A Uniformly Convergent Difference Scheme for the Singularly Perturbed Periodic Problem

Year 2025, Volume: 14 Issue: 3, 1348 - 1361, 30.09.2025

Erkan Cimen

https://doi.org/10.17798/bitlisfen.1600715

Abstract

In this article, a novel numerical scheme is suggested to solve periodical boundary value problem for linear first order singularly perturbed equation. This scheme is constructed by the finite difference method on a special non-uniform mesh (Shishkin mesh) using quadrature rules with the remaining terms in integral form. It is proven that the scheme achieves almost first-order convergence on the discrete maximum norm. Finally, two test problems are considered to demonstrate the accuracy and performance of the method.

Keywords

Singular perturbation , Periodic problem , Finite difference method , Uniform convergence

Ethical Statement

The study is complied with research and publication ethics.

Singüler Pertürbe Olmuş Periyodik Problem için Bir Düzgün Yakınsak Fark Şeması

Abstract

Bu çalışmada, birinci mertebeden singüler pertürbe olmuş periyodik sınır değeri problemini çözmek için yeni bir sayısal şema önerilmiştir. Bu şema, kalan terimleri integral formda olan quadratür kuralları kullanılarak düzgün olmayan özel bir şebeke (Shishkin şebeke) üzerinde sonlu farklar yöntemi ile oluşturulmuştur. Şemanın ayrık maksimum normda neredeyse birinci dereceden yakınsamaya ulaştığı kanıtlanmıştır. Yöntemin doğruluğunu ve performansını göstermek için sayısal bir örnek ele alınmıştır.

Keywords

Singüler perturbasyon , Periyodik problem , Sonlu fark metodu , Düzgün yakınsaklık

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