非线性反应扩散方程的高精度有限差分方法  

High-order Finite Difference Scheme for the Nonlinear Reaction Diffusion Equation

作　　者：刘圣恩 葛永斌 祁应楠[2] LIU Shengen;GE Yongbin;QI Yingnan(School of Mathematics and Statistics,Ningxia University,Yinchuan 750021,China;School of Mathematics and Computer Science,Ningxia Normal University,Guyuan 756000,China)

机构地区：[1]宁夏大学数学统计学院,宁夏银川750021 [2]宁夏师范学院数学与计算机科学学院,宁夏固原756000

出　　处：《应用数学》2023年第3期747-755,共9页Mathematica Applicata

基　　金：国家自然科学基金资助项目(12161067);宁夏自然科学基金资助项目(2022AAC02023,2022AAC03301,2022AAC03314);宁夏自治区青年拔尖人才培养工程项目;宁夏大学研究生创新项目(CXXM2022-12)。

摘　　要：首先,针对空间二阶导数,提出了一种五点六阶差分公式.然后,针对一维非线性反应扩散方程,空间导数项采用该差分公式离散,时间导数项采用Crank-Nicolson方法进行离散,再利用Richardson外推方法将时间精度提高到四阶,提出了一种时间四阶空间六阶精度的有限差分格式.由于每一个时间层上所形成的线性方程组是五对角形的,因此采用五对角追赶法进行计算,计算简单且高效.最后通过数值实验验证了格式的精确性和可靠性.Firstly,for the second-order derivative of space,a five-point sixth-order difference formula is proposed.Then,for the one-dimensional nonlinear reaction-diffusion equation,the spatial derivative term is discretized by the five-point sixth-order difference formula,and the temporal derivative term is discretized by the Crank-Nicolson method.The Richardson extrapolation method is used to improve the temporal accuracy from the second order to the fourth order.A difference scheme of the fourth order temporal accuracy and sixth-order spatial accuracy is proposed.Since the linear systems formed at each time layer are penta-diagonal,the penta-diagonal matrix algorithm is used for calculations,which is simple and efficient.Finally,the accuracy and reliability of the scheme are verified by some numerical experiments.

关 键 词：非线性反应扩散方程 五点六阶差分公式 高精度 五对角追赶法 RICHARDSON外推 

分 类 号：O241.82[理学—计算数学]