Meshless Local Discontinuous Galerkin Method

AMeshlessLocalDiscontinuousGalerkinMethodwithSatelliteSlopeLimiterforConvection-dominatedproblemsGAOWei-ran*,QIANGHong-fu(FacultyofMechanical&PropulsionEngineering,Xi’anHi-TechInstitute,Xi’an,Shaanxi,PRC,PC710025)Abstract:Thispaperderivesanewclassofstabilizedmeshlessmethodforthesimulationsoftheconvection-dominatedproblems,basedonthegeneralDiscontinuousGalerkin(DG)frameworkandthelocalsymmetricweakform(LSWF).IncontrasttotheusualmeshbasedDGmethodswhichapproximatethepartialdifferentialequationsdependonthenon-overlappingmesh,thepresentedmethodrepresentsthesolutionslocallyonasetofindependentoverlappingsub-domains,andthestandardnumericalfluxfunctionsareemployedtodeterminetheinformationexchangedbetweeneachpairsofoverlappedsub-domains.Forthestabilizationreasonofhigh-orderMeshlessLocalDiscontinuousGalerkin(MLDG)schemes,acompatiblemeshlessSlopeLimiterisfirstlydevelopedforonedimensionalcasesbasedontheoreticalstabilityanalysisoftheresultingMLDGschemes.Finally,thefirstandsecondorderMLDGschemesareverifiedbyseveralclassicalone-dimensionalnumericaltestswiththeexistenceofdiscontinues,allthenumericalresultsarefairlyagreedwiththeexactsolutionswhateversolvingproblemswiththeequidistantnodesorrandomdistributionnodes.ItshowsthecapacityofMLDGtopreservetheaccuracyofthesolutionsinsmoothregions,andtocapturestrongshocks.ThepresentedMLDGmethodisatrulymeshlessmethodforsolvingtheconvection-dominatedproblems,andfluidmechanicsproblems,whichinheritboththeadvantagesofDGmethodandmeshlessmethod..Keywords:DiscontinuousGalerkinmethods,meshlessmethods,localsymmetricweakform,slopelimiters,convection-dominatedproblems,shockwaves1IntroductionConventionalComputationalFluidDynamics(CFD)methodsneedanaprioridefinitionoftheconnectivityofnodes,i.e.theyrelyonamesh.However,thegenerationofgoodqualitynon-overlappingmeshespresentsfatefuldifficultiesintheanalysisofengineeringsystems,especiallyinthemulti-dimensionalproblemswithcomplexgeometrycharactersandproblemsinvolvingthefluid-structureinteractions.Inrecentyears,agrowingattentionhasbeenpaidtotheso-calledMeshlessMethods(MMs),asthepartialdifferentialequationsaresolvedonlybasedonascattersetofnodes,withouttheneedforanadditionalmesh.Itisexpectedthatforfluidproblemswithcomplexgeometrycharacters,especiallyforfluid-structureinteractionproblems,MMswillhaveasignificantadvantageovertraditionalmeshbasedmethods.Nevertheless,tillnow,mostofthoseMMsweredevotedtosolvethesolidmechanicsproblems,onlyveryfewworkswerereportedbyusingtheMMsforconvection-dominatedflows.Inthenumericalsimulationofconvection-dominatedflows,theexistenceoftheconvectiontermmakesthatthesolutionwilldevelopstrongdiscontinuitiesaswellasshockwaves.Inordertohandlethiskindofweaksolutionsnumerically,oneneedstointroduceaspecialtreatmenttostabilizethenumericalapproximation.Thecurrentliteratureonthesolutionoftheconvection-dominatedequationusingMMsisratherlimited.TheSmoothedParticleHydrodynamics(SPH)[1,2]employsanartificialviscositytoimproveitsstabilityinsimulationofshockphenomenon,butpossessesverylowaccuracyandtheadditionalchangeableartificialparametersareneeded;afewofsophisticatedMMsintroducetheupwindalgorithmsforcalculationofconvection-dominatedproblemsincludingtheFiniteParticleMethod(FPM)[3],ReproducingKernelParticleMethod(RKPM)[4]andMeshlessLocalPetrov-Galerkin(MLPG)[5],nevertheless,theMMswithupwindalgorithmsfornon-linearconservationlawsdonotenforcetheconservationlawlocally,whichrendersthemquiteinefficientwhenstrongshockwavesareexisted,furthermore,theuseofmovingleastsquaresinterpolationfunctionsorreproducingkernelapproximationandthehighordernumericalintegrationschememakethosemethodaverycomputationallyexpensivemethod;TheFinite-VolumeParticlemethod(FVPM)[6-8],developedbyHietelandhisco-workers,incorporateselementsoftheFiniteVolumeMethod(FVM)intoameshlessmethodandinheritsmostadvantagesofbothFVMandMMs,whatsoever,thismethodpossessesveryloworderapproximationsassameasthefirstorderFVM,anditlimitsitsfartherapplicationsextraordinary.Recently,weproposedatrulymeshlessmethodforcompressibleflows,whichnamesMeshlessLocalDiscontinuousGalerkinMethod(MLDG)[9].ThismeshlessmethodincorporatesoneoftheflexibleandrobustmeshbasedComputationalFluidDynamics(CFD)method,Runge–KuttaDiscontinuousGalerkin(RKDG)[10-15],intothemeshlessmethod,byusingalocaloverlappingdiscontinuousGalerkinspatialdiscretizationinsteadofthenon-overlappingspatialdiscretizationusedinmeshbasedRKDG,andsubstitutingthetestfunctionswithanappropriateforminaanalogicalwayofFVPM[6-8],TheMLDGinheritsboththeadvantagesofDGmethodandMMs,asmeshless,stableforconvection-dominatedproblems,high-orderapproximation,andparticularwellsuitedforthehp-adaptivity.Imperfectly,thepreviousversionofhigh-orderMLDGschemesneedsanartificialviscosityforsolvingtheproblemswithstrongshock-waves,andtheartificialparametersareneeded.ForthemeshbasedDGmethod,theslopelimitingalgorithmisthemostappliedtechniquefortheshock-captureswithouttheneedofchangeableartificialparameters,however,thistechniquecannotbeimplementeddirectlytothemeshlessMLDGduetoitsmeshbasedcharacters.I