A robust higher-order fitted mesh numerical method for solving singularly perturbed parabolic reaction-diffusion problems
In this study, a robust higher-order numerical method for solving singularly perturbed parabolic reaction-diffusion problems is presented. The Crank-Nicolson method is applied to discretize the time derivative on a uniform mesh. On a Shishkin mesh, the space derivative is discretized using a hybrid...
| Main Authors: | , , |
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| Format: | Article |
| Language: | English |
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Elsevier
2023-11-01
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| Series: | Results in Applied Mathematics |
| Subjects: | |
| Online Access: | http://www.sciencedirect.com/science/article/pii/S2590037423000511 |
| _version_ | 1797405357160529920 |
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| author | Gemechis File Duressa Fasika Wondimu Gelu Guta Demisu Kebede |
| author_facet | Gemechis File Duressa Fasika Wondimu Gelu Guta Demisu Kebede |
| author_sort | Gemechis File Duressa |
| collection | DOAJ |
| description | In this study, a robust higher-order numerical method for solving singularly perturbed parabolic reaction-diffusion problems is presented. The Crank-Nicolson method is applied to discretize the time derivative on a uniform mesh. On a Shishkin mesh, the space derivative is discretized using a hybrid numerical method that combines the cubic spline in tension method for the boundary layer regions with the central difference method for the outer region. Theoretically, we proved that the proposed hybrid numerical method is second-order in the outer region and fourth-order in the boundary layer regions in the space direction. As a result of this, the proposed method produces an almost second-order rate of convergence in the time domain and a higher-order rate of convergence in the space domain. The newly developed method is numerically demonstrated to be uniformly convergent at higher-order, independent of the perturbation parameter. Three numerical examples are computed to support the theoretical results. |
| first_indexed | 2024-03-09T03:08:48Z |
| format | Article |
| id | doaj.art-2cb3bd34fe8b4e2ba871bc4629ec9973 |
| institution | Directory Open Access Journal |
| issn | 2590-0374 |
| language | English |
| last_indexed | 2024-03-09T03:08:48Z |
| publishDate | 2023-11-01 |
| publisher | Elsevier |
| record_format | Article |
| series | Results in Applied Mathematics |
| spelling | doaj.art-2cb3bd34fe8b4e2ba871bc4629ec99732023-12-04T05:24:01ZengElsevierResults in Applied Mathematics2590-03742023-11-0120100405A robust higher-order fitted mesh numerical method for solving singularly perturbed parabolic reaction-diffusion problemsGemechis File Duressa0Fasika Wondimu Gelu1Guta Demisu Kebede2Department of Mathematics, Jimma University, Jimma, EthiopiaDepartment of Mathematics, Dilla University, Dilla, Ethiopia; Corresponding author.Department of Mathematics, Dilla University, Dilla, EthiopiaIn this study, a robust higher-order numerical method for solving singularly perturbed parabolic reaction-diffusion problems is presented. The Crank-Nicolson method is applied to discretize the time derivative on a uniform mesh. On a Shishkin mesh, the space derivative is discretized using a hybrid numerical method that combines the cubic spline in tension method for the boundary layer regions with the central difference method for the outer region. Theoretically, we proved that the proposed hybrid numerical method is second-order in the outer region and fourth-order in the boundary layer regions in the space direction. As a result of this, the proposed method produces an almost second-order rate of convergence in the time domain and a higher-order rate of convergence in the space domain. The newly developed method is numerically demonstrated to be uniformly convergent at higher-order, independent of the perturbation parameter. Three numerical examples are computed to support the theoretical results.http://www.sciencedirect.com/science/article/pii/S2590037423000511Singularly perturbedHigher-orderShishkin meshTension method |
| spellingShingle | Gemechis File Duressa Fasika Wondimu Gelu Guta Demisu Kebede A robust higher-order fitted mesh numerical method for solving singularly perturbed parabolic reaction-diffusion problems Results in Applied Mathematics Singularly perturbed Higher-order Shishkin mesh Tension method |
| title | A robust higher-order fitted mesh numerical method for solving singularly perturbed parabolic reaction-diffusion problems |
| title_full | A robust higher-order fitted mesh numerical method for solving singularly perturbed parabolic reaction-diffusion problems |
| title_fullStr | A robust higher-order fitted mesh numerical method for solving singularly perturbed parabolic reaction-diffusion problems |
| title_full_unstemmed | A robust higher-order fitted mesh numerical method for solving singularly perturbed parabolic reaction-diffusion problems |
| title_short | A robust higher-order fitted mesh numerical method for solving singularly perturbed parabolic reaction-diffusion problems |
| title_sort | robust higher order fitted mesh numerical method for solving singularly perturbed parabolic reaction diffusion problems |
| topic | Singularly perturbed Higher-order Shishkin mesh Tension method |
| url | http://www.sciencedirect.com/science/article/pii/S2590037423000511 |
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