求解带非线性边界条件的抛物型方程组的数值解法

Numerical Methods for Coupled Nonlinear Parabolic Equations with Nonlinear Boundary Conditions

研究了一类含有非线性边界条件的非线性反应扩散方程组的数值解法。把上下解方法应用到相应有限差分系统上, 得到两个迭代序列。可以证明, 当反应项以及边界条件为拟单调函数时, 这两个序列均单调收敛到差分系统的唯一解, 并且, 当网格结点的间距趋于0 时,该解收敛到相应微分方程组的解。

We study a system of nonlinear finite difference equations corresponding to a class of coupled parabolic equations with nonlinear boundary conditions in a boundary domain. Using the method of upper lower solutions we construct two monotone sequences for the finite difference equations. It is shown that when the reaction functions and boundary conditions are quasimonotone nondecreasing, nonincreasing or mixed quasimonotone, these two sequences converge to a unique solution of the finite difference system. 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.