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An efficient computational technique for time dependent semilinear parabolic problems involving two small parameters

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Abstract

The present work discusses a robust numerical scheme for singularly perturbed semilinear parabolic problems with two small parameters. Unlike many works in the literature, the manuscript contemplates a unified solution for these model problems whose solutions can have multiscale behavior depending upon the ratio of both perturbation parameters. Further, the robustness of the proposed scheme is tested over the problems having a large time lag as well. At first, the semilinearity is treated with the help of Newton’s linearization technique. Then, a first-order global accurate scheme is employed combining the implicit Euler scheme in time direction over a uniform mesh and the upwind scheme in spatial direction over two-layer rectifying meshes namely, the Shishkin mesh and the Bakhvalov-Shishkin mesh. The use of Bakhvalov-Shishkin mesh overrides the drawback of order deduction because of the logarithmic term in Shishkin mesh. Thomas algorithm is employed for calculation which computationally is more robust than the usual matrix inversion methods. The stability and consistency are discussed thoroughly. The truncation error and barrier function approach is used for the evaluation of error bounds. The analytical claims are proved through numerical simulations.

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The data sets generated during and/or analyzed during the current study are available from the corresponding author on reasonable request.

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Acknowledgements

The first author Ms. S. Priyadarshana conveys her profound gratitude to the Department of Science and Technology, Govt. of India for providing INSPIRE fellowship (IF 180938).

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  1. Department of Mathematics, National Institute of Technology, Rourkela, Odisha, 769008, India

    S. Priyadarshana & J. Mohapatra

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Priyadarshana, S., Mohapatra, J. An efficient computational technique for time dependent semilinear parabolic problems involving two small parameters. J. Appl. Math. Comput. 69, 3721–3754 (2023). https://doi.org/10.1007/s12190-023-01900-9

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