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All-at-once method for variable-order time fractional diffusion equations

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Abstract

We propose a fast solver for the variable-order (VO) time-fractional diffusion equation. Due to the impact of the time-dependent VO function, the resulting coefficient matrix of the large linear system assembling discrete equations of all time levels is a block lower triangular matrix without the block Toeplitz structure. Here, we approximate the off-diagonal blocks by low-rank matrices based on the polynomial interpolation, which can be constructed in \(\mathcal {O}(M\log ^{2} M)\) operations with the same number storage requirement, where M is the number of time steps. Furthermore, a divide-and-conquer method is developed to fast solve the approximated linear system. The proposed solver can be implemented in \(\mathcal {O} (NM\log ^{2} M)\) complexity with N being the degree of freedom in space. The accuracy of approximation is theoretically studied, and the stability and convergence of the proposed fast method are also investigated. Numerical experiments are carried out to exemplify the accuracy and efficiency of the proposed method.

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Acknowledgements

We thank the anonymous referees for valuable comments and suggestions which lead to a significant improvement of the presentation.

Funding

This work was supported in part by the National Natural Science Foundation of China (Grant Nos. 11771189, 12090011, 11501562), the Natural Science Foundation of Jiangsu Province (Grant No. BK20171162), the Qing-Lan Project of Jiangsu Higher Education Institutions, the Fundamental Research Funds for the Central Universities (Grant No. 2015QNA49), the Science and Technology Development Fund, Macau SAR (file no. 0118/2018/A3), and MYRG2018-00015-FST from University of Macau.

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Authors and Affiliations

  1. School of Mathematics and Statistics, Jiangsu Normal University, Xuzhou, 221116, Jiangsu, People’s Republic of China

    Hong-Kui Pang

  2. School of Mathematics, China University of Mining and Technology, Xuzhou, 221116, Jiangsu, People’s Republic of China

    Hai-Hua Qin

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

    Hai-Wei Sun

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  1. Hong-Kui Pang
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Pang, HK., Qin, HH. & Sun, HW. All-at-once method for variable-order time fractional diffusion equations. Numer Algor 90, 31–57 (2022). https://doi.org/10.1007/s11075-021-01178-7

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