Second order parameter-uniform convergence for a finite difference method for a singularly perturbed linear parabolic system

TL;DR

This paper develops a finite difference method for a singularly perturbed parabolic system, achieving uniform convergence in space and time despite overlapping layers caused by small perturbation parameters.

Contribution

It introduces a Shishkin mesh-based finite difference scheme that ensures first order temporal and essentially second order spatial convergence uniformly across parameters.

Findings

Method achieves uniform convergence in space and time.

Numerical scheme handles overlapping layers effectively.

Convergence is proven theoretically for the proposed method.

Abstract

A singularly perturbed linear system of second order partial differential equations of parabolic reaction-diffusion type with given initial and boundary conditions is considered. The leading term of each equation is multiplied by a small positive parameter. These singular perturbation 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 a numerical method for solving this problem. It is proved that the numerical approximations obtained with this method are first order convergent in time and essentially second order convergent in the space variable uniformly with respect to all of the parameters.