Recent Developments in the Numerics of Nonlinear Hyperbolic Conservation Laws: Lectures Presented at a Workshop at the Mathematical Research Institute Oberwolfach, Germany, Jan 15 – 21, 2012 PDF
Preview Recent Developments in the Numerics of Nonlinear Hyperbolic Conservation Laws: Lectures Presented at a Workshop at the Mathematical Research Institute Oberwolfach, Germany, Jan 15 – 21, 2012
Notes on Numerical Fluid Mechanics and Multidisciplinary Design 120 SeriesEditors Prof.Dr.WolfgangSchröder (GeneralEditor),RWTHAachen,LehrstuhlfürStrömungslehreundAerodynamisches Institut,Wüllnerstr.5a,52062Aachen,Germany E-mail:offi[email protected] Prof.Dr.Ir.BendiksJanBoersma ChairofEnergytechnology,DelftUniversityofTechnology,Leeghwaterstraat44, 2628CADelft,TheNetherlands E-mail:[email protected] Prof.Dr.KozoFujii SpaceTransportationResearchDivision,TheInstituteofSpaceandAstronautical Science, 3-1-1,Yoshinodai,Sagamihara,Kanagawa,229-8510,Japan E-mail:fujii@flab.eng.isas.jaxa.jp Dr.WernerHaase HöhenkirchenerStr.19d,D-85662Hohenbrunn,Germany E-mail:[email protected] Prof.Dr.MichaelA.Leschziner AeronauticsDepartment,ImperialCollegeofScienceTechnologyandMedicine, PrinceConsortRoad,LondonSW72BY,UK E-mail:[email protected] Prof.Dr.JacquesPeriaux 38,BoulevarddeReuilly,F-75012Paris,France E-mail:[email protected] Prof.Dr.SergioPirozzoli DipartimentodiMeccanicaeAeronautica,UniversitàdiRoma“LaSapienza”, ViaEudossiana18,00184,Roma,Italy E-mail:[email protected] Prof.Dr.ArthurRizzi DepartmentofAeronautics,KTHRoyalInstituteofTechnology,Teknikringen8, S-10044Stockholm,Sweden E-mail:[email protected] Dr.BernardRoux L3M-IMTLaJetée,TechnopoledeChateau-Gombert,F-13451MarseilleCedex20,France E-mail:[email protected] Prof.Dr.YuriiI.Shokin InstituteofComputationalTechnologies,SiberianBranchoftheRussianAcademy ofSciences,Ac.LavrentyevaAve.6,630090Novosibirsk,Russia E-mail:[email protected] Forfurthervolumes: http://www.springer.com/series/4629 Recent Developments in the Numerics of Nonlinear Hyperbolic Conservation Laws Lectures Presented at a Workshop at the Mathematical Research Institute Oberwolfach, Germany, Jan 15–21, 2012 Rainer Ansorge, Hester Bijl, Andreas Meister, and Thomas Sonar (Eds.) ABC Editors Prof.RainerAnsorge Prof.AndreasMeister DepartmentofMathematics DepartmentofMathematicsand UniversityofHamburg NaturalSciences Hamburg UniversityofKassel Germany Kassel Germany Prof.HesterBijl DepartmentofAerospaceEngineering Prof.ThomasSonar DelftUniversityofTechnology InstituteforComputationalMathematics Delft TechnicalUniversityofBraunschweig TheNetherlands Braunschweig Germany ISSN1612-2909 e-ISSN1860-0824 ISBN978-3-642-33220-3 e-ISBN978-3-642-33221-0 DOI10.1007/978-3-642-33221-0 SpringerHeidelbergNewYorkDordrechtLondon LibraryofCongressControlNumber:2012947385 (cid:2)c Springer-VerlagBerlinHeidelberg2013 Thisworkissubjecttocopyright.AllrightsarereservedbythePublisher,whetherthewholeorpartof thematerialisconcerned,specificallytherightsoftranslation,reprinting,reuseofillustrations,recitation, broadcasting,reproductiononmicrofilmsorinanyotherphysicalway,andtransmissionorinformation storageandretrieval,electronicadaptation,computersoftware,orbysimilarordissimilarmethodology nowknownorhereafterdeveloped.Exemptedfromthislegalreservationarebriefexcerptsinconnection with reviews or scholarly analysis or material supplied specifically for the purpose of being entered and executed on a computer system, for exclusive use by the purchaser of the work. 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Printedonacid-freepaper SpringerispartofSpringerScience+BusinessMedia(www.springer.com) Preface AworkshoponMethodsofveryhighorderfornonlinearhyperbolicconservation laws andtheiruse in science andengineeringtookplaceatthe MathematicalRe- searchInstituteOberwolfachfromJan15toJan21,2012. The development of Spectral Element methods in recent years as well as Discontinuous Galerkin and Essentially Non-Oscillatory or Weighted Essentially Non-Oscillatory Finite Volume methodshave opened new fields for very high or- dernumericalmethods,i.e.methodsofordersstartingatfourorfive,andtheiruse in science and engineering.In case of such methods, convergenceresults can not yetbeatissue.Instead,methodsofveryhighorderingeneraldemandresultsfrom