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Higher-order monotone iterative methods for finite difference systems of nonlinear reaction- diffusion-convection equations

机译：非线性反应扩散对流方程有限差分系统的高阶单调迭代方法

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摘要

This paper is concerned with the computational algorithms for finite difference discretizations of a class of nonlinear reaction-diffusion-convection equations with nonlinear boundary conditions. A higher-order monotone iterative method is presented for solving the finite difference discretizations of both the time-dependent problem and the corresponding steady-state problem. This method leads to an efficient linear iterative algorithm which yields two sequences of iterations that converge monotonically to a unique solution of the system. The monotone property of the iterations gives concurrently improved upper and lower bounds of the solution in each iteration. It is shown that the rate of convergence for the sum of the two produced sequences is of order p + 2, where p ≥ 1 is a positive integer depending on the construction of the method, and under an additional requirement, the higher-order rate of convergence is attained for one of these two sequences. An application is given to an enzyme-substrate reaction-diffusion problem, and some numerical results are presented to illustrate the effectiveness of the proposed method.
机译：本文涉及一类具有非线性边界条件的非线性反应扩散对流方程的有限差分离散化的计算算法。提出了一种高阶单调迭代方法来求解时变问题和相应的稳态问题的有限差分离散化。这种方法导致了一种有效的线性迭代算法，该算法产生两个迭代序列，这些迭代序列单调收敛到系统的唯一解。迭代的单调属性可在每次迭代中同时提高解决方案的上限和下限。结果表明，两个产生的序列之和的收敛速度为p + 2，其中p≥1是一个正整数，具体取决于方法的构造，并且在附加要求下，更高的频率对于这两个序列之一，可以达到收敛收敛。将其应用于酶-底物反应扩散问题，并给出了一些数值结果，说明了该方法的有效性。

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