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三对角四阶紧致差分格式的优化和初步应用.docx

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三对角四阶紧致差分格式的优化和初步应用 Title:OptimizationandPreliminaryApplicationsofTridiagonalFourth-OrderCompactDifferenceScheme Abstract: Thetridiagonalfourth-ordercompactdifferenceschemeisanumericalmethodwidelyusedforsolvingpartialdifferentialequations.Thispaperfocusesontheoptimizationtechniquesandpreliminaryapplicationsofthisscheme.Theoptimizationincludesreducingcomputationalcomplexity,improvingaccuracy,andenhancingstability.Thepreliminaryapplicationscovervarioustypesofpartialdifferentialequations,suchasadvection-diffusion,reaction-diffusion,andconvection-diffusionequations.Throughthestudyandimplementationoftheseoptimizationtechniques,itisdemonstratedthatthetridiagonalfourth-ordercompactdifferenceschemecanefficientlysolvepartialdifferentialequationswithimprovedaccuracyandstability. 1.Introduction Partialdifferentialequations(PDEs)playacrucialroleinmathematicalmodelingofvariousscientificandengineeringproblems.SolvingPDEsnumericallyisoftennecessaryduetothecomplexityofanalyticalsolutions.Thetridiagonalfourth-ordercompactdifferenceschemehasgainedpopularityduetoitsaccuracyandefficiency.However,itstillhasroomforoptimizationintermsofcomputationalcomplexity,accuracy,andstability.Thispaperaimstoexploretheseoptimizationtechniquesanddemonstratetheireffectivenessthroughpreliminaryapplications. 2.OptimizationofTridiagonalFourth-OrderCompactDifferenceScheme 2.1.Reducingcomputationalcomplexity -Reorderingtheunknowns:Rearrangingtheunknownscanreducethebandwidthofthecoefficientmatrix,leadingtofastercomputations. -Matrixfactorization:EmployingLUorCholeskyfactorizationtechniquescanfurtherreducethecomputationalcomplexityofsolvinglinearsystems. -Parallelcomputing:Utilizingparallelalgorithms,suchasOpenMPorMPI,canspeeduptheimplementationofthescheme. 2.2.Improvingaccuracy -Higher-orderspatialapproximation:Incorporatinghigher-orderspatialapproximationscansignificantlyimprovetheaccuracyofthescheme. -Non-uniformgrids:Implementingnon-uniformgridscanprovideabetterresolutioninregionswherethesolutionhasrapidva