areaslikeapproximationtheoryandfilterdesign.InDiscontinuousGalerkinmeth- odsandinSpectralElementmethodsonsimplicialmeshes,forexample,orthogonal polynomialsonsimplicesseemtobetheappropriatechoice. Besidesapproximationtheorythereisalsoagrowingneedconcerninghighorder quadraturerulesonsimplices.Althoughthisareaseemstobequiteolditshouldbe lookedatagainwithaparticularviewonrobustandefficienthighordermethods. Moreover,afurtheringredientnecessaryforthestablenumericalapproximation ofconservationlawswithveryhighordermethodsliesinfilterdesign.Oneofthe centralquestionstobeansweredconcernstheinformationonoscillationswhichcan bederivedfromthecoefficientsoforthogonalpolynomialsonsimplices. The aim of the workshop was to bring together experts from the fields of nu- mericalmethodsfor conservationlaws, approximationtheory,filter design, image recovery,andengineeringapplications. We,theorganisers,arereallysatisfiedthatmanyofthecontributionstothiswork- shop can be publishedin this volume of the NNFM series of SpringerVerlag and we thank the publishersas well as the General Editor of this series, W. Schro¨der, Aachen,fortheirhelpandsupport. VI Preface We alsothanktheOberwolfachInstituteanditsDirector,G.-M.Greuel,forthe interestinourworkshop. Hamburg,Delft,Kassel,Brunswick RainerAnsorge April2012 HesterBijl AndreasMeister ThomasSonar Contents A SecondOrder Accurate Kinetic RelaxationScheme forInviscid CompressibleFlows ............................................. 1 K.R.Arun,M.Luka´cˇova´-Medvid’ova´,PhoolanPrasad,S.V.RaghuramaRao 1 Introduction.............................................. 2 2 RelaxationSystemforEulerEquations ....................... 3 3 KineticRelaxationScheme ................................. 6 3.1 ConservationPropertyoftheScheme................. 8 3.2 PositivityPreservingProperty ....................... 9 3.3 EntropyStabilityoftheScheme ..................... 11 4 SecondOrderAccurateKineticRelaxationScheme............. 12 4.1 SecondOrderAccuracyinTime ..................... 13 4.2 SecondOrderAccuracyinSpace .................... 16 5 NumericalCaseStudies.................................... 18 6 ConcludingRemarks ...................................... 22 References..................................................... 23 OnOne-DimensionalLowMachNumberApplications................ 25 MariaBauer,ElisabettaFelaco,IngenuinGasser 1 Introduction.............................................. 25 1.1 GeneralSetting ................................... 26 1.2 InitialandBoundaryConditions ..................... 27 1.3 AsymptoticRegimes............................... 28 2 TheGasDynamicsinaChimney ............................ 28 2.1 Asymptotics...................................... 29 2.2 NumericalSimulations............................. 31 3 ModellinganEnergyTower ................................ 33 3.1 AsymptoticsandNumericalSimulations .............. 35 4 Conclusion............................................... 38 References..................................................... 38 VIII Contents HighOrderandUnderresolution .................................. 41 AndreaBeck,GregorGassner,Claus-DieterMunz 1 Introduction.............................................. 41 2 NumericalModel ......................................... 42 2.1 PhysicalModel ................................... 42 2.2 DiscontinuousGalerkin ............................ 43 2.3 StabilityandDe-aliasing ........................... 45 3 HighOrderSimulationofUnderresolvedTurbulence ........... 50 3.1 Highvs.LowOrderSimulationsatModerate Resolutions ...................................... 50 3.2 Underresolution:StabilitybyOverintegration.......... 51 3.3 AccuracyandEfficiency............................ 52 4 Conclusion............................................... 54 References..................................................... 55 SolvingNonlinearSystemsInsideImplicitTimeIntegrationSchemesfor UnsteadyViscousFlows .......................................... 57 PhilippBirken 1 Introduction.............................................. 57 2 Navier-StokesEquations ................................... 59 3 ImplicitTimeDiscretization ................................ 59 4 SolvingNonlinearEquationSystems......................... 60 4.1 MultigridSchemes ................................ 60 4.2 NewtonSchemes.................................. 