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变网格波动方程高阶有限差分正演模拟研究.docx

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变网格波动方程高阶有限差分正演模拟研究 Title:AHigh-OrderFiniteDifferenceMethodforSimulatingtheWaveEquationinaVaryingGrid Abstract: Thewaveequationplaysacrucialroleinvariousfieldsofscienceandengineering,includingseismicimaging,medicalimaging,andacousticsimulations.Accurateandefficientnumericalmethodsareessentialforsimulatingwaveequationsincomplexmedia.Inthispaper,weproposeahigh-orderfinitedifferencemethodtosimulatethewaveequationinavaryinggrid.Themethodtakesintoaccountthespatialvariationsofthewavepropagationvelocity,resultinginmoreaccurateandrealisticsimulations.Wepresentthemathematicalformulationofthemethodanddiscussitsimplementation.Numericalexperimentsdemonstratetheeffectivenessandefficiencyoftheproposedmethodincapturingwavephenomenaincomplexmedia. 1.Introduction Thewaveequationisafundamentalmathematicalmodelthatdescribesthepropagationofwavesinvariousphysicalsystems.Ithasapplicationsinawiderangeoffields,includinggeophysics,medicalimaging,andacoustics.Inmanypracticalscenarios,thewavepropagationvelocityisnotconstantandvariesinspace.Simulatingwavepropagationinsuchmediarequiresnumericalmethodsthatcanaccuratelyhandlethespatialvariationsofthewaveequation.Inthispaper,weproposeahigh-orderfinitedifferencemethodthatcansimulatethewaveequationinavaryinggrid. 2.MathematicalFormulation Westartbyderivingthewaveequationinavaryinggridsetting.Weconsidera2Dor3Dspatialdomainwithanon-uniformgridthatisdefinedbyafunctiondescribingthevariationsinthewavepropagationvelocity.Thewaveequationisthenmodifiedtoincludetermsaccountingforthegridvariations.Wepresentthemathematicalformulationofthemodifiedwaveequationanddiscusstheimplicationsofthenon-uniformgridonthenumericalsimulation. 3.High-OrderFiniteDifferenceMethod Tosolvethemodifiedwaveequation,wedevelopahigh-orderfinitedifferencemethod.Themethodemploysahigher-orderaccuratediscretizationschemethatcanaccuratelyrepresentthespatialvariationsofthegrid.Wediscussthedetailsofthefinitedifferenceapproximationanddescribethenumericalimplementationofthemethod.Wealsoaddressth