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Banded Preconditioners for Riesz Space Fractional Diffusion Equations

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Abstract

In this paper, we consider numerical methods for Toeplitz-like linear systems arising from the one- and two-dimensional Riesz space fractional diffusion equations. We apply the Crank–Nicolson technique to discretize the temporal derivative and apply certain difference operator to discretize the space fractional derivatives. For the one-dimensional problem, the corresponding coefficient matrix is the sum of an identity matrix and a product of a diagonal matrix and a symmetric Toeplitz matrix. We transform the linear systems to symmetric linear systems and introduce symmetric banded preconditioners. We prove that under mild assumptions, the eigenvalues of the preconditioned matrix are bounded above and below by positive constants. In particular, the lower bound of the eigenvalues is equal to 1 when the banded preconditioner with diagonal compensation is applied. The preconditioned conjugate gradient method is applied to solve relevant linear systems. Numerical results are presented to verify the theoretical results about the preconditioned matrices and to illustrate the efficiency of the proposed preconditioners.

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Acknowledgements

We thank the referees for providing valuable comments and suggestions, which are very helpful for us to improve our paper.

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Authors and Affiliations

  1. Department of Mathematics, Shantou University, Shantou, 515063, Guangdong, China

    Zi-Hang She, Cheng-Xue Lao, Hong Yang & Fu-Rong Lin

  2. Department of Mathematics, Hanshan Normal University, Chaozhou, 521041, Guangdong, China

    Zi-Hang She

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Correspondence to Fu-Rong Lin.

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This research was supported by National Natural Science Foundation of China (11771265), key research projects of general universities in Guangdong Province (2019KZDXM034), basic research and applied basic research projects in Guangdong Province (Projects of Guangdong-Hong Kong-Macao Center for Applied Mathematics) (2020B1515310018), Natural Science Foundation of Guangdong Provincial Department of Education (2018KQNCX156, 2020KZDZX1147).

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She, ZH., Lao, CX., Yang, H. et al. Banded Preconditioners for Riesz Space Fractional Diffusion Equations. J Sci Comput 86, 31 (2021). https://doi.org/10.1007/s10915-020-01398-4

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  • DOI: https://doi.org/10.1007/s10915-020-01398-4

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