二维抛物方程的有限差分法

华北电力大学本科毕业设计（论文）I二维抛物方程的有限差分法摘要二维抛物方程是一类有广泛应用的偏微分方程，由于大部分抛物方程都难以求得解析解，故考虑采用数值方法求解。有限差分法是最简单又极为重要的解微分方程的数值方法。本文介绍了二维抛物方程的有限差分法。首先，简单介绍了抛物方程的应用背景，解抛物方程的常见数值方法，有限差分法的产生背景和发展应用。讨论了抛物方程的有限差分法建立的基础，并介绍了有限差分方法的收敛性和稳定性。其次，介绍了几种常用的差分格式，有古典显式格式、古典隐式格式、Crank-Nicolson隐式格式、Douglas差分格式、加权六点隐式格式、交替方向隐式格式等，重点介绍了古典显式格式和交替方向隐式格式。进行了格式的推导，分析了格式的收敛性、稳定性。并以热传导方程为数值算例，运用差分方法求解。通过数值算例，得出古典显式格式计算起来较简单，但稳定性条件较苛刻；而交替方向隐式格式无条件稳定。关键词：二维抛物方程；有限差分法；古典显式格式；交替方向隐式格式华北电力大学本科毕业设计（论文）IIFINITEDIFFERENCEMETHODFORTWO-DIMENSIONALPARABOLICEQUATIONAbstractTwo-dimensionalparabolicequationisawidelyusedclassofpartialdifferentialequations.Becausethiskindofequationissocomplex,weconsidernumericalmethodsinsteadofobtaininganalyticalsolutions.finitedifferencemethodisthemostsimpleandextremelyimportantnumericalmethodsfordifferentialequations.Thepaperintroducesthefinitedifferencemethodfortwo-dimensionalparabolicequation.Firstly,thispaperintroducesthebackgroundandcommonnumericalmethodsforParabolicEquation,Backgroundanddevelopmentofapplications.DiscussesthebasementfortheestablishmentofthefinitedifferencemethodforparabolicequationAnddescribestheconvergenceandstabilityforfinitedifferencemethod.Secondly,Introducessomeofthemorecommonsimpledifferentialformat,forexample,theclassicalexplicitscheme,theclassicalimplicitscheme,Crank-Nicolsonimplicitscheme,Douglasdifferencescheme,weightedsiximplicitschemeandthealternatingdirectionimplicitformat.Thepaperfocusesontheclassicalexplicitschemeandthealternatingdirectionimplicitformat.Thepapertakesdiscussesthederivationconvergence,andstabilityoftheformat.ThepapertakesAndtheheatconductionequationforthenumericalexample,usingthedifferentialmethodtosolve.Throughnumericalexamples,theclassicalexplicitschemeisrelativelysimpleforcalculation,withmorestringentstabilityconditions;andalternatingdirectionimplicitschemeisunconditionallystable.Keywords:Two-dimensionalParabolicEquation;Finite-DifferenceMethod;EclassicalExplicitScheme;AlternatingDirectionImplicitScheme华北电力大学本科毕业设计（论文）目录摘要..................................................................................................................................................IAbstract..........................................................................................................................................II1绪论..............................................................................................................................................11.1课题背景...................................................................................................................................11.2发展概况...................................................................................................................................11.2.1抛物型方程的常见数值解法................................................................................................11.2.2有限差分方法的发展............................................................................................................21.3差分格式建立的基础...............................................................................................................31.3.1区域剖分................................................................................................................................31.3.2差商代替微商........................................................................................................................31.3.3差商代替微商格式的误差分析............................................................................................41.4本文主要研究内容...................................................................................................................52显式差分格式..............................................................................................................................72.1常系数热传导方程的古典显式格式.......................................................................................72.1.1古典显式格式格式的推导....................................................................................................72.1.3古典显式格式的算法步骤....................................................................................................83隐式差分格式............................................................................................................................103.1古典隐式格式.........................................................................................................................103.2Crank-Nicolson隐式格式......................................................................................................123.3Douglas差分格式...................................................................................................................133.4加权六点隐式格式.................................................................................................................143.5交替方向隐式格式.................................................................................................................153.5.1Peaceman-Rachford格式....................................................................................................153.5.2Rachford-Mitchell格式................................