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Preconditioning for symmetric positive definite systems in balanced fractional diffusion equations

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Abstract

In this paper, we study the finite volume discretization method for balanced fractional diffusion equations where the fractional differential operators are comprised of both Riemann-Liouville and Caputo fractional derivatives. The main advantage of this approach is that new symmetric positive definite Toeplitz-like linear systems can be constructed for solving balanced fractional diffusion equations when diffusion functions are non-constant. It is different from non-symmetric Toeplitz-like linear systems usually obtained by existing numerical methods for fractional diffusion equations. The preconditioned conjugate gradient method with circulant and banded preconditioners can be applied to solve the proposed symmetric positive definite Toeplitz-like linear systems. Numerical examples, for both of one- and two- dimensional cases, are given to demonstrate the good accuracy of the finite volume discretization method and the fast convergence of the preconditioned conjugate gradient method.

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Authors and Affiliations

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

    Zhi-Wei Fang

  2. Shenzhen JL Computational Science and Applied Research Institute, Shenzhen, China

    Xue-Lei Lin

  3. Beijing Computational Science Research Center, Beijing, 100193, China

    Xue-Lei Lin

  4. Department of Mathematics, The University of Hong Kong, Pokfulam, Hong Kong

    Michael K. Ng

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

    Hai-Wei Sun

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

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The research is funded in part by The Science and Technology Development Fund, Macau SAR (file no. 0118/2018/A3), University of Macau (file no. MYRG2018-00015-FST), HKRGC GRF (12306616, 12200317, 12300218 and 12300519), NSAF U930402, and Guangdong Basic and Applied Basic Research Foundation (Grant No. 2019A1515110893).

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Fang, ZW., Lin, XL., Ng, M.K. et al. Preconditioning for symmetric positive definite systems in balanced fractional diffusion equations. Numer. Math. 147, 651–677 (2021). https://doi.org/10.1007/s00211-021-01175-x

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  • DOI: https://doi.org/10.1007/s00211-021-01175-x

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