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An Efficient Second-Order Convergent Scheme for One-Side Space Fractional Diffusion Equations with Variable Coefficients

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Abstract

In this paper, a second-order finite-difference scheme is investigated for time-dependent space fractional diffusion equations with variable coefficients. In the presented scheme, the Crank–Nicolson temporal discretization and a second-order weighted-and-shifted Grünwald–Letnikov spatial discretization are employed. Theoretically, the unconditional stability and the second-order convergence in time and space of the proposed scheme are established under some conditions on the variable coefficients. Moreover, a Toeplitz preconditioner is proposed for linear systems arising from the proposed scheme. The condition number of the preconditioned matrix is proven to be bounded by a constant independent of the discretization step-sizes, so that the Krylov subspace solver for the preconditioned linear systems converges linearly. Numerical results are reported to show the convergence rate and the efficiency of the proposed scheme.

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Acknowledgements

The authors would like to thank the editor and referees for valuable comments and suggestions, which helped to improve the quality of the manuscript.

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Authors and Affiliations

  1. Department of Mathematics, Hong Kong Baptist University, Hong Kong, China

    Xue-lei Lin

  2. School of Economic Mathematics, Southwestern University of Finance and Economics, Chengdu, 611130, China

    Pin Lyu

  3. Department of Mathematics, The University of Hong Kong, Hong Kong, China

    Michael K. Ng

  4. Department of Mathematics, University of Macau, Macao, China

    Hai-Wei Sun & Seakweng Vong

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  1. Xue-lei Lin
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  2. Pin Lyu
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  3. Michael K. Ng
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  4. Hai-Wei Sun
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  5. Seakweng Vong
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Corresponding author

Correspondence to Pin Lyu.

Additional information

This research was supported by research Grants, 12306616, 12200317, 12300519, 12300218 from HKRGC GRF, 11801479 from NSFC, MYRG2018-00015-FST from University of Macau, and 0118/2018/A3 from FDCT of Macao, Macao Science and Technology Development Fund 0005/2019/A, 050/2017/A, and the Grant MYRG2017-00098-FST and MYRG2018-00047-FST from University of Macau.

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Lin, Xl., Lyu, P., Ng, M.K. et al. An Efficient Second-Order Convergent Scheme for One-Side Space Fractional Diffusion Equations with Variable Coefficients. Commun. Appl. Math. Comput. 2, 215–239 (2020). https://doi.org/10.1007/s42967-019-00050-9

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  • DOI: https://doi.org/10.1007/s42967-019-00050-9

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