时滞Fisher方程的保结构Du Fort-Frankel差分格式及其分析  

STRUCTURE-PRESERVING DU FORT-FRANKEL FINITE DIFFERENCE SCHEMES AND THEIR ANALYSIS FOR DELAYED FISHER EQUATION

作　　者：熊小红 邓定文[1] Xiong Xiaohong;Deng Dingwen(School of Mathematics and Information Science,Nanchang Hangkong University,Nanchang 330063,China)

机构地区：[1]南昌航空大学数学与信息科学学院,南昌330063

出　　处：《计算数学》2024年第2期189-212,共24页Mathematica Numerica 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)资助。

摘　　要：本文首先对一维时滞Fisher方程建立了保非负性的DuFort-Frankel差分格式。运用数学归纳法证明了当网格比r_(x)=(ε△t)/h^(2)_(x)≤1/2时,它的数值解大于或者等于零.这里ε,△t和h分别是扩散系数,时间和空间方向上的网格步长其次,运用截断技巧修正由保非负性的Du FortFrankel差分格式获得的数值解,从而设计了一类既保非负性又保最大界的差分方法.运用数学归纳法证明了当r_(x)≤1/2时,它的数值解落在区间[0,1]内.运用能量分析法,我们证明这两类方法在最大范数下均有O(△t+(△t/h_(x))^(2)+h^(2)_(x))的收敛阶.再次,类似地,我们对二维问题建立了保非负性的Du Fort-Frankel差分格式和既保非负性又保最大界的差分法,及其理论.最后,数值结果验证了理论的正确性和新算法的高效性。To begin with,a non-negativity-preserving Du Fort-Frankel finite difference method(FDM)is derived for one-dimensional(1D)delayed Fisher's equation.By applying mathematical induction,we can prove that its numerical solutions are all larger than zero as long as r_(x)=(ε△t)/h^(2)_(x)≤1/2.Here,ε,△t and h_(x) are diffusion coefficient,time-step size and spatial meshsize in a-direction,respectively.Secondly,by using cut-off technique to adjust numerical solutions obtained using this non-negativity-preserving Du Fort Frankel FDM,an improved FDM,which can inherit the non-negativity and boundedness of the exact solutions,is designed.Also,by applying mathematical induction,it is shown that its numerical solutions locate in[0,1].By using the discrete energy method,it is shown that both of the proposed algorithms possess the convergence rates of O(△t+(△t/h_(x))^(2)+h^(2)_(x))in the maximum norm.Thirdly,by using the techniques similar to 1D case,a non-negativity-preserving Du Fort-Frankel FDM and a non-negativity-and boundedness-preserving FDM are developed for two-dimensional Fisher's equation with delay.Also,theoretical findings can be obtained,similarly.Finally,numerical results confirm the exactness of theoretical results,and high efficiency of the proposed methods.

关 键 词：时滞Fisher方程 DuFortFrankel差分格式 非负性 有界性 收敛性 

分 类 号：O241.3[理学—计算数学]