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

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.