65 5 SummaryandConclusions ................................. 69 References..................................................... 69 DiscreteFlux-CorrectedTransport: NumericalAnalysis,Tensor-ValuedExtensionandApplication inImageProcessing ............................................. 73 MichaelBreuß,BernhardBurgeth,LuisPizarro 1 Introduction.............................................. 73 2 ThePDEsofMathematicalMorphology...................... 74 3 ReviewoftheDFCTMethod ............................... 75 4 AnalysisoftheDFCTScheme .............................. 76 5 MathematicsofMatrixFields ............................... 78 6 PDE-BasedMorphologyforMatrixFields .................... 81 7 NumericalExperiments .................................... 83 8 Conclusion............................................... 86 References..................................................... 86 QuantificationofNumericalandPhysicalMixinginCoastalOcean ModelApplications.............................................. 89 HansBurchard,UlfGra¨we 1 Introduction.............................................. 89 2 ModelEquations.......................................... 90 Contents IX 2.1 ContinuousEquations.............................. 90 2.2 DiscretisationandNumericalMixingAnalysis ......... 92 3 Applications ............................................. 96 3.1 GeneralEstuarineTransportModel(GETM)........... 96 3.2 FreshwaterLens .................................. 96 3.3 WesternBalticSea ................................ 99 4 Conclusions..............................................101 References.....................................................102 DealingwithParasiticBehaviourinG-SymplecticIntegrators.......... 105 J.C.Butcher 1 Introduction..............................................105 2 DissipativeandConservativeProblems .......................106 3 Runge–KuttaMethods .....................................108 3.1 AlgebraicallyStable andSymplecticRunge–Kutta Methods .........................................108 3.2 Runge–KuttaSimulations...........................109 4 GeneralLinearMethods....................................111 4.1 AlgebraicallyStableandG-SymplecticGeneralLinear Methods .........................................113 4.2 OrderandStartingMethods.........................114 5 CorruptionofLong-TermSolutionbyParasitism...............115 5.1 CancellationUsingCompositions....................116 5.2 CancellationUsingScaledSteps.....................116 5.3 NumericalExperiments ............................117 5.4 Parasitism-FreeMethods ...........................119 6 DerivationofaNewMethod................................120 6.1 StartingMethod...................................122 References.....................................................123 AnAdaptiveArtificialViscosityMethodfortheSaint-VenantSystem.... 125 YunlongChen,AlexanderKurganov,MinlanLei,YuLiu 1 Introduction..............................................125 2 AdaptiveArtificialViscosityMethod.........................127 2.1 SourceTermQuadrature............................130 2.2 CorrectionoftheReconstructedPointValues ..........130 2.3 Desingularization .................................131 2.4 PositivityPreservingProperty .......................132 3 NumericalExamples ......................................133 References.....................................................140 DiscontinuousGalerkinMethod–ARobustSolverforCompressible Flow .......................................................... 143 MiloslavFeistauer,JanCˇesenek,Va´clavKucˇera 1 Introduction..............................................143 2 DescriptionofCompressibleFlow ...........................144 X Contents 3 DiscreteFlowProblem.....................................147 3.1 SpaceDiscretizationbytheDiscontinuousGalerkin Method..........................................147 3.2 TimeDiscretizationbytheBDFMethod ..............150 3.3 Space-TimeDiscontinuousGalerkinMethod...........151 4 NumericalExperiments ....................................152 4.1 InviscidStationaryFlowwithLowMachNumber ......152 4.2 FlowInducedAirfoilVibrations .....................154 5 Conclusion ..............................................158 References.....................................................159 A RigorousApplicationoftheMethodofVerticalLines toCoupled SystemsinFiniteElementAnalysis................................. 