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A Preconditioning Technique for an All-at-once System from Volterra Subdiffusion Equations with Graded Time Steps

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Abstract

Volterra subdiffusion problems with weakly singular kernel describe the dynamics of subdiffusion processes well. The graded L1 scheme is often chosen to discretize such problems since it can handle the singularity of the solution near \(t = 0\). In this paper, we propose a modification. We first split the time interval [0, T] into \([0, T_0]\) and \([T_0, T]\), where \(T_0\) (\(0< T_0 < T\)) is reasonably small. Then, the graded L1 scheme is applied in \([0, T_0]\), while the uniform one is used in \([T_0, T]\). Our all-at-once system is derived based on this strategy. In order to solve the arising system efficiently, we split it into two subproblems and design two preconditioners. Some properties of these two preconditioners are also investigated. Moreover, we extend our method to solve semilinear subdiffusion problems. Numerical results are reported to show the efficiency of our method.

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  1. A parallel version of FFT is available at http://www.fftw.org/parallel/parallel-fftw.html.

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Acknowledgements

This research is supported by the National Natural Science Foundation of China (No. 11801463) and the Applied Basic Research Program of Sichuan Province (No. 2020YJ0007). The first author is also supported by the China Scholarship Council. We would like to express our sincere thanks to the referees for insightful comments and invaluable suggestions that greatly improved the representation of this paper.

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Authors and Affiliations

  1. School of Mathematics, Sichuan Normal University, Chengdu, 610068, Sichuan, People’s Republic of China

    Yong-Liang Zhao

  2. School of Economic Mathematics/Institute of Mathematics, Southwestern University of Finance and Economics, Chengdu, 611130, Sichuan, People’s Republic of China

    Xian-Ming Gu

  3. Department of Mathematics, University of Innsbruck, Technikerstraße 13, 6020, Innsbruck, Austria

    