求解一类非线性偏微分方程的高精度紧致差分方法  

A high-order compact difference method for solving a class of nonlinear partial differential equations

作　　者：武莉莉 WU Li-li(School of Mathematics and Statistics,Ningxia University,Yinchuan 750021,Ningxia,China)

机构地区：[1]宁夏大学数学与统计学院,宁夏银川750021

出　　处：《西北师范大学学报（自然科学版）》2021年第3期26-31,共6页Journal of Northwest Normal University(Natural Science)

基　　金：国家自然科学基金资助项目(11772165,11701306);宁夏师范学院校级科研资助项目(NXSFZDA2001)。

摘　　要：针对一类非线性偏微分方程,提出了一种新的高精度紧致差分方法.首先对内部网格节点处的空间一阶和二阶导数项采用四阶精度的Padé紧致差分格式进行离散,然后对时间导数项采用泰勒级数展开并使用截断误差余项修正法进行离散,最终得到了求解该非线性方程的一种三层隐式高精度紧致差分格式,其截断误差为O(τ2+τh 2+h 4),即当τ=O(h 2)时,该格式在空间上具有四阶精度.最后通过对广义Burgers-Fisher方程和广义Burgers-Huxley方程的数值求解,验证了本文方法的精确性和可靠性.A new high-order compact difference method is proposed for solving a class of nonlinear partial differential equations in this paper.At first,for the first and second spatial derivatives of the interior grid points,the fourth-order Padéformula is employed to discretize them;then the time derivative is discretized by the forward difference scheme and truncation error remainder correction approach.Furthermore,a three-layer implicit high-order compact difference scheme is proposed.The local truncation error of the scheme is O(τ2+τh 2+h 4),i.e.,the scheme is the fourth-order accuracy for space whenτ=O(h 2).Finally,numerical solutions of the general Burgers-Fisher and Burgers-Huxley equations are solved to verify the accuracy and reliability of the present method.

关 键 词：非线性偏微分方程 BURGERS-FISHER方程 Burgers-Huxley方程 紧致差分格式 高精度 

分 类 号：O241.82[理学—计算数学]