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arXiv · 1411.4323

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

Enes Duvnjaković·Samir Karasuljić·Vedad Pasic·Helena Zarin
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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.

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Enes Duvnjaković, Samir Karasuljić, Vedad Pasic, Helena Zarin. 2014-11-16. A uniformly convergent difference scheme on a modified Shishkin mesh for the singular perturbation boundary value problem. https://arxiv.org/abs/1411.4323

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