Non-standard finite-difference scheme for singularly perturbed parabolic convection-diffusion problem with boundary turning points

Document Type : Research Article

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Department of Mathematics, College of Science, Bahir Dar University, Bahir Dar, Ethiopia

Abstract

This paper presents a numerical method for solving singularly perturbed parabolic convection-diffusion problems with boundary turning points. As the perturbation parameter ε approaches zero, the solution shows rapid changes on the left side of the spatial domain, forming a small boundary layer. The classical finite difference methods on uniform meshes fail to capture these oscillations without using a large number of mesh points. To solve this, we use the implicit Euler method for time discretization and a non-standard finite difference method in space. The method satisfies stability, the discrete minimum principle, and ε-uniform convergence. Error estimates show that the proposed method is first-order convergence in time and space. The order of convergence is improved by applying the Richardson extrapolation method. Two model examples are provided to show the scheme's applicability. It demonstrates that the numerical results are in agreement with the theoretical findings.

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Journal of Mathematical Modeling
Volume 13, Issue 1
March 2025
Pages 85-103

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