重心Lagrange插值配点法求解二维双曲电报方程

计算物理 ›› 2016, Vol. 33 ›› Issue (3): 341-348.

重心Lagrange插值配点法求解二维双曲电报方程

刘婷, 马文涛

宁夏大学数学计算机学院, 银川 750021

收稿日期:2015-03-15 修回日期:2015-06-10 出版日期:2016-05-25 发布日期:2016-05-25
作者简介:马文涛(1977-),男(回族),宁夏人,副教授,博士,硕导,从事岩土工程数值计算研究,E-mail:wt-ma2002@163.com;刘婷(1990-),女,甘肃人,硕士研究生,主要从事计算力学及其工程应用研究,E-mail:keke900117@163.com
基金资助:
国家自然科学基金(51269024,51468053)资助项目

Barycentric Lagrange Interpolation Collocation Method for Two-dimensional Hyperbolic Telegraph Equation

LIU Ting, MA Wentao

Department of Mathematics & Computer Science, Ningxia University, Yinchuan 750021, China

Received:2015-03-15 Revised:2015-06-10 Online:2016-05-25 Published:2016-05-25

摘要/Abstract

摘要： 提出一种求解二维双曲电报方程的高精度重心Lagrange插值配点法.采用重心Lagrange插值构造包含时间和空间变量的近似函数.在给定Chebyshev-Gauss-Lobatto节点上,将多变量重心Lagrange插值近似函数代入双曲电报方程及其定解条件,得到离散代数方程组.包含狄里克雷和诺依曼边界条件的数值算例表明,本文方法程序实现方便并具有高精度,可应用于求解高维问题.

关键词: 双曲电报方程, 重心Lagrange插值, 配点法, Chebyshev-Gauss-Lobatto节点

Abstract: We propose a numerical scheme for two-dimensional hyperbolic telegraph equation, in which Chebyshev-Gauss-Lobatto collocation nodes and approximate solutions with multi-variable barycentric Lagrange interpolation functions are used. Multi-variable barycentric Lagrange interpolation functions are given for spatial, temporal variable and their derivatives. Accuracy of the method is demonstrated with test examples with Dirichlet and Neumann boundary conditions. Numerical results are more accurate than numerical solutions in literatures. In additional, the method is easy to implement for multidimensional problems.

Key words: hyperbolic telegraph equation, barycentric Lagrange interpolation, collocation method, Chebyshev-Gauss-Lobatto nodes

中图分类号:

TB115.1

刘婷, 马文涛. 重心Lagrange插值配点法求解二维双曲电报方程[J]. 计算物理, 2016, 33(3): 341-348.

LIU Ting, MA Wentao. Barycentric Lagrange Interpolation Collocation Method for Two-dimensional Hyperbolic Telegraph Equation[J]. CHINESE JOURNAL OF COMPUTATIONAL PHYSICS, 2016, 33(3): 341-348.

参考文献

[1] BÜIBÜl R, SEZER M. Taylor polynomial solution of hyperbolic equation with constant coefficient[J]. Int J Comput Math,2011, 88(3):533-544.
[2] DEHGHAN M, GHESMATI A. Solution of the second-order one-dimensional hyperbolic telegraph equation by using the dual reciprocity boundary integral equation (DRBIE) method[J]. Eng Anal Bound Elem, 2010, 34(1):51-59.
[3] GAO F, CHI C. Unconditionally stable difference scheme for a one-space-dimensional linear hyperbolic telegraph equation[J]. J Comput, 2007, 187(2):1272-1276.
[4] MITTAL R.C, BHATIA R. Numerical solution of second order one dimensional hyperbolic telegraph equation by cubic B-spline collocation method[J]. Appl Math Comput, 2013, 220:496-506.
[5] SAADATMANDI A, DEHGHAN M. Numerical solution of hyperbolic telegraph eqution using Chebyshev tau method[J]. Numer Methods Partial Differ Equ,2010, 26(1):239-252.
[6] LAKSTANI M, SARAY B. N. Numerical solution of telegraph equation using interpolating scaling functions[J]. Comput Math Appl, 2010, 60(7):1964-1972.
[7] 齐振民,等.无界区域上电报方程初边值问题的数值求解[D].北京:清华大学科学数学系,2012:1-5.
[8] 苏令德.基于无网格方法的几类电报偏微分方程的数值解[D].济南:山东师范大学数学科学学院,2014:33-11.
[9] BÜIBÜI R, SEZER M. A taylor matrix method for the solution of a two-dimensional linear hyperbolic equation[J]. Appl Math Lett,2011, 24(10):1716-1720.
[10] DEHGHAN M, AREZOU G. Combination of meshless local weak and strong (MWLS) forms to solve the two-dimensional hyperbolic telegraph equation[J]. Eng Anal Bound Elem, 2010, 34(4):324-336.
[11] DEHGHAN M, MOHEBBI A. High order implicit collocation method for the solution of two-dimensional linear hyperbolic equation[J]. Numer Meth Partial Diff Eq, 2009, 25:232-43.
[12] JIWARI R, PANDIT S, MITTAL R. C. A differential quadrature algorithm to solve the two-dimensional linear hyperbolic telegraph equation with Dirichlet and Neumann boundary conditions[J]. Appl Math Comput, 2012, 218:7279-7294.
[13] BERRUT J P, TREFETHEN L N. Barycentric Lagrange interpolation[J]. SIAM Review, 2004, 46(3):501-517.
[14] 李树忱, 王兆清. 高精度无网格重心插值配点法[M]. 北京:科学出版社, 2012.