一类非线性反应-扩散方程的间断Galerkin谱元方法

doi:10.3969/j.issn.1007-2861.2013.07.029

上海大学学报(自然科学版) ›› 2014, Vol. 20 ›› Issue (6): 757-768.doi: 10.3969/j.issn.1007-2861.2013.07.029

一类非线性反应-扩散方程的间断Galerkin谱元方法

吴华, 韩晓飞

上海大学理学院, 上海200444

收稿日期:2013-05-27 出版日期:2014-12-23 发布日期:2014-12-23
通讯作者: 吴华(1970—), 女, 副教授, 博士, 研究方向为偏微分方程数值解. E-mail: hwu@staff.shu.edu.cn E-mail:hwu@staff.shu.edu.cn
作者简介:吴华(1970—), 女, 副教授, 博士, 研究方向为偏微分方程数值解. E-mail: hwu@staff.shu.edu.cn
基金资助:
国家自然科学基金资助项目(11171209); 教育部留学回国人员科研启动基金资助项目; 上海市自然科学基金资助项目(13ZR1416700)

Discontinuous Galerkin Spectral Element Methods for Nonlinear Reaction-Diffusion Equations

WU Hua, HAN Xiao-fei

College of Sciences, Shanghai University, Shanghai 200444, China

Received:2013-05-27 Online:2014-12-23 Published:2014-12-23

摘要/Abstract

摘要： 提出了一类非线性反应-扩散方程的间断Galerkin 谱元方法, 在每个子区间上, 基本格式采用Legendre-Galerkin 方法, 非线性项采用Chebyshev-Gauss-Lobatto 插值, 跳跃项利用中心数值流量处理, 时间方向应用4阶低存储Runge-Kutta 格式离散. 该方法处理某些初值间断问题有效, 并可并行实现; 给出了该方法半离散格式下的稳定性和收敛性分析, 利用Chebyshev-Gauss-Lobatto 插值算子在不带权意义下的逼近结果, 获得了按L2-模的最优误差估计; 最后, 给出了连续问题和间断问题的数值算例.

关键词: Chebyshev-Gauss-Lobatto插值, 反应-扩散方程, 间断Galerkin方法, 谱元法

Abstract: Discontinuous Galerkin spectral element methods for nonlinear reaction-diffusion equations are considered. The schemes are basically in the Legendre-Galerkin form. The nonlinear term is interpolated through the Chebyshev-Gauss-Lobatto points. The jump term tackled by the central numerical flux in space variation. The fourth-order low-storage Runge-Kutta scheme is applied for time discrete inside each subinterval. The methods can be used to solve discontinuous initial value problems and implemented in a parallel way. Stability and the optimal rate of convergence in L2-norm for the semi-discrete scheme are shown using the approximate results of the Chebyshev-Gauss-Lobatto interpolation operator without a weight function. Numerical results for the continuous and discontinuous problems are given.

Key words: Chebyshev-Gauss-Lobatto interpolation, discontinuous Galerkin method, reaction-diffusion equation, spectral element method

中图分类号:

O 241.82

吴华, 韩晓飞. 一类非线性反应-扩散方程的间断Galerkin谱元方法[J]. 上海大学学报(自然科学版), 2014, 20(6): 757-768.

WU Hua, HAN Xiao-Fei. Discontinuous Galerkin Spectral Element Methods for Nonlinear Reaction-Diffusion Equations[J]. Journal of Shanghai University（Natural Science Edition）, 2014, 20(6): 757-768.

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一类非线性反应-扩散方程的间断Galerkin谱元方法

Discontinuous Galerkin Spectral Element Methods for Nonlinear Reaction-Diffusion Equations

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