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1、COMPUTATIONALHIGH-FREQUENCYWAVEPROPAGATIONUSINGTHELEVELSETMETHOD,WITHAPPLICATIONSTOTHESEMI-CLASSICALLIMITOFSCHRODINGEREQUATIONS¨LI-TIENCHENG,HAILIANGLIU,ANDSTANLEYOSHERAbstract.Weintroducealevelsetmethodforcomputationalhighfrequencywavepropagationindispersivemediaand
2、considertheapplicationtolinearSchro¨odingerequationswithhighfrequencyinitialdata.Highfrequencyasymptoticsofdispersiveequationsoftenleadtothewell-knownWKBsystemwherethephaseoftheplanewaveevolvesaccordingtoanonlinearHamilton-Jacobiequationandtheintensityisgovernedbyali
3、nearconservationlaw.FromtheHamilton-Jacobiequation,wavefrontswithmul-tiplephasesareconstructedbysolvingalinearLiouvilleequationofavectorvaluedlevelsetfunctioninthephasespace.Themulti-valuedphaseitselfcanbeconstructedeitherfromanadditionallinearhyperbolicequationinpha
4、sespaceoranadditionallinearhomogeneousequationandcomponenttothelevelsetfunctioninanaugmentedphasespace.Thisphaseisinfactvalidintheentirephysicaldomain,butoneofthecomponentsofthelevelsetfunctioncanbeusedtorestrictittoawavefrontofinter-est.Theuseofthelevelsetmethodinth
5、isnumericalapproachprovidesanEulerianframeworkthatautomaticallyresolvesthemulti-valuedwavefrontsandphasefromthesuperpositionofsolutionsoftheequationsinphasespace.KeyWords:Levelsetmethod,Schr¨odinger’sequation,semiclassicallimit,wavefront,multi-valuedphasesAMSsubjectc
6、lassification:Primary35Q55;Secondary65M25Contents1.Introduction12.LevelSetFormulation63.ReductionofthePhaseSpace104.NumericalTechniques125.NumericalExamples16Acknowledgments20References291.IntroductionInthiswork,weareinterestedinthecomputationofhighfrequencypropagatin
7、gwavesforequationsarisingindispersivemedia.Theintentionofthispaperistointroduceagenerallevelsetmethodandframeworkformulti-phasecomputationsofproblemsofthistype.However,theapplicationthatwefocusonisthetime-dependentmulti-dimensional12L.-T.Cheng,H.-L.Liu,andS.OsherSchr
8、¨odingerequation,2n(1.1)i∂tψ=−∆ψ+V(x)ψ,x∈IR,2subjecttothehighfrequencyinitialdata,S0(x)(1.2)ψ(x,0)=A0(x)expi,whereψisthec
