A parameter uniform fitted mesh method for a weakly coupled system of two singularly perturbed convection-diffusion equations

TL;DR

This paper introduces a numerical method using a fitted mesh for solving a weakly coupled system of singularly perturbed convection-diffusion equations, achieving uniform convergence despite boundary layers.

Contribution

It develops a first-order convergent upwind finite difference scheme on a parameter-uniform Shishkin mesh for coupled singularly perturbed systems.

Findings

Method is proven to be first order convergent uniformly in parameters.

Numerical examples confirm theoretical convergence and effectiveness.

The approach effectively captures boundary layers in the solution.

Abstract

In this paper, a boundary value problem for a singularly perturbed linear system of two second order ordinary differential equations of convection- diffusion type is considered on the interval [0, 1]. The components of the solution of this system exhibit boundary layers at 0. A numerical method composed of an upwind finite difference scheme applied on a piecewise uniform Shishkin mesh is suggested to solve the problem. The method is proved to be first order convergent in the maximum norm uniformly in the perturbation parameters. Numerical examples are provided in support of the theory.