求解二维Fisher-KPP方程的一类保正保界差分格式及其Richardson外推法

本文研究求解二维Fisher-Kolmogorov-Petrovsky-Piscounov (Fisher-KPP)方程的一类保正保界差分格式.运用能量分析法证明了当网格比满足$R_{x}+R_{y}+b\tau (p-1)]/2\leq\frac{1}{2}$时差分解具有一系列数学性质,包括保正性、保界性和单调性,且在无穷范数意义下有$O (\tau+h_{x}^{2}+h_{y}^{2})$的收敛阶.然后通过发展Richardson外推法得到收敛阶为$O (\tau^{2}+h_{x}^{4}+h_{y}^{4})$的外推解.最后数值实验表明数值结果与理论结果相吻合.值得提及的是在运用本文构造的Richardson外推法时对时空网格比没有增加更严格的条件.

A POSITIVITY AND BOUNDEDNESS PRESERVING DIFFERENCE SCHEME AND ITS RICHARDSON EXTRAPOLATION METHOD FOR SOLVING A TWO-DIMENSIONAL FISHER-KPP EQUATION

In this paper, a positivity-preserving difference scheme is studied for two-dimensional Fisher-Kolmogorov-Petrovsky-Piscounov equation (Fisher-KPP). The energy analysis method is used to prove that the difference scheme has a series of mathematical properties, including preserving positivity, preserving boundedness, preserving monotonicity, and has a convergence order of $O(\tau+h_x^2+h_y^2)$ in maximum norm, when the ratio of temporal meshsize to spatial meshsizes satisfy $R_{x}+R_{y}+b\tau(p-1)]/2\leq \frac{1}{2}$. Then by developing Richardson extrapolation methods (REMs), the extrapolation solutions with convergence orders of $O(\tau^{2}+h_{x}^{4}+h_{y}^{4})$ are obtained. Finally, two numerical examples are given to confirm that the numerical results agree well with the theoretical results. It is worthwhile meaning that the condition of the ratios of temporal meshsize to spatial meshsizes as the current REMs are used are the same as the one of the original numerical method.