A fitted second--order difference scheme on a modified Shishkin mesh for a semilinear singularly-perturbed boundary-value problem

TL;DR

This paper introduces a second-order difference scheme on a modified Shishkin mesh for solving semilinear singularly-perturbed boundary-value problems with boundary layers, demonstrating theoretical convergence and numerical validation.

Contribution

A novel second-order difference scheme on a modified Shishkin mesh for semilinear singular perturbation problems, with proven convergence and numerical confirmation.

Findings

Second-order convergence of the scheme on the modified Shishkin mesh

Numerical experiments confirm theoretical convergence rates

Effective handling of boundary layers in singular perturbation problems

Abstract

In the present paper we consider the numerical solving of a semilinear singular--perturbation reaction--diffusion boundary--value problem having boundary layers. A new difference scheme is constructed, the second order of convergence on a modified Shishkin mesh is shown. The numerical experiments are included in the paper, which confirm the theoretical results.