Uniformly Convergent Difference Scheme for a Semilinear Reaction-Diffusion Problem on Shishikin mesh

TL;DR

This paper develops and proves the uniform convergence of two difference schemes for a one-dimensional singularly perturbed reaction-diffusion problem on a Shishkin mesh, supported by numerical experiments.

Contribution

It introduces two difference schemes with proven epsilon-uniform convergence on a Shishkin mesh for singularly perturbed problems.

Findings

Both schemes achieve epsilon-uniform convergence.

Numerical experiments confirm theoretical convergence results.

The methods are effective for singularly perturbed boundary value problems.

Abstract

In this paper we consider two difference schemes for numerical solving of a one--dimensional singularly perturbed boundary value problem. We proved an $\varepsilon$--uniform convergence for both difference schemes on a Shiskin mesh. Finally, we present four numerical experiments to confirm the theoretical results.