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R. Chan’s circulant-based approximate inverse preconditioning iterative method for solving second-order space fractional advection–dispersion equations with variable coefficients

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Abstract

In this paper, we consider a second-order approximation scheme for the space fractional advection–diffusion equations with variable coefficients. This scheme employs the weighted and shifted Grünwald difference formulas to approximate the Riemann–Liouville fractional derivatives and the Crank–Nicolson scheme for the temporal derivative, respectively. The stability and convergence of this approximation scheme are analyzed under certain conditions satisfied by the coefficients. This numerical scheme results in a sequence of nonsymmetric Toeplitz-like linear systems whose coefficient matrix can be written as a sum of an identity matrix and four diagonal-times-Toeplitz matrices. To improve the computational efficiency of the Krylov subspace iteration methods used to solve the scheme, we develop an approximate inverse preconditioner. This preconditioner is based on the inverse of the weighted R. Chan’s circulant matrix and the interpolation method. We also demonstrate that the preconditioned matrix can be written as a sum of an identity matrix, a low-rank matrix, and a small-norm matrix, resulting in a clustered spectrum around one. The numerical results demonstrate that the numerical scheme and its approximate inverse preconditioner are robust and efficient.

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Acknowledgements

This work is supported by the National Natural Science Foundation of China [Grant No. 12161030] and Hainan Provincial Natural Science Foundation of China [Grant Nos. 121RC537, 523MS039].

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Authors and Affiliations

  1. School of Mathematics and Statistics, Hainan Normal University, Haikou, 570000, People’s Republic of China

    Shi-Ping Tang & Ai-Li Yang

  2. School of Mathematics and Statistics, Lanzhou University, Lanzhou, 730000, People’s Republic of China

    Shi-Ping Tang, Jian-Lin Zhou & Yu-Jiang Wu

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  1. Shi-Ping Tang
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  2. Ai-Li Yang
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Correspondence to Ai-Li Yang.

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Communicated by Zhong-Zhi Bai.

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Tang, SP., Yang, AL., Zhou, JL. et al. R. Chan’s circulant-based approximate inverse preconditioning iterative method for solving second-order space fractional advection–dispersion equations with variable coefficients. Comp. Appl. Math. 43, 97 (2024). https://doi.org/10.1007/s40314-024-02592-y

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