求解二维Fisher-KPP方程的一组加权结构保持差分格式的分析及其Richardson外推法  

The Analyses and Richardsdon Extrapolation Methods of a Class of Weighted Structure-preserving Difference Schemes for Fisher-KPP Equations

作　　者：赵紫琳 邓定文[1] ZHAO ZILIN;DENG DINGWEN(College of Mathematics and Information Science,Nanchang Hangkong University,Nanchang 330063,China;Science Teaching Department,Jiangxi University of Technology,Nanchang 330100,China)

机构地区：[1]南昌航空大学数学与信息科学学院,南昌330063 [2]江西科技学院理学教学部,南昌330100

出　　处：《应用数学学报》2024年第1期101-123,共23页Acta Mathematicae Applicatae Sinica

基　　金：江西省杰出青年基金(20212ACB211006);国家自然科学基金(11861047);江西省自然科学基金(20202BABL201005)。

摘　　要：本文对二维Fisher-Kolmogorov-Petrovsky-Piscounov(Fisher-KPP)方程建立了一组加权的结构保持有限差分方法.运用能量分析法证明了当网格步长,参数α,p及θ满足一定条件时差分解具有保正性,保界性,保单调性等一系列数学性质,且在无穷范数意义下有O(τ+h_(x)^(2)+h_(y)^(2))的收敛阶.然后,依据差分解的渐进展式,建立了一类Richardson外推法,获得了收敛阶为O(τ^(2)+h_(x)^(4)+h_(y)^(4))的外推解,提高了计算效率.最后数值实验表明,数值结果与理论结果相吻合.值得提及的是本文构造的Richardson外推法无需对时、空网格比增加额外的条件.This study is concerned with numerical solutions of the two-dimensional Fisher-Kolmogorov-Petrovsky-Piscounov equation(Fisher-KPP)by a class of weighted structure-preserving finite difference methods(W-SP-FDMs)combined with Richardson extrapolation methods(REMs).By using the discrete energy analysis method,it is shown that as the parametersα,p andθ,and the ratios of temporal meshsize to spatial meshsizes satisfy certain conditions,the current W-SP-FDMs possess many properties,such as,preserving positivity,preserving boundedness,preserving monotonicity,and has a convergence order of O(τ+h_x~2+h_y~2)in L~∞-norm.Also,by using the discrete energy analysis method,it is shown that the REMs,which are developed by the asymptotic expansion formula of the numerical solutions,can make the final solutions convergent with an order of O(τ~2+h_x~4+h_y~4)in L~∞-norm,thus improving computational efficiency.Finally,numerical results confirm the correctness of theoretical findings and high performance of the current methods.It is worthwhile to mention that additional condition for the ratios of temporal meshsize to spatial meshsizes is not supplemented as REMs are used.

关 键 词：二维Fisher-KPP方程 结构保持差分格式 收敛性 RICHARDSON外推法 

分 类 号：O175[理学—数学]