A fitted parameter convergent finite difference scheme for two-parameter singularly perturbed parabolic differential equations
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Date
2025
Journal Title
Journal ISSN
Volume Title
Publisher
Tamkang University
Abstract
The objective of this paper is to develop a numerical scheme that is uniform in its parameters for a specific type of time-dependent parabolic problem with two perturbation parameters. The existence of these two parameters in the terms with the highest-order derivatives results in the formation of boundary layer(s) in the solution of such problems. Solving these model problems using classical methods does not yield satisfactory results due to the layer behavior. Therefore, nonstandard finite difference schemes have been developed as a means to obtain numerical solutions for these problems. To develop the scheme, we employ the Crank-Nicolson discretization on a uniform time mesh and apply a fitted operator method with a uniform spatial mesh. We have established the stability and convergence of the proposed scheme. The proposed scheme exhibits uniform convergence of second order in the temporal direction and first order in the spatial direction. However, temporal mesh refinements is employed to enhance the order to two in both directions.. Model examples are provided to validate the practicality of the proposed numerical scheme.
Description
Keywords
Fitted operator scheme, Nonstandard finite difference method, Singularly perturbation, Uniform convergence, Numerical scheme.
Citation
Ali, M., Munyakazi, J.B. and Dinka, T.G., 2025. A fitted parameter convergent finite difference scheme for two-parameter singularly perturbed parabolic differential equations. Tamkang Journal of Mathematics, 56(1), pp.73-91.