On construction of a global numerical solution for a semilinear singularly--perturbed reaction diffusion boundary value problem

TL;DR

This paper develops and analyzes stable numerical schemes for semilinear singularly-perturbed reaction-diffusion boundary value problems, proving uniform convergence and constructing global solutions with spline methods, supported by numerical experiments.

Contribution

It introduces new stable difference schemes with proven uniform convergence on modified Shishkin meshes and constructs global solutions using spline methods, advancing numerical analysis for singular perturbation problems.

Findings

Proved stability and uniqueness of the schemes.

Established uniform convergence on modified Shishkin mesh.

Validated theoretical results with numerical experiments.

Abstract

A class of different schemes for the numerical solving of semilinear singularly--perturbed reaction--diffusion boundary--value problems was constructed. The stability of the difference schemes was proved, and the existence and uniqueness of a numerical solution were shown. After that, the uniform convergence with respect to a perturbation parameter $\varepsilon$ on a modified Shishkin mesh of order 2 has been proven. For such a discrete solution, a global solution based on a linear spline was constructed, also the error of this solution is in expected boundaries. Numerical experiments at the end of the paper, confirm the theoretical results. The global solutions based on a natural cubic spline, and the experiments with Liseikin, Shishkin and modified Bakhvalov meshes are included in the numerical experiments as well.