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A τ-preconditioner for a non-symmetric linear system arising from multi-dimensional Riemann-Liouville fractional diffusion equation

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Abstract

In this paper, we study a τ-preconditioner for non-symmetric linear system arising from a steady-state multi-dimensional Riemann-Liouville (RL) fractional diffusion equation. The generalized minimal residual (GMRES) method is applied to solve the preconditioned linear system. Theoretically, we show that the GMRES solver for the preconditioned linear system has a convergence rate independent of discretization stepsizes. To the best of our knowledge, this is the first iterative solver with stepsize-independent convergence rate for the non-symmetric linear system. The proposed τ-preconditioner is diagonalizable by the sine transform matrix, thanks to which the matrix-vector multiplication in each iteration step can be fast implemented by the fast sine transform (FST). Hence, the total operation cost of the proposed solver for the non-symmetric problem is linearithmic. Numerical results are reported to show the efficiency of the proposed preconditioner.

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Data availability statement

The data of codes involved in this paper are available from the corresponding author on reasonable request.

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Funding

This research is supported by RGC GRF 12300519, 17201020, and 17300021.

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Authors and Affiliations

  1. Shenzhen JL Computational Science and Applied Research Institute, Shenzhen, People’s Republic of China

    Xue-lei Lin

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

    Xue-lei Lin

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

    Xin Huang & Hai-Wei Sun

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

    Michael K. Ng

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  1. Xue-lei Lin
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  2. Xin Huang
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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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Lin, Xl., Huang, X., Ng, M.K. et al. A τ-preconditioner for a non-symmetric linear system arising from multi-dimensional Riemann-Liouville fractional diffusion equation. Numer Algor 92, 795–813 (2023). https://doi.org/10.1007/s11075-022-01342-7

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  • DOI: https://doi.org/10.1007/s11075-022-01342-7

Keywords

Mathematics Subject Classification (2010)