Polynomial Chaos Methods for Hyperbolic Partial Differential Equations: Numerical Techniques for Fluid Dynamics Problems in the Presence of Uncertainties

217 Pages·2015·5.286 MB·English

by  Mass Per Pettersson, Gianluca Iaccarino, Jan Nordström (auth.)

Mathematical Engineering
Mass Per Pettersson
Gianluca Iaccarino
Jan Nordström
Polynomial
Chaos Methods
for Hyperbolic
Partial Diff erential
Equations
Numerical Techniques for Fluid Dynamics
Problems in the Presence of
Uncertainties
Mathematical Engineering
Serieseditors
ClausHillermeier:UniversitätderBundeswehrMünchen,Germany
JörgSchröder:UniversitätDuisburg-Essen,Germany
BernhardWeigand:InstituteofAerospaceThermodynamics,
UniversitätStuttgart,Germany.
Moreinformationaboutthisseriesathttp://www.springer.com/series/8445
Mass Per Pettersson (cid:129) Gianluca Iaccarino
Jan Nordström
Polynomial Chaos Methods
for Hyperbolic Partial
Differential Equations
Numerical Techniques for Fluid Dynamics
Problems in the Presence of Uncertainties
123
MassPerPettersson GianlucaIaccarino
UniResearch DepartmentofMechanicalEngineering
Bergen,Norway andInstituteforComputational
andMathematicalEngineering
JanNordström StanfordUniversity,USA
DepartmentofMathematics
ComputationalMathematics
LinköpingUniversity
Sweden
Additionalmaterialtothisbookcanbedownloadedfromhttp://extras.springer.com.
ISSN2192-4732 ISSN2192-4740 (electronic)
MathematicalEngineering
ISBN978-3-319-10713-4 ISBN978-3-319-10714-1 (eBook)
DOI10.1007/978-3-319-10714-1
LibraryofCongressControlNumber:2014950671
SpringerChamHeidelbergNewYorkDordrechtLondon
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Preface
Uncertainty quantification in computational physics is a broad research field that
has spurred increasing interest in the last two decades, partly due to the growth
of computer power. The objective of this textbook is the analysis and design of
numericaltechniquesforsolvingequationsrepresentingconservationlawssubject
touncertainty.Inparticular,thefocusisonstochasticGalerkinmethodsthatrequire
non-trivial development of new numerical solvers for hyperbolic and mixed type
problems. There are already textbookscovering the stochastic Galerkin and other
polynomial chaos methods from a general perspective, cf. [1–3]: this textbook is
morespecializedinitsscope.Toenhanceunderstandingofthematerialpresented,
we provideexercisesandcodescriptsandbuildingblocksthatcanbeextendedto
newproblemsettings.
The interest in stochastic Galerkin methods has burgeoned because of the
availability of ever more powerful computers that can handle the computational
cost inherent to large system implementations. Moreover, these methods have
positive numerical properties that make them attractive for handling complex
situations.Specifically,themathematicalformulationleadstosystemsofequations
thatresembletheoriginalconservationlaws,allowingustomakeextensiveuseof
availablenumericalanalysistoolsandtechniques.Atthesametime,thestochastic
Galerkinmethodisanattractivealternativeforcomplexproblemsinvolvingpartial
differentialequationsandmultipleuncertainvariables(hereinreferredtoasstochas-
ticdimensions).
Chapters 1–3 introduce and give a brief overview of the basic concepts of
uncertaintyquantificationandthestochasticGalerkinmethod.Chapter4isdevoted
to spatial discretization methods for conservation laws under uncertainty. In par-
ticular, we introduce the so-called SBP-SAT finite difference technique based
on summation-by-parts operators (SBP) and weak boundary conditions using
simultaneous approximation terms (SAT). The SBP-SAT schemes allow for the
design of stable high-order accurate schemes. Summation by parts is the discrete
equivalentofintegrationbypartsandthematrixoperatorsthatarepresentedleadto
energyestimatesthat,inturn,leadtoprovablestabilityincombinationwiththeSAT
terms.Thesemidiscretestabilityfollowsnaturallyfromthecontinuousanalysisof
v
vi Preface
well-posednesswhichprovidestheboundaryconditionsintheSBP-SATtechnique.
