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A uniformly convergent nonstandard finite difference scheme for a system of convection–diffusion equations

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Abstract

We propose a nonstandard finite difference method to integrate coupled systems of singularly perturbed convection–diffusion equations. We prove that the underlying discrete operator satisfies a stability property in the maximum norm. This fact is used to prove that the proposed method converges uniformly with respect to the singular perturbation parameters and the convergence rate is linear with respect to the step size. Numerical investigations are presented to confirm our theoretical findings. We also show that the proposed approach compares well with existing methods.

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Acknowledgments

This work was completed while the author was a visiting scholar on the University of Michigan African Presidential Scholars (UMAPS) program. He wishes to thank Professor Robert Krasny for reading this work and suggesting improvements. He is also indebted to Professor Kailash C. Patidar for introducing him to the field of nonstandard finite difference methods.

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Authors and Affiliations

  1. Department of Mathematics and Applied Mathematics, University of the Western Cape, Private Bag X17, Bellville , 7535, South Africa

    Justin B. Munyakazi

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  1. Justin B. Munyakazi
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Correspondence to Justin B. Munyakazi.

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Communicated by Raphaèle Herbin.

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Munyakazi, J.B. A uniformly convergent nonstandard finite difference scheme for a system of convection–diffusion equations. Comp. Appl. Math. 34, 1153–1165 (2015). https://doi.org/10.1007/s40314-014-0171-6

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  • DOI: https://doi.org/10.1007/s40314-014-0171-6

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