二维半线性扩散反应方程的高精度全隐格式及其多重网格方法

doi:10.19596/j.cnki.1001-246x.8042

计算物理 ›› 2020, Vol. 37 ›› Issue (3): 307-319.DOI: 10.19596/j.cnki.1001-246x.8042

二维半线性扩散反应方程的高精度全隐格式及其多重网格方法

张林, 葛永斌

宁夏大学数学统计学院, 宁夏银川 750021

收稿日期:2019-01-18 修回日期:2019-05-28 出版日期:2020-05-25 发布日期:2020-05-25
通讯作者: 葛永斌,男,教授,博士生导师,研究方向为偏微分方程数值解、科学计算可视化,E-mail:gyb@nxu.edu.cn
作者简介:张林,男,博士研究生,研究方向为偏微分方程数值解、科学计算可视化,E-mail:zl_linzhang@sina.com
基金资助:
国家自然科学基金（11772165，11361045）、宁夏自然科学基金重点项目（2018AAC02003）及宁夏自治区重点研发项目（2018BEE03007）资助

High-order Fully Implicit Scheme and Multigrid Method for Two-dimensional Semilinear Diffusion Reaction Equations

ZHANG Lin, GE Yongbin

School of Mathematics and Statistics, Ningxia University, Yinchuan, Ningxia 750021, China

Received:2019-01-18 Revised:2019-05-28 Online:2020-05-25 Published:2020-05-25

摘要/Abstract

摘要： 针对二维非定常半线性扩散反应方程，空间导数项采用四阶紧致差分公式离散，时间导数项采用四阶向后Euler公式进行离散，提出一种无条件稳定的高精度五层全隐格式.格式截断误差为O（τ⁴+τ²h²+h⁴），即时间和空间均具有四阶精度.对于第一、二、三时间层采用Crank-Nicolson方法进行离散，并采用Richardson外推公式将启动层时间精度外推到四阶.建立适用于该格式的多重网格方法，加快在每个时间层上迭代求解代数方程组的收敛速度，提高计算效率.最后通过数值实验验证格式的精确性和稳定性以及多重网格方法的高效性.

关键词: 半线性扩散反应方程, 高精度全隐格式, 紧致格式, 多重网格方法, Richardson外推

Abstract: A finite difference method is used for high-order numerical solution of two-dimensional unsteady semilinear diffusion reaction equation. The spatial derivative term is discretized by a fourth-order compact difference formula, and the time derivative term is discretized by a fourth-order backward Euler formula. An unconditionally stable high-order five-level fully implicit scheme is proposed. Truncation error of the scheme is O(τ⁴+τ²h²+h⁴), that is, the time and space have fourth-order accuracy. In calculation of start-up steps, the first, second and third time levels are discretized by Crank-Nicolson method. Richardson extrapolation formula was used to extrapolate startup time accuracy to the fourth-order. A multigrid method based on the scheme is established, which accelerates convergence speed of the algebraic equations on each time level and improves computational efficiency. Finally, accuracy, stability and efficiency of the scheme and multigrid approach are verified with numerical experiments.

Key words: semilinear diffusion reaction equation, high-oder fully implicit scheme, compact scheme, multigrid method, Richardson extrapolation

中图分类号:

O241.82

张林, 葛永斌. 二维半线性扩散反应方程的高精度全隐格式及其多重网格方法[J]. 计算物理, 2020, 37(3): 307-319.

ZHANG Lin, GE Yongbin. High-order Fully Implicit Scheme and Multigrid Method for Two-dimensional Semilinear Diffusion Reaction Equations[J]. CHINESE JOURNAL OF COMPUTATIONAL PHYSICS, 2020, 37(3): 307-319.

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二维半线性扩散反应方程的高精度全隐格式及其多重网格方法

High-order Fully Implicit Scheme and Multigrid Method for Two-dimensional Semilinear Diffusion Reaction Equations

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