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Efficient parameter-robust numerical methods for singularly perturbed semilinear parabolic PDEs of convection-diffusion type

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Abstract

This paper is concerned with a class of singularly perturbed semilinear parabolic convection-diffusion initial-boundary-value problems exhibiting a boundary layer. This type of model problem often appears in modeling various physical phenomena, particularly, in mathematical biology; and thus, it requires effective numerical techniques for analyzing them computationally. For this purpose, we approximate the considered nonlinear problem by developing two efficient fitted mesh methods followed by the extrapolation technique. The first one is the fully implicit fitted mesh method which utilizes the implicit-Euler method for the temporal discretization; and the other one is the implicit-explicit (IMEX) fitted mesh method which utilizes the IMEX-Euler method for the temporal discretization. The spatial discretization for both the numerical methods is based on a new hybrid finite difference scheme. To accomplish this, the spatial domain is discretized by an appropriate layer-adapted mesh and the time domain by an equidistant mesh. At first, we analyze stability and study the asymptotic behavior of the analytical solution of the governing nonlinear problem. Then, we perform stability analysis and establish the parameter-uniform convergence of both the newly proposed methods in the discrete supremum norm. Thereafter, we analyze the Richardson extrapolation technique solely for the time variable to improve the order of convergence in the temporal direction. Hereby, we provide a comparative error analysis to achieve parameter-robust higher-order numerical approximations (concerning both space and time) for the considered nonlinear problem utilizing two new algorithms on a nonuniform grid. The theoretical outcomes are finally supported by the extensive numerical experiments, which also include comparison of the proposed numerical methods along with the fully-implicit upwind method in terms of the order of accuracy and the computational cost.

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Acknowledgements

The authors would like to express their sincere thanks to the anonymous reviewers for their constructive comments and valuable suggestions. The first author wishes to thank Indian Institute of Space Science and Technology (IIST), for the financial support during his Ph.D.

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Authors and Affiliations

  1. Department of Mathematics, SRM University AP, Amaravati, Andhra Pradesh, India

    Narendra Singh Yadav

  2. Department of Mathematics, Indian Institute of Space Science and Technology (IIST), Thiruvananthapuram, 695547, Kerala, India

    Kaushik Mukherjee

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  1. Narendra Singh Yadav
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Yadav, N.S., Mukherjee, K. Efficient parameter-robust numerical methods for singularly perturbed semilinear parabolic PDEs of convection-diffusion type. Numer Algor 96, 925–973 (2024). https://doi.org/10.1007/s11075-023-01670-2

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