High order parameter-robust numerical method for a system of (M>=2) coupled singularly perturbed parabolic reaction-diffusion problems

TL;DR

This paper introduces a high-order, parameter-robust numerical method for solving coupled singularly perturbed parabolic reaction-diffusion systems, achieving high accuracy uniformly with respect to the small perturbation parameter.

Contribution

It develops a second-order in time and nearly fourth-order in space numerical scheme using Crank-Nicolson and hybrid finite differences on a Shishkin mesh, proven to be psilon-uniform and validated numerically.

Findings

Method is psilon-uniform and second order in time.

Achieves almost fourth order accuracy in space.

Numerical results confirm theoretical convergence and practical robustness.

Abstract

We present a high order parameter-robust numerical method for a system of (M>=2) coupled singularly perturbed parabolic reaction-diffusion problems. A small perturbation parameter {\epsilon} is multiplied with the second order spatial derivatives in all the equations. The parabolic boundary layer appears in the solution of the problem when the perturbation parameter {\epsilon} tends to zero. To obtain a high order approximation to the solution of this problem, we propose a numerical method that employs the Crank-Nicolson method on an uniform mesh in time direction, together with a hybrid finite difference scheme on a generalized Shishkin mesh in spatial direction. We prove that the resulting method is parameter-robust or \epsilon-uniform of second order in time and almost fourth order in spatial variable, if the discretization parameters satisfy a non-restrictive relation. Numerical experiments are presented to validate the theoretical results and also indicate that the relation between the discretization parameters is not necessary in practice.