fouriermethods).Laterthisdifficultywaspartlyovercomebytheuse
ofspectralelementmethods,wherethedomainispartitionedintomanysubdomains
withorthogonalpolynomialsusedforapproximationoneachoneofthem.
During the last decade, discontinuous Galerkin methods have arisen as a new
interestingclassofmethods.TheycanbeseenasageneralizationofFEM,andhave
thepotentialofleadingtofasteralgorithms.
Inthefinalchapter,wegiveabriefintroductiontoallofthemethodsmentioned
here.
Theavailablecommercialandpublicsoftwareistodaydominatedbyalgorithms
based on finite element methods (or finite volume methods for problems in fluid
problems). One reason for this is that the development of unstructured grid gen-
eratorshasmadeenormousprogressduringthelast decades,andthisis important
formanyapplications.Ontheotherhand,forproblemswherestructuredgridspro-
vide an acceptable representation of the computational domain, it is hard to beat
a goodhighorderdifferencemethodwhenitcomesto implementation,speedand
effectiveness.
Wehavelimitedthepresentationinthisbookalmostexclusivelytouniformgrids.
TheuseofCartesiangridsisforconvenience,wecouldofcourseuseanyotherof
theclassiccoordinatesystemslikeforexamplecylindricalcoordinates.Ingeneral,
the Cartesian uniformgrid can be seen as the modelfor all cases where a smooth
mappingtakesthephysicaldomaintoarectanglein2-Dorhyper-rectangleinhigher
dimensions.
Ifasmoothmappingcannotbeusedfortransformationofthewholedomain,one
can use Cartesian coordinates together with some sort of interpolation procedure
neartheboundary,andconstructthefinitedifferenceslocallyonunstructuredgrids.
Anotherwayistoconstructonelocalstructuredgridthatfitstheirregularboundary,
and anothergrid thatis used in the main partof the domain.These two (or more)
gridsarethenconnectedviainterpolation.Athirdwayistocoupleafinitedifference
method with a finite element or finite volume method near the boundary, and in
this way arriving at a hybrid method. Still another way is to embed the irregular
boundaryinalargerdomainwithregularboundaries,andtoenforcetheboundary
conditionsbysomemodificationofthePDE.
Preface ix
If the need fora nonstructuredgrid arises fromthe fact thatthe solution varies
on very different scales, the most general finite difference technique is based on
piecewiseuniformgrids,thatarecoupledbyaninterpolationprocedure.
Thereisalso the possibilityto constructFDM directlyon unstructuredgridsin
thewholecomputationaldomain.However,thestabilityisthenaharderissue,and
furthermore,theeffectivenesswillnotbemuchbetterthanwithFEM.
Acknowledgment
The main part of this book was written after my retirement from the chair at the
DivisionofScientificComputingatUppsalaUniversity.However,theuniversityhas
providedallthenecessaryinfrastructure,suchaslibrariesandcomputersduringthe
wholeproject.Furthermore,thestaffattheDepartmentofInformationTechnology
have been very helpful when needed. During the writing period I have also spent
some time as a visitor at Stanford University, and I want to express my gratitude
to the Center for Turbulence Research (CTR) and the Institute for Computational
MathematicsinEngineering(ICME)forprovidingverygoodworkingconditions.
FinallyIwouldliketothankmywifeMargaretaforenduringstillanothermajor
undertakingfrommysideafterformalretirement.
Uppsala,Sweden,November2007 BertilGustafsson
Contents
1 WhenareHighOrderMethodsEffective? ........................ 1
1.1 Preliminaries .............................................. 1
1.2 WavePropagationProblems.................................. 2
1.3 ParabolicEquations......................................... 8
1.4 Schro¨dingerTypeEquations ................................. 11
1.5 Summary ................................................. 12
2 Well-posednessandStability .................................... 13
2.1 WellPosedProblems ....................................... 13
2.2 PeriodicProblemsandFourierAnalysis........................ 16
2.2.1 ThePDEProblem.................................... 17
2.2.2 DifferenceApproximations............................ 21
2.3 Initial–BoundaryValueProblemsandtheEnergyMethod......... 29
2.3.1 ThePDEProblem.................................... 29
2.3.2 SemidiscreteApproximations.......................... 33
2.3.3 FullyDiscreteApproximations......................... 38
2.4 Initial–BoundaryValueProblemsandNormalModeAnalysisfor
HyperbolicSystems ........................................ 41
2.4.1 SemidiscreteApproximations.......................... 41
2.4.2 FullyDiscreteApproximations......................... 59
2.5 Summary ................................................. 66
3 OrderofAccuracyandtheConvergenceRate ..................... 69
3.1 PeriodicSolutions.......................................... 69
3.2 Initial–BoundaryValueProblems ............................. 72
3.3 Summary ................................................. 79
4 ApproximationinSpace ........................................ 81
4.1 HighOrderFormulasonStandardGrids ....................... 81
4.2 HighOrderFormulasonStaggeredGrids ...................... 85
4.3 CompactPade´TypeDifferenceOperators ...................... 87
xi
xii Contents
4.4 OptimizedDifferenceOperators .............................. 91
4.5 Summary ................................................. 93
5 ApproximationinTime ......................................... 95
5.1 StabilityandtheTestEquation ............................... 95
5.2 Runge–KuttaMethods ...................................... 97
5.3 LinearMultistepMethods ...................................102
5.4 DeferredCorrection ........................................108
5.5 RichardsonExtrapolation....................................111
5.6 Summary .................................................113
6 CoupledSpace-TimeApproximations ............................115
6.1 TaylorExpansionsandtheLax–WendroffPrinciple ..............115
6.2 ImplicitFourthOrderMethods ...............................117
6.3 Summary .................................................124
7 BoundaryTreatment ...........................................127
7.1 NumericalBoundaryConditions..............................127
7.2 SummationbyParts(SBP)DifferenceOperators ................130
7.3 SBPOperatorsandProjectionMethods ........................140
7.4 SBPOperatorsandSimultaneousApproximationTerm(SAT)
Methods ..................................................147
7.5 Summary .................................................155
8 TheBoxScheme ...............................................157
8.1 TheOriginalBoxScheme ...................................157
8.2 TheShiftedBoxScheme ....................................161
8.3 TwoSpaceDimensions......................................165
8.4 NonuniformGrids..........................................169
8.5 Summary .................................................176
9 WavePropagation..............................................177
9.1 TheWaveEquation.........................................177
9.1.1 OneSpaceDimension ................................178
9.1.2 TwoSpaceDimensions ...............................185
9.2 DiscontinuousCoefficients ..................................192
9.2.1 TheOriginalOneStepScheme.........................193
9.2.2 ModifiedCoefficients.................................201
9.2.3 AnExamplewithDiscontinuousSolution................206
9.3 BoundaryTreatment ........................................209
9.3.1 HighOrderBoundaryConditions.......................209
9.3.2 EmbeddedBoundaries................................210
9.4 Summary .................................................216