161 StefanHartmann,SteffenRothe 1 Introduction..............................................161 2 UncoupledSituationandBasicInitialBoundary-ValueProblem ..164 3 CoupledProblems.........................................167 4 Examples................................................168 4.1 Electro-ThermalCoupling ..........................168 4.2 Thermo-Viscoplasticity.............................170 5 Conclusions..............................................172 References.....................................................173 Monotonicity Conditions for Multirate and Partitioned Explicit Runge-KuttaSchemes ........................................... 177 WillemHundsdorfer,AnnaMozartova,ValeriuSavcenco 1 Introduction..............................................177 2 SomeMultirateSchemesofOrderOneandTwo ...............178 2.1 ExamplesofSimpleSchemesfortheAdvection Equation.........................................178 2.2 SomeSchemeswithOneRefinementLevelforGeneral Semi-discreteProblems ............................179 3 PartitionedRunge-KuttaMethods............................184 3.1 GeneralProperties.................................184 4 MonotonicityandConvexEulerCombinations.................188 4.1 Maximum-NormMonotonicity......................189 4.2 MonotonicityunderAssumption(27).................190 4.3 MonotonicityunderAssumption(26).................191 4.4 Application:MultirateSchemeswithOneLevelof Refinement.......................................192 5 ConcludingRemarks ......................................193 References.....................................................194 Contents XI OntheConstructionofKernel-BasedAdaptiveParticleMethodsin NumericalFlowSimulation ....................................... 197 ArminIske 1 Introduction..............................................197 2 HyperbolicConservationLaws..............................198 3 FiniteVolumeParticleMethod(FVPM) ......................199 4 WENOReconstruction.....................................200 5 Kernel-BasedReconstructioninParticleFlowSimulations.......201 6 ReconstructionbyPolyharmonicSplines......................204 7 NumericalAspectsofPolyharmonicSplineReconstruction ......205 7.1 SpectralConditionNumberofReconstructionMatrix ...205 7.2 ConditioningofReconstructionProblem ..............207 7.3 Scale-InvarianceoftheLebesgueConstant ............208 7.4 StableEvaluationoftheReconstruction...............210 7.5 LocalApproximationOrder.........................211 7.6 AdvantagesofPolyharmonicSplineReconstruction.....212 8 AdaptionRules ...........................................213 8.1 ErrorIndication...................................214 8.2 CoarseningandRefinement .........................215 9 OilReservoirSimulation:TheFive-SpotProblem ..............216 9.1 TheFive-SpotProblem.............................216 9.2 AdaptiveParticleFlowSimulation ...................219 References.....................................................220 An Assessment of the Efficiency of Nodal Discontinuous Galerkin SpectralElementMethods........................................ 223 DavidA.Kopriva,EdwinJimenez 1 Introduction..............................................223 2 TheDGSEM .............................................224 3 RelativeEfficiencyoftheDGSEMApproximation .............226 4 Implicitvs.ExplicitComputation............................229 4.1 TheEffectofParallelism ...........................233 5 Conclusions..............................................234 References.....................................................235 Sub-cyclingStrategiesforMaritimeTwo-PhaseFlowSimulations....... 237 ManuelManzke,Jan-PatrickVoss,ThomasRung 1 Introduction..............................................237 2 ComputationalMethod.....................................238 2.1 GoverningEquationsandComputationalAlgorithm ....238 2.2 Free-SurfaceModel................................239 2.3 DiscreteMixture-FractionEquation ..................240 3 Sub-cyclingStrategy ......................................241 4 Results ..................................................243 4.1 Two-DimensionalDamBreak .......................243 4.2 DragPredictionofaTanker .........................245
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