An efficient numerical method for 2D elliptic singularly perturbed systems with different magnitude parameters in the diffusion and the convection terms, part Ⅱ
This work is the continuation of [11], where a two-dimensional elliptic singularly perturbed weakly system, for which small parameters affected both the diffusion and the convection terms, was solved; moreover, all perturbation parameters could have different orders of magnitude, which is the most i...
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AIMS Press
2024-12-01
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| Online Access: | https://www.aimspress.com/article/doi/10.3934/math.20241688 |
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| author | Ram Shiromani Carmelo Clavero |
| author_facet | Ram Shiromani Carmelo Clavero |
| author_sort | Ram Shiromani |
| collection | DOAJ |
| description | This work is the continuation of [11], where a two-dimensional elliptic singularly perturbed weakly system, for which small parameters affected both the diffusion and the convection terms, was solved; moreover, all perturbation parameters could have different orders of magnitude, which is the most interesting and difficult case for this type of problem. It is well known that then, in general, overlapping regular or parabolic boundary layers appear in the solution of the continuous problem. To solve numerically the problem, the classical upwind finite difference scheme, defined on special piecewise uniform Shsihkin meshes, was used, proving its uniform convergence, with respect to all parameters, for four different ratios between them. In this paper, we complete the previous analysis, considering the two cases for these possible ratios, that were not considered in [11]. To see in practice the efficiency of the numerical method, we show the numerical results obtained with our algorithm for a test problem, when the cases analyzed in this work are fixed; from those results, the uniform convergence of the numerical algorithm follows, in agreement with the theoretical results. |
| format | Article |
| id | doaj-art-b82b3873ec664406be28bce5813f6201 |
| institution | Kabale University |
| issn | 2473-6988 |
| language | English |
| publishDate | 2024-12-01 |
| publisher | AIMS Press |
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| series | AIMS Mathematics |
| spelling | doaj-art-b82b3873ec664406be28bce5813f62012025-01-23T07:53:25ZengAIMS PressAIMS Mathematics2473-69882024-12-01912355703559810.3934/math.20241688An efficient numerical method for 2D elliptic singularly perturbed systems with different magnitude parameters in the diffusion and the convection terms, part ⅡRam Shiromani0Carmelo Clavero1Department of Mathematics and Statistics, Indian Institute of Technology Kanpur, Uttar Pradesh, IndiaDepartment of Applied Mathematics, IUMA, University of Zaragoza, Zaragoza, SpainThis work is the continuation of [11], where a two-dimensional elliptic singularly perturbed weakly system, for which small parameters affected both the diffusion and the convection terms, was solved; moreover, all perturbation parameters could have different orders of magnitude, which is the most interesting and difficult case for this type of problem. It is well known that then, in general, overlapping regular or parabolic boundary layers appear in the solution of the continuous problem. To solve numerically the problem, the classical upwind finite difference scheme, defined on special piecewise uniform Shsihkin meshes, was used, proving its uniform convergence, with respect to all parameters, for four different ratios between them. In this paper, we complete the previous analysis, considering the two cases for these possible ratios, that were not considered in [11]. To see in practice the efficiency of the numerical method, we show the numerical results obtained with our algorithm for a test problem, when the cases analyzed in this work are fixed; from those results, the uniform convergence of the numerical algorithm follows, in agreement with the theoretical results.https://www.aimspress.com/article/doi/10.3934/math.202416882d elliptic coupled systemssingularly perturbed problemsupwind schemeshishkin meshesuniform convergence |
| spellingShingle | Ram Shiromani Carmelo Clavero An efficient numerical method for 2D elliptic singularly perturbed systems with different magnitude parameters in the diffusion and the convection terms, part Ⅱ AIMS Mathematics 2d elliptic coupled systems singularly perturbed problems upwind scheme shishkin meshes uniform convergence |
| title | An efficient numerical method for 2D elliptic singularly perturbed systems with different magnitude parameters in the diffusion and the convection terms, part Ⅱ |
| title_full | An efficient numerical method for 2D elliptic singularly perturbed systems with different magnitude parameters in the diffusion and the convection terms, part Ⅱ |
| title_fullStr | An efficient numerical method for 2D elliptic singularly perturbed systems with different magnitude parameters in the diffusion and the convection terms, part Ⅱ |
| title_full_unstemmed | An efficient numerical method for 2D elliptic singularly perturbed systems with different magnitude parameters in the diffusion and the convection terms, part Ⅱ |
| title_short | An efficient numerical method for 2D elliptic singularly perturbed systems with different magnitude parameters in the diffusion and the convection terms, part Ⅱ |
| title_sort | efficient numerical method for 2d elliptic singularly perturbed systems with different magnitude parameters in the diffusion and the convection terms part ii |
| topic | 2d elliptic coupled systems singularly perturbed problems upwind scheme shishkin meshes uniform convergence |
| url | https://www.aimspress.com/article/doi/10.3934/math.20241688 |
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