A parameter--uniform finite difference method for a singularly perturbed linear system of second order ordinary differential equations of reaction-diffusion type

TL;DR

This paper develops finite difference methods on Shishkin meshes for solving singularly perturbed reaction-diffusion systems, achieving uniform convergence regardless of small parameters.

Contribution

It introduces two finite difference schemes on Shishkin meshes that are proven to be uniformly convergent for systems with multiple small parameters.

Findings

Methods are essentially first and second order convergent uniformly.

Numerical schemes effectively handle overlapping layers in solutions.

Convergence proofs validate the robustness of the approaches.

Abstract

A singularly perturbed linear system of second order ordinary differential equations of reaction-diffusion type with given boundary conditions is considered. The leading term of each equation is multiplied by a small positive parameter. These parameters are assumed to be distinct. The components of the solution exhibit overlapping layers. Shishkin piecewise-uniform meshes are introduced, which are used in conjunction with a classical finite difference discretisation, to construct two numerical methods for solving this problem. It is proved that the numerical approximations obtained with these methods are essentially first, respectively second, order convergent uniformly with respect to all of the parameters.