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An efficient numerical technique for two-parameter singularly perturbed problems having discontinuity in convection coefficient and source term

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Abstract

We devise a spline-based numerical technique for a class of two-parameter singularly perturbed problems having discontinuous convection and source terms. The problem is discretized using the Crank–Nicolson formula in the temporal direction, and the trigonometric -spline basis functions are used in the spatial direction. The presence of perturbation parameters and the discontinuous convection/source terms result in the interior and boundary layers in the solution to the problem. Our primary focus is to resolve these layers and develop a uniformly convergent scheme. Initially, the proposed method gives almost first and second-order convergence in the spatial and temporal directions, respectively. Then, to improve the accuracy in the spatial direction, we have used the Richardson extrapolation technique. Two numerical examples are taken to demonstrate the layer phenomenon and confirm the theoretical proofs. It is evident from the tables that the Richardson extrapolation technique increases the accuracy from one to two in the spatial direction.

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Data Availability

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

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Acknowledgements

The authors are grateful to the unknown reviewers for their insightful observations leading to the improvement of the manuscript. The first author thanks UGC, New Delhi, India, for its financial support (Award letter no. 1078/(CSIR-UGC NET JUNE 2019), and the second author thanks DST-SERB, New Delhi, for providing the financial support (Award letter no. CRG/2019/002528) under the CRG scheme.

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Authors and Affiliations

  1. Department of Mathematics, Birla Institute of Technology and Science, Pilani, Rajasthan, 333031, India

    Satpal Singh, Renu Choudhary & Devendra Kumar

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  1. Satpal Singh
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  2. Renu Choudhary
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  3. Devendra Kumar
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Correspondence to Renu Choudhary.

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Communicated by Baisheng Yan.

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Singh, S., Choudhary, R. & Kumar, D. An efficient numerical technique for two-parameter singularly perturbed problems having discontinuity in convection coefficient and source term. Comp. Appl. Math. 42, 62 (2023). https://doi.org/10.1007/s40314-023-02196-y

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  • DOI: https://doi.org/10.1007/s40314-023-02196-y

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