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Additive schemes based domain decomposition algorithm for solving singularly perturbed parabolic reaction-diffusion systems

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Abstract

In this paper, we examine a 1D parabolic coupled system of singularly perturbed reaction-diffusion problems in which perturbation parameters can be of distinct magnitude. To solve this system numerically we develop additive (or splitting) schemes based domain decomposition algorithm of Schwarz waveform relaxation type. On each subdomain we consider two additive schemes on a uniform mesh in time and the standard central difference scheme on a uniform mesh in space. We provide convergence analysis of the algorithm using some auxiliary problems and the algorithm is shown to be uniformly convergent. The additive schemes make the computation more efficient as they decouple the components of the approximate solution at each time level. Numerical results for two test problems are given in support of the theoretical convergence result and as well as to illustrate the efficiency of the additive schemes.

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Acknowledgements

This research was supported by the Science and Engineering Research Board (SERB) under the Project No. MTR/2017/001036. The first author was supported by Council of Scientific and Industrial Research (CSIR) under the grant having file no. 09/1217(0035)/2018-EMR-I. The authors gratefully acknowledge the valuable comments and suggestions from the anonymous referees.

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Authors and Affiliations

  1. Department of Mathematical Sciences, Indian Institute of Technology (BHU) Varanasi, Varanasi, Uttar Pradesh, India

    Aakansha, Joginder Singh & Sunil Kumar

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  1. Aakansha
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  2. Joginder Singh
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  3. Sunil Kumar
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Correspondence to Sunil Kumar.

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Communicated by José R Fernández.

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Aakansha, Singh, J. & Kumar, S. Additive schemes based domain decomposition algorithm for solving singularly perturbed parabolic reaction-diffusion systems. Comp. Appl. Math. 40, 82 (2021). https://doi.org/10.1007/s40314-021-01457-y

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  • DOI: https://doi.org/10.1007/s40314-021-01457-y

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