A numerical approach for singularly perturbed parabolic reaction-diffusion problem on a modified graded mesh
This paper addresses the numerical approximations of solutions for one dimensional parabolic singularly perturbed problems of reaction-diffusion type. The solution of this class of problems exhibit boundary layers on both sides of the domain. The proposed numerical method involves combining the bac...
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| Format: | Article |
| Language: | English |
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Publishing House of the Romanian Academy
2025-06-01
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| Series: | Journal of Numerical Analysis and Approximation Theory |
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| Online Access: | https://ictp.acad.ro/jnaat/journal/article/view/1513 |
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| author | Kishun Kumar Sah Subramaniam Gowrisankar |
| author_facet | Kishun Kumar Sah Subramaniam Gowrisankar |
| author_sort | Kishun Kumar Sah |
| collection | DOAJ |
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This paper addresses the numerical approximations of solutions for one dimensional parabolic singularly perturbed problems of reaction-diffusion type. The solution of this class of problems exhibit boundary layers on both sides of the domain. The proposed numerical method involves combining the backward Euler method on a uniform mesh for temporal discretization and an upwind finite difference scheme for spatial discretization on a modified graded mesh. The numerical solutions presented here are calculated using a modified graded mesh and the error bounds are rigorously assessed within the discrete maximum norm. The primary focus of this study is to underscore the crucial importance of utilizing a modified graded mesh to enhance the order of convergence in numerical solutions. The method demonstrates uniform convergence, with first-order accuracy in time and nearly second-order accuracy in space concerning the perturbation parameter. Theoretical findings are supported by numerical results presented in the paper.
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| format | Article |
| id | doaj-art-d937d4fa070f4430a279b97994989e33 |
| institution | Kabale University |
| issn | 2457-6794 2501-059X |
| language | English |
| publishDate | 2025-06-01 |
| publisher | Publishing House of the Romanian Academy |
| record_format | Article |
| series | Journal of Numerical Analysis and Approximation Theory |
| spelling | doaj-art-d937d4fa070f4430a279b97994989e332025-08-22T15:39:20ZengPublishing House of the Romanian AcademyJournal of Numerical Analysis and Approximation Theory2457-67942501-059X2025-06-0154110.33993/jnaat541-1513A numerical approach for singularly perturbed parabolic reaction-diffusion problem on a modified graded meshKishun Kumar Sah0https://orcid.org/0009-0005-2887-664XSubramaniam Gowrisankar1Department of Mathematics, National Institute of Technology Patna, IndiaDepartment of Mathematics, National Institute of Technology Patna, India This paper addresses the numerical approximations of solutions for one dimensional parabolic singularly perturbed problems of reaction-diffusion type. The solution of this class of problems exhibit boundary layers on both sides of the domain. The proposed numerical method involves combining the backward Euler method on a uniform mesh for temporal discretization and an upwind finite difference scheme for spatial discretization on a modified graded mesh. The numerical solutions presented here are calculated using a modified graded mesh and the error bounds are rigorously assessed within the discrete maximum norm. The primary focus of this study is to underscore the crucial importance of utilizing a modified graded mesh to enhance the order of convergence in numerical solutions. The method demonstrates uniform convergence, with first-order accuracy in time and nearly second-order accuracy in space concerning the perturbation parameter. Theoretical findings are supported by numerical results presented in the paper. https://ictp.acad.ro/jnaat/journal/article/view/1513Singular perturbation problem,Parabolic reaction-diffusion problems,Finite difference methods,Modified graded mesh,Boundary layers,Uniform convergence |
| spellingShingle | Kishun Kumar Sah Subramaniam Gowrisankar A numerical approach for singularly perturbed parabolic reaction-diffusion problem on a modified graded mesh Journal of Numerical Analysis and Approximation Theory Singular perturbation problem, Parabolic reaction-diffusion problems, Finite difference methods, Modified graded mesh, Boundary layers, Uniform convergence |
| title | A numerical approach for singularly perturbed parabolic reaction-diffusion problem on a modified graded mesh |
| title_full | A numerical approach for singularly perturbed parabolic reaction-diffusion problem on a modified graded mesh |
| title_fullStr | A numerical approach for singularly perturbed parabolic reaction-diffusion problem on a modified graded mesh |
| title_full_unstemmed | A numerical approach for singularly perturbed parabolic reaction-diffusion problem on a modified graded mesh |
| title_short | A numerical approach for singularly perturbed parabolic reaction-diffusion problem on a modified graded mesh |
| title_sort | numerical approach for singularly perturbed parabolic reaction diffusion problem on a modified graded mesh |
| topic | Singular perturbation problem, Parabolic reaction-diffusion problems, Finite difference methods, Modified graded mesh, Boundary layers, Uniform convergence |
| url | https://ictp.acad.ro/jnaat/journal/article/view/1513 |
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