摘要对流方程是一类重要的偏微分方程.因此,数值求解该类方程具有非常重要的理论价值和实际意义.本文建立了求解对流方程的高阶紧致差分格式.首先,假设方程在点成立,将(xi,tn1/2)方程在时间方向和空间方向上均采用泰勒级数展开及对截断误差余项中的三阶导数进行修正的方法对时间和空间导数进行离散,得到一种求解一维对流方程的两层高精度紧致全隐格式.该格式在时间和空间上均具有四阶精度.再将方程在处展开,得到一种求解一维HOC1(xi,tn)对流方程的三层高精度紧致差分格式HOC2.采用VonNeumann方法分析了两种格式的稳定性.然后通过几个具有精确解的数值算例进行数值验证,可以看出,本文提出的HOC1格式具有较好的稳定性和精确性.其次,针对二维、三维对流方程,利用局部一维化(LOD)方法分裂为一维问题进行求解.并将分裂后的一维对流方程在时间和空间上均采用泰勒级数展开及对截断误差余项中的三阶导数进行修正的方法对时间和空间导数进行离散,得到二维、三维对流方程的高精度紧致LOD格式,运用VonNeumann方法分析了该格式的稳定性,通过数值算例验证了格式的精确性和可靠性.最后,将本文所推导的格式接入到“PHOEBESolver”[1]求解软件,使得偏微分方程数值解的相关学者更加方便地使用本文格式.关键词:对流方程;高精度;紧致差分格式;LOD方法;有限差分法IAbstractConvectionequationsisakindofpartialdifferentialequations.Therefore,solvingtheseequationshasveryimportanttheoreticalandpracticalsignificance.Thispaperestablisheshigh-ordercompactdifferenceschemeforsolvingconvectionequations.Firstofall,assumingthattheonedimensionalequationisestablishedat(xi,tn1/2),Taylorseriesexpansionandcorrectionforthethirdderivativeinthetruncationerrorremainderofthecentraldifferenceschemeareusedfordiscretizationoftimeandspace.Soatwo-levelimplicitcompactdifferenceschemeforsolvingtheone-dimensionalconvectionequationisproposed.ItisnamedinHOC1.itisthefourth-orderaccuracyinbothtimeandspace.Secondly,expendingtheonedimensionalequationat(xi,tn),athree-levelcompactdifferenceschemeforsolvingtheone-dimensionalconvectionequationisproposed.ItisnamedinHOC2.ThesestabilitiesareobtainedbythevonNeumannmethod.Theaccuracyandthestabilityofthepresentschemearevalidatedbysomenumericalexperiments.WecandrawaconclusionthatHOC1isbetterthanHOC2inthestabilityandtheaccuracy.Then,Forthetwo-dimensionalandthree-dimensionalconvectionequations,usingtheLODmethodtomakingthetwo-dimensionalandthree-dimensionalproblemssplitone-dimensionalequations.Theone-dimensionalconvectionequationsusetaylorseriesexpansionandcorrectionforthethirdderivativeinthetruncationerrorremainderofthecentraldifferenceschemeindiscretizationoftimeandspace.Wecanestablishsomehigh-ordercompactLO