数值求解一维波动方程的四阶紧致差分方法  

A Fourth-order Compact Difference Method for Solving the 1D Wave Equation

作　　者：陆静颖 葛永斌 Lu Jingying;Ge Yongbin(School of Mathematics and Statistics, Ningxia University, Yinchuan 750021, China)

机构地区：[1]宁夏大学数学统计学院,宁夏银川750021

出　　处：《宁夏大学学报（自然科学版）》2020年第1期17-22,共6页Journal of Ningxia University(Natural Science Edition)

基　　金：国家自然科学基金资助项目(11772165,11361045);宁夏自然科学基金重点资助项目(2018AAC02003);宁夏回族自治区重点研发计划项目(2018BEE03007)。

摘　　要：针对一维波动方程提出了一种有限差分方法.首先,采用泰勒级数展开公式和原方程代入的方法推导出了第一个时间层未知函数值的四阶紧致差分格式.然后,用四阶紧致差分公式近似空间导数项,采用中心差分格式截断误差余项修正的方法处理时间导数项,推导出了第二个时间层以后未知函数的四阶紧致差分格式.该方法时间和空间具有整体四阶精度.利用Fourier方法分析了所提格式的稳定性.由于本文格式在未知时间层仅涉及3个网格点,因此可采用追赶法求解离散化后所得到的线性方程组.最后,用数值算例验证了本文格式的精确性和稳定性.In this paper,a finite difference method for solving the one-dimensional wave equation is proposed.Firstly,the Taylor series expansion and the original equation substitution are employed to get a fourth order compact difference scheme for computing the first time level.Then,the fourth order compact difference formula is used to approximate spatial derivate and correction for local truncation error remainder of the central difference scheme for temporal derivate is employed to derive a fourth order compact difference scheme for other time levels.The whole accuracy of the present method is the fourth order in both time and space.The stability of the scheme is analyzed by the Fourier method.Because only three unknown grid points are involved,the forward elimination and backward substitution algorithm can be used to solve the arising linear system.Numerical experiments are carried out to verify the accuracy and the stability of the present method.

关 键 词：波动方程 紧致差分格式 四阶精度 有限差分方法 稳定性 

分 类 号：O241.82[理学—计算数学]