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Circulant preconditioners for a kind of spatial fractional diffusion equations

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Abstract

In this paper, circulant preconditioners are studied for discretized matrices arising from finite difference schemes for a kind of spatial fractional diffusion equations. The fractional differential operator is comprised of left-sided and right-sided derivatives with order in \((\frac {1}{2},1)\). The resulting discretized matrices preserve Toeplitz-like structure and hence their matrix-vector multiplications can be computed efficiently by the fast Fourier transform. Theoretically, the spectra of the circulant preconditioned matrices are shown to be clustered around 1 under some conditions. Numerical experiments are presented to demonstrate that the preconditioning technique is very efficient.

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Authors and Affiliations

  1. School of Mathematics and Big Data, Foshan University, Foshan, Guangdong, China

    Zhi-Wei Fang

  2. Department of Mathematics, Hong Kong Baptist University, Kowloon Tong, Hong Kong

    Michael K. Ng

  3. Department of Mathematics, University of Macau, Taipa, Macao

    Zhi-Wei Fang & Hai-Wei Sun

Authors
  1. Zhi-Wei Fang
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  2. Michael K. Ng
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  3. Hai-Wei Sun
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Correspondence to Hai-Wei Sun.

Additional information

The research is supported in part by research grants 0118/2018/A3 from FDCT of Macao, MYRG2016-00063-FST from University of Macau, HKRGC GRF 12302715, 12306616, 12200317 and 12300218, and HKBU RC-ICRS/16-17/03.

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Fang, ZW., Ng, M.K. & Sun, HW. Circulant preconditioners for a kind of spatial fractional diffusion equations. Numer Algor 82, 729–747 (2019). https://doi.org/10.1007/s11075-018-0623-y

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  • DOI: https://doi.org/10.1007/s11075-018-0623-y

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