A class of difference schemes uniformly convergent on a modified Bakhvalov mesh

TL;DR

This paper develops a class of difference schemes for singularly perturbed reaction-diffusion problems, proving their uniform convergence and validating the results with numerical experiments.

Contribution

It introduces a new class of differential schemes with proven ε-uniform second-order convergence on a modified Bakhvalov mesh, ensuring robustness for singular perturbation problems.

Findings

Proved existence and uniqueness of the numerical solution.

Established ε-uniform convergence of order 2.

Numerical experiments confirm theoretical results.

Abstract

In this paper we consider the numerical solution of a singularly perturbed one-dimensional semilinear reaction-diffusion problem. A class of differential schemes is constructed. There is a proof of the existence and uniqueness of the numerical solution for this constructed class of differential schemes. The central result of the paper is an $\varepsilon$--uniform convergence of the second order $\mathcal{O}\left(1/N^2 \right),$ for the discrete approximate solution on the modified Bakhvalov mesh. At the end of the paper there are numerical experiments, two representatives of the class of differential schemes are tested and it is shown the robustness of the method and concurrence of theoretical and experimental results.