Chapters5–9presentin-depthanalysisoflinearandnonlinearstochasticGalerkin
conservationlaws, complementedby exercises and scripts. We providethe reader
with computercodesfor solvingthe advection-diffusionequationand the inviscid
Burgers’ equation with the stochastic Galerkin method. These codes can also be
usedastemplatesforextensiontomorecomplexproblems.
Thistextbookisintendedforanaudiencewithsomepriorknowledgeofuncer-
taintyquantification.Basic conceptsofprobabilitytheory,statistics andnumerical
analysisarealsoassumedtobefamiliartothereader.Foramoregeneralexposition
and further details on the basic concepts, we refer to the existing literature in the
field.
Thistextbookhasbenefitedfromnumerouscollaborationsanddiscussionswith
Alireza Doostan (who co-authored the material contained in Chap.5), Antony
Jameson, Xiangyu Hu, Rémi Abgrall and Paul Constantine. We would like to
thankMargotGerritsenforconstructivefeedbackandsuggestionsforimprovement.
Financial support was partially provided by KAUST under the Stanford/KAUST
AcademicExcellenceAlliance(AEA)collaboration(UDGIAAward48803).Gian-
lucaIaccarinowishestothanktheBorristercrewforsupportincompletingthefinal
revisionofthetext.
Bergen,Norway MassPerPettersson
Stanford,USA GianlucaIaccarino
Linköping,Sweden JanNordström
References
1.Xiu D (2010) Numerical methods for stochastic computations: a spectral method approach.
PrincetonUniversityPress,Princeton.http://www.worldcat.org/isbn/9780691142128
2.Le Maître OP, Knio OM (2010) Spectral methods for uncertainty quantification, 1st edn.
Springer,Dordrecht/NewYork
3.SmithR(2013)Uncertaintyquantification:theory,implementation,andapplications.Computa-
tionalscienceandengineering.SIAM,Philadelphia
Contents
PartI IntroductoryConceptsandBackground
1 Introduction .................................................................. 3
1.1 TheoryforInitialBoundaryValueProblems......................... 4
1.1.1 TheContinuousProblem..................................... 5
1.1.2 TheSemidiscreteProblem.................................... 6
1.2 Outline ................................................................. 7
References..................................................................... 8
2 RandomFieldRepresentation.............................................. 11
2.1 Karhunen-LoèveExpansion........................................... 12
2.2 GeneralizedChaosExpansions ....................................... 13
2.2.1 GeneralizedPolynomialChaosExpansion.................. 13
2.2.2 HaarWaveletExpansion...................................... 16
2.2.3 MultiwaveletExpansion...................................... 17
2.2.4 ChoiceofBasisFunctionsforGeneralizedChaos.......... 20
2.3 Exercises............................................................... 20
References..................................................................... 20
3 PolynomialChaosMethods................................................. 23
3.1 IntrusiveMethods...................................................... 23
3.1.1 StochasticGalerkinMethods................................. 23
3.1.2 Semi-intrusiveMethods ...................................... 24
3.2 Non-intrusiveMethods................................................ 25
3.2.1 InterpolationandIntegrationApproaches................... 25
3.2.2 SpectralProjection............................................ 26
3.2.3 StochasticMulti-elements.................................... 27
3.3 Exercises............................................................... 28
References..................................................................... 28
vii
viii Contents
4 NumericalSolutionofHyperbolicProblems.............................. 31
4.1 Summation-by-PartsOperators ....................................... 32
4.1.1 RecipeforConstructingaScheme........................... 32
4.1.2 TheContinuousProblem..................................... 33
4.2 Analysis................................................................ 33
4.2.1 Well-Posedness ............................................... 33
4.2.2 Stability........................................................ 34
4.2.3 ConvergenceforFiniteTime................................. 36
4.2.4 AnErrorBoundinTime...................................... 37
4.2.5 ArtificialDissipationOperators.............................. 38
4.3 Shock-CapturingMethods ............................................ 38
4.3.1 MUSCLScheme.............................................. 38
4.3.2 HLLRiemannSolver......................................... 40
4.4 TimeIntegration....................................................... 40
4.5 Exercises............................................................... 42
