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 back...

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Published in:Journal of numerical analysis and approximation theory Vol. 54; no. 1; pp. 117 - 139
Main Authors: Sah, Kishun Kumar, Gowrisankar, Subramaniam
Format: Journal Article
Language:English
Published: Publishing House of the Romanian Academy 30.06.2025
Subjects:
Boundary layers
Finite difference methods
Modified graded mesh
Parabolic reaction-diffusion problems
Singular perturbation problem
Uniform convergence
ISSN:2457-6794, 2501-059X
Online Access:Get full text
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Abstract 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.
AbstractList 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.
Author Gowrisankar, Subramaniam
Sah, Kishun Kumar
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Snippet This paper addresses the numerical approximations of solutions for one dimensional parabolic singularly perturbed problems of reaction-diffusion type. The...
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StartPage 117
SubjectTerms Boundary layers
Finite difference methods
Modified graded mesh
Parabolic reaction-diffusion problems
Singular perturbation problem
Uniform convergence
Title A numerical approach for singularly perturbed parabolic reaction-diffusion problem on a modified graded mesh
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