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A new parameter-convergent nonstandard finite difference method for two-parameter singularly perturbed problems

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Titel: A new parameter-convergent nonstandard finite difference method for two-parameter singularly perturbed problems
Autoren: Mekashaw Ali Mohye, Justin B. Munyakazi, Tekle Gemechu Dinka, Yusuf Hussen Haji, Abe Nura Ware, Jemal Muhammed Ahmed
Quelle: Discover Applied Sciences, Vol 7, Iss 11, Pp 1-19 (2025)
Verlagsinformationen: Springer, 2025.
Publikationsjahr: 2025
Bestand: LCC:Science (General)
Schlagwörter: Singular perturbation problems, Uniform convergence, Shishkin mesh, Boundary layers, Fitted mesh, Science (General), Q1-390
Beschreibung: Abstract This article focuses on the numerical solution of a time-dependent parabolic problem that exhibits singular perturbations and involves two perturbation parameters. To address this problem, a fitted mesh finite difference method is developed. In numerical discretization, the implicit Crank-Nicolson technique is employed to discretize the time derivative using a uniform mesh. As for the spatial derivative, a hybrid finite difference scheme known as the adaptive fitted mesh of the Shishkin type is utilized. The study also includes a discussion on a priori bounds for the continuous solution and its derivatives. The proposed method is proven to be uniformly convergent of order two in both time and space. Theoretical analysis and simulations on various test examples confirm the scheme’s accuracy and convergence properties.
Publikationsart: article
Dateibeschreibung: electronic resource
Sprache: English
ISSN: 3004-9261
Relation: https://doaj.org/toc/3004-9261
DOI: 10.1007/s42452-025-07721-8
Zugangs-URL: https://doaj.org/article/5ed02e1acddb4668b4556226050e0b76
Dokumentencode: edsdoj.5ed02e1acddb4668b4556226050e0b76
Datenbank: Directory of Open Access Journals
Beschreibung
Abstract:Abstract This article focuses on the numerical solution of a time-dependent parabolic problem that exhibits singular perturbations and involves two perturbation parameters. To address this problem, a fitted mesh finite difference method is developed. In numerical discretization, the implicit Crank-Nicolson technique is employed to discretize the time derivative using a uniform mesh. As for the spatial derivative, a hybrid finite difference scheme known as the adaptive fitted mesh of the Shishkin type is utilized. The study also includes a discussion on a priori bounds for the continuous solution and its derivatives. The proposed method is proven to be uniformly convergent of order two in both time and space. Theoretical analysis and simulations on various test examples confirm the scheme’s accuracy and convergence properties.
ISSN:30049261
DOI:10.1007/s42452-025-07721-8