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A RBF-WENO Finite Difference Scheme for Non-linear Degenerate Parabolic Equations

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Abstract

In this research, a radial basis functions weighted essentially non-oscillatory (RBF-WENO) scheme in the framework of finite difference is developed for solving non-linear degenerate parabolic (NDP) equations which may contain discontinuous solutions. The traditional WENO schemes for solving NDP equations are based on the polynomial interpolation. In this paper, a non-polynomial WENO finite difference scheme in order to enhance the local accuracy is proposed. For a non-polynomial interpolation basis, the infinitely smooth RBFs such as the multi-quadratic are adopted. By optimizing the free parameter in RBFs, the local error generated by the RBFs interpolation can be easily controlled. The formulation of scheme will be described in detail and numerical tests have been prepared to demonstrate the efficiency of the scheme in one and two-dimension. The numerical results demonstrate that the developed non-polynomial WENO schemes enhance the local accuracy and give sharper solution profile than the original WENO based on the polynomial interpolation for solving NDP equations.

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Acknowledgements

The authors are very thankful to the reviewers for carefully reading the paper, their comments and suggestions have improved the quality of the paper.

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Authors and Affiliations

  1. School of Engineering Science, College of Engineering, University of Tehran, Tehran, Iran

    Rooholah Abedian

  2. Department of Applied Mathematics, Faculty of Mathematics and Computer Science, Amirkabir University of Technology, No. 424, Hafez Ave., Tehran, Iran

    Mehdi Dehghan

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  1. Rooholah Abedian
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Correspondence to Rooholah Abedian.

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Abedian, R., Dehghan, M. A RBF-WENO Finite Difference Scheme for Non-linear Degenerate Parabolic Equations. J Sci Comput 93, 60 (2022). https://doi.org/10.1007/s10915-022-02022-3

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  • DOI: https://doi.org/10.1007/s10915-022-02022-3

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