DFR法和LDG法解线性三阶KdV方程的等价性  

Equivalence Between DFR Method and LDG Method for Solving Linear Third-Order KdV Equation

作　　者：毕卉 李晓彤 BI Hui;LI Xiaotong(School of Sciences,Harbin University of Science and Technology,Harbin 150080,China)

机构地区：[1]哈尔滨理工大学理学院,哈尔滨150080

出　　处：《哈尔滨理工大学学报》2024年第5期142-148,共7页Journal of Harbin University of Science and Technology

基　　金：国家自然科学基金青年基金(12201157);黑龙江省自然科学基金联合引导项目(LH2020A015)。

摘　　要：研究了直接通量重构方法(Direct Flux Reconstruction,简称DFR)和局部间断Galerkin方法(Local discontinuous Galerkin,简称LDG)求解线性三阶KdV方程的等价性问题。首先分别利用DFR法和LDG法对线性三阶KdV方程进行空间离散并给出两种数值算法的空间离散格式,其次用两种方法证明了DFR法和LDG法求解线性三阶KdV方程的等价性。第一种方法借助高斯求积的性质:M点高斯求积具有2M-1阶代数精度;第二种方法利用洛巴托多项式的特殊性质:M+1次洛巴托多项式导数的零点是M个高斯点。最后以M=1为例,给出了两种数值算法求解线性三阶KdV方程的离散常微分方程组,验证了两种方法的等价性。The equivalence between direct flux reconstruction method and local discontinuous Galerkin method in solving linear third-order KdV equation is studied.Firstly,the linear third-order KdV equation is spatially dispersed by DFR method and LDG method respectively,and the spatial discretization schemes of two numerical algorithms are given.Secondly,the equivalence of DFR and LDG methods is proved by two methods.The first proof relies on the property of Gauss quadrature:the M-point Gauss quadrature has 2M-1 order algebraic accuracy;the second proof takes advantage of the special property of the Lobatto polynomial:the zeros of the derivative of Lobatto polynomial of degree M+1 are M Gauss points.At last,taking the value of M to be 1 as an example,it is shown that the two numerical algorithms are equivalent to the ordinary differential equations used for programming when solving the linear third-order KdV equation.

关 键 词：直接通量重构法 局部间断Galerkin法 线性三阶KdV方程 高斯求积 洛巴托多项式 

分 类 号：O241.3[理学—计算数学]