A uniformly convergent difference scheme on a modified Shishkin mesh for the singular perturbation boundary value problem

TL;DR

This paper develops a stable, uniformly convergent difference scheme on a modified Shishkin mesh for semilinear singular perturbation boundary value problems, demonstrating near second-order accuracy and robustness through theoretical analysis and numerical examples.

Contribution

It introduces a new difference scheme on a modified Shishkin mesh that achieves uniform convergence for semilinear singular perturbation problems, which was not previously established.

Findings

The scheme is stable and has a unique solution.

The scheme achieves ε-uniform near second-order convergence.

Numerical examples confirm theoretical accuracy and robustness.

Abstract

In this paper we are considering a semilinear singular perturbation reaction -- diffusion boundary value problem, which contains a small perturbation parameter that acts on the highest order derivative. We construct a difference scheme on an arbitrary nonequidistant mesh using a collocation method and Green's function. We show that the constructed difference scheme has a unique solution and that the scheme is stable. The central result of the paper is $\epsilon$-uniform convergence of almost second order for the discrete approximate solution on a modified Shishkin mesh. We finally provide two numerical examples which illustrate the theoretical results on the uniform accuracy of the discrete problem, as well as the robustness of the method.