high-order adaptive and paralleldiscontinuous galerkin m：高阶自适应paralleldiscontinuous galerkin m

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1、High-OrderAdaptiveandParallelDiscontinuousGalerkinMethodsforHyperbolicConservationLawsJ.E.Flaherty,L.Krivodonova,J.F.Remacle,andM.S.ShephardScientificComputationResearchCenterDiscontinuousGalerkinMethodArbitraryorder:extendsfinitevolumemethodStructuredorunstructuredmeshesNoneedfor

2、inter-elementcontinuitySimplifiesadaptiveh-andp-refinementDiscontinuousGalerkinMethodFace-basedcommunicationSimplifiesparallelcomputationSharpcapturingofdiscontinuitiesElementlevelconservationAposteriorierrorestimatesDiscontinuousGalerkinMethodFace-basedcommunicationSimplifiesparall

3、elcomputationSharpcapturingofdiscontinuitiesElementlevelconservationAposteriorierrorestimatesHowever:MoremeshunknownsthanFEMforsameorderPossiblyOKwithparallelcomputationMonotonictycontrol(limiting)isdifficultDGFormulationConservationlawConstructaGalerkinproblemonjcf.CockburnandShu(

4、1989)DGSolutionSolvingtheGalerkinproblemIntegralevaluationTimeintegrationFluxevaluation,limitingApproximationHigherorderequationsDiscontinuousapproximationsneedsregularizationforgradientsExampleApproximationuUjPjpL2(j)OrthogonalbasisTimeIntegrationExplicitRunge-KuttaTVBmethodofC

5、ockburnandShu(1989)Localtimestepping,Remacleetal.(2002)xtFluxEvaluationApproximatefn(Uj)byanumericalfluxFn(Uj,Unbj)DefineFn(Uj,Unbj)byaRiemannproblemPossibilities:Upwind:fluxfrominflowneighborLax-Friedrichs:

6、max

7、isthemaximumabsoluteeigenvalueoffuRoe:linearizedRiemannproblemvanLeer:

8、fluxvectorsplittingColella-Woodward:contactsurfaceresolutionLimitingLimiting:suppressspuriousoscillationswhenp>0whilemaintainingorderSlopelimiter:CockburnandShu(1989)Curvaturelimiter:Barth(1990)Momentlimiter:Biswasetal.(1994)Filtering:Gottliebetal.(1999)Norobustproceduresformulti-di

9、mensionalsituationsSlopevs.MomentLimitingSlopeLimitingMomentLimitingKinematicwaveequation:ut+ux=0p=2SuperconvergenceOne-dimensionalconservationlawSuperconvergenceatRadaupointsAdjeridetal.(1995)Biswasetal.(1994)SuperconvergenceTheorem:Ifp>0,thespatialdiscretizationerroroftheDGmethodw

10、ithUjPpon[xj-1,xj]