带非线性边界条件的反应扩散方程的数值方法

１引言近年来关于非线性抛物型方程数值解法的研究取得了许多好的结果，其中以Ｃ．Ｖ．Ｐａｏ为主的研究者们利用上、下解方法对带线性边界条件的半线性抛物型方程的有限差分系统进行了广泛的研究，提出了一系列有效的迭代算法（见［１］、［２］、［３］、［４］）．但对带非线性边界条件的半线性抛物型方程初边值问题，作者至今尚未见到有研究者将上、下解方法用在相应的差分系统上，求得数值解．其主要原因是由于边界上函数的非线性，解在边界网格点上的值未知且无法用内部网格点上的值直接表示，相应的差分系统表示形式受到影响，边界网…

NUMERICAL METHODS FOR REACTION-DIFFUSION EQUATIONS WITH NONLINEAR BOUNDARY CONDITIONS

In this paper we study the finite difference equations corresponding to a class of semilinear parabolic equations with nonlinear boundary conditions in a bounded domain. Using the method of upper-lower solutions we construct two monotone sequences for finite difference systems. It is shown that these two sequences converge monotonically from above and below, respectively, to a unique solution of the nonlinear discrete equations. Moreover, the monotone convergence property is used to prove the convergence of the finite difference solution to the corresponding solution of the differential system as the mesh size decreases to zero.