Asian Journal of Pure and Applied Mathematics

Richardson Extrapolation Technique for Singularly Perturbed Parabolic Convection-diffusion Problems with a Discontinuous Initial Condition

Desta Sodano Sheiso *

Department of Mathematics, Indian Institute of Technology Guwahati, Guwahati - 781 039, India.

*Author to whom correspondence should be addressed.

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Abstract

This article presents the Richardson extrapolation techniques for solving singularly perturbed parabolicconvection-diffusion problems with discontinuous initial conditions (DIC). The scheme uses backward-Eulerfor temporal derivatives on a uniform mesh and classical upwind finite difference method (FDM) for spatialderivatives on a piecewise-uniform (Shishkin) mesh. This scheme provides almost a first-order convergence solution in both space and time variables. The method employs an upwind finite difference operator on a piecewise- uniform mesh to approximate the gap between the analytic function and the parabolic issue solution. The numerical solution’s accuracy is improved by using Richardson extrapolation techniques, which raises it from O(N⁻¹ ln N+ ∆t) to O(N⁻² ln² N+ ∆t²) in the discrete maximum norm, where N is the number of spatial mesh intervals, and ∆t is the size of the temporal step size. Parameter uniformerror estimates, stability results, and bounds for the truncation errors are all addressed. Finally, numerical experiments are presented to validate our theoretical results.      

Keywords: Singularly perturbed parabolic problems, Upwind scheme, discontinuous initial condition, interior layer, Richardson extrapolation technique, Piecewise-uniform Shishkin mesh

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