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Uniformly convergent additive schemes for 2d singularly perturbed parabolic systems of reaction-diffusion type

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Abstract

In this work, we consider parabolic 2D singularly perturbed systems of reaction-diffusion type on a rectangle, in the simplest case that the diffusion parameter is the same for all equations of the system. The solution is approximated on a Shishkin mesh with two splitting or additive methods in time and standard central differences in space. It is proved that they are first-order in time and almost second-order in space uniformly convergent schemes. The additive schemes decouple the components of the vector solution at each time level of the discretization which makes the computation more efficient. Moreover, a multigrid algorithm is used to solve the resulting linear systems. Numerical results for some test problems are showed, which illustrate the theoretical results and the efficiency of the splitting and multigrid techniques.

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Acknowledgements

The authors thank the referees for their valuable suggestions which have helped to improve the presentation of this paper.

Funding

This research was partially supported by the Instituto Universitario de Investigación en Matemáticas y Aplicaciones (IUMA), the projects MTM2017-83490-P, and MTM2016-75139-R and the Diputación General de Aragón.

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Authors and Affiliations

  1. Department of Applied Mathematics and IUMA, University of Zaragoza, Zaragoza, Spain

    C. Clavero & J. L. Gracia

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  1. C. Clavero
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  2. J. L. Gracia
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Correspondence to C. Clavero.

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Clavero, C., Gracia, J.L. Uniformly convergent additive schemes for 2d singularly perturbed parabolic systems of reaction-diffusion type. Numer Algor 80, 1097–1120 (2019). https://doi.org/10.1007/s11075-018-0518-y

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