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求解含有高阶导数偏微分方程的局部间断Petrov-Galerkin方法（英文）.docx

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求解含有高阶导数偏微分方程的局部间断Petrov-Galerkin方法（英文） Title:AStudyonLocallyDiscontinuousPetrov-GalerkinMethodsforPartialDifferentialEquationswithHigher-OrderDerivativePDEs Abstract: Partialdifferentialequations(PDEs)arewidelyusedtomodelvariousphysicalphenomenainthefieldsofphysics,engineering,andappliedmathematics.SolvingPDEsaccuratelyandefficientlyisachallengingtask,especiallywhendealingwithPDEsthatinvolvehigher-orderderivatives.Inthispaper,weinvestigatetheapplicationoflocallydiscontinuousPetrov-Galerkin(LD-PG)methodsforsolvingPDEswithhigher-orderderivativeterms.TheLD-PGmethodsprovidearobustandflexibleframeworkforhandlingsuchcomplexPDEsbycombiningthebenefitsofPetrov-GalerkinmethodsandlocaldiscontinuousGalerkinmethods. Introduction: Partialdifferentialequationswithhigher-orderderivativetermsareencounteredinmanyscientificandengineeringproblems.Examplesincludethewaveequation,heatdiffusionequation,andgeneralwavepropagationproblems.Traditionalnumericalmethods,suchasfinitedifferenceorfiniteelementapproaches,facechallengesinaccuratelyandefficientlysolvingthesePDEs,especiallywhenthesolutionpresentsdiscontinuities,sharpgradients,orboundarylayers.ThelocallydiscontinuousPetrov-Galerkinmethodsofferapromisingsolutiontoovercomethesedifficulties. Methodology: TheLD-PGmethodscombinetheflexibilityofPetrov-GalerkinmethodsandthelocaladaptabilityofthediscontinuousGalerkin(DG)methods.Theideaistoselectdifferentfiniteelementspacesforthetestandtrialfunctions,whichallowsfordiscontinuitiesinthesolutionwithineachelement.TheLD-PGmethodsintroducelocalstabilityandsmoothnessintothesolutionbyusingalocalstabilizationtermthatpenalizesjumpsinthesolutionacrosselementboundaries. ResultsandDiscussion: WeapplytheLD-PGmethodstoseveralPDEscontaininghigher-orderderivativeterms,includingthewaveequationanddiffusionequation.ThenumericalresultsdemonstratethatLD-PGmethodsarecapableofhandlingdiscontinuoussolutionsandcapturingsharpgradientsaccurately.Furthermore,themethodsshowimprovedstabilityandconvergencepropertiescomparedtot