Second-Order Robust Numerical Method for a Partially Singularly Perturbed Time-Dependent Reaction–Diffusion System

Abstract

This article aims at the development and analysis of a numerical scheme for solving a singularly perturbed parabolic system of n reaction–diffusion equations where m of the equations (with m<n) contain a perturbation parameter while the rest do not contain it. The scheme is based on a uniform mesh in the temporal variable and a piecewise uniform Shishkin mesh in the spatial variable, together with classical finite difference approximations. Some analytical properties and error analyses are derived. Furthermore, a bound of the error is provided. Under certain assumptions, it is proved that the proposed scheme has almost second-order convergence in the space direction and almost first-order convergence in the time variable. Errors do not increase when the perturbation parameter ε→0, proving the uniform convergence. Some numerical experiments are presented, which support the theoretical results.