五阶偏微分方程的时空谱方法  

SPACE-TIME SPECTRAL METHOD FOR SOLVING FIFTH-ORDER PARTIAL DIFFERENTIAL EQUATION

作　　者：余荣玉 吴华[1] Yu Rongyu;Wu Hua(College of Sciences,Shanghai University,Shanghai 200444)

机构地区：[1]上海大学理学院,上海200444

出　　处：《高等学校计算数学学报》2022年第2期159-174,共16页Numerical Mathematics A Journal of Chinese Universities

基　　金：国家自然科学基金(11971016)。

摘　　要：1引言在实际问题中,五阶偏微分方程有着广泛的应用,这些方程出现在材料、力学、光学、热传导、振动、流体运动、控制以及生物系统等方面,例如:广义Korteweg-de Vries(KdV)方程[1,2,3,4,5,6].在此之前,国内外的一些科研工作者已经对五阶偏微分方程进行了大量的研究,在[7]中Xu等提出了局部间断Galerkin(LDG)方法来求解五阶波动方程,在[8]中Yan等提出了新的LDG方法来求解五阶方程,在[9]中Cheng等提出了间断Galerkin(DG)有限元方法来求解时间依赖的高阶方程,在[10]中赵小红等针对一类组合的五阶方程的柯西问题提出了一种显式的差分格式,在[11]等针对五阶方程的变系数问题提出了Jacobi-Jacobi dual-Petrov-Galerkin谱方法,在[12]中Tao等提出了ultraweak局部间断Galerkin方法来求解五阶方程.In this paper,we propose the space-time spectral method for the fifth-order partial differential equation.The method uses Legendre-Petrov-Galerkin spectral method in the direction of space,that is:the test function space choosed is different from the trial function space,and makes the coefficient matrix in the direction of space sparse by choosing the appropriate basis function;the method uses Legendre-tau spectral method in the direction of time.At the same time,error analysis of the fully discrete scheme for the fifth order linear partial differential equation is given by this paper,the nonlinear term for the nonlinear equation is computed by interpolation on Chebyshev-Gauss-Lobatto points;finally,some numerical examples are given to verify the effectiveness of the proposed algorithm.

关 键 词：Galerkin谱方法 偏微分方程 柯西问题 高阶方程 流体运动 差分格式 波动方程 变系数 

分 类 号：O241.82[理学—计算数学]