References..................................................................... 43
PartII ScalarTransportProblems
5 LinearTransportUnderUncertainty...................................... 47
5.1 ProblemDefinition .................................................... 48
5.1.1 UncertaintyandSolutionProcedure......................... 49
5.1.2 StochasticGalerkinProjection............................... 49
5.1.3 DiagonalizationoftheStochasticGalerkinSystem......... 52
5.2 TheEigenvaluesoftheDiffusionMatrixB .......................... 53
5.2.1 GeneralBoundsontheEigenvaluesofB.................... 53
5.2.2 LegendrePolynomialRepresentation........................ 54
5.2.3 HermitePolynomialRepresentation......................... 54
5.3 BoundaryConditionsforWell-Posedness............................ 55
5.4 MonotonicityoftheSolution ......................................... 56
5.4.1 Second-OrderOperators...................................... 56
5.4.2 Fourth-OrderOperators....................................... 59
5.5 StabilityoftheSemidiscretizedProblem............................. 60
5.5.1 TheInitialValueProblem:vonNeumannAnalysis......... 60
5.5.2 TheInitialBoundaryValueProblem......................... 62
5.5.3 EigenvaluesoftheTotalSystemMatrix..................... 66
5.5.4 ConvergencetoSteady-State................................. 69
5.6 NumericalResults ..................................................... 71
5.6.1 TheInviscidLimit ............................................ 72
5.6.2 Steady-StateCalculations .................................... 76
5.7 SummaryandConclusions............................................ 77
5.8 SupplementaryCodes ................................................. 78
5.9 Exercises............................................................... 78
References..................................................................... 79
Contents ix
6 NonlinearTransportUnderUncertainty.................................. 81
6.1 PolynomialChaosExpansionofBurgers’Equation................. 82
6.1.1 EntropyandEnergyEstimatesfortheM D2Case ........ 83
6.1.2 DiagonalizationoftheSystemMatrixA.uM/ ............... 84
6.2 AReferenceSolution.................................................. 85
6.2.1 RegularityDeterminedbythegPCExpansionOrder....... 87
6.3 Well-Posedness ........................................................ 88
6.3.1 TheImportanceofBoundaryConditions.................... 90
6.4 EnergyEstimatesforStability ........................................ 90
6.4.1 ArtificialDissipationforEnhancedStability................ 92
6.5 TimeIntegration....................................................... 93
6.6 EigenvalueApproximation............................................ 93
6.7 EfficiencyofthePolynomialChaosMethod......................... 95
6.7.1 NumericalConvergence...................................... 96
6.8 TheoreticalResultsandInterpretation................................ 98
6.8.1 AnalysisofCharacteristics:DisturbedCosineWave........ 98
6.9 SummaryandConclusions............................................ 106
6.10 SupplementaryCodes ................................................. 107
6.11 Exercises............................................................... 107
References..................................................................... 108
7 BoundaryConditionsandData ............................................ 111
7.1 DependenceonAvailableData ....................................... 111
7.1.1 CompleteSetofData......................................... 112
7.1.2 IncompleteSetofBoundaryData............................ 112
7.1.3 DiscussionoftheResultswithIncompleteSetofData..... 118
7.2 SummaryandConclusions............................................ 120
7.3 Exercises............................................................... 120
Reference...................................................................... 121
PartIII EulerEquationsandTwo-PhaseFlow
8 gPCfortheEulerEquations................................................ 125
8.1 EulerEquationswithInputUncertainty ............................. 126
8.1.1 FormulationinRoeVariables ................................ 127
8.1.2 StochasticGalerkinFormulationoftheEulerEquations ... 128
8.2 NumericalMethod..................................................... 130
8.2.1 ExpansionofConservativeVariables........................ 130
8.2.2 ExpansionofRoe’sVariables ................................ 131
8.2.3 StochasticGalerkinRoeAverageMatrixfor
RoeVariables.................................................. 132
8.3 NumericalResults ..................................................... 135
8.3.1 SpatialConvergence.......................................... 136
8.3.2 InitialConditionsandDiscontinuousSolutions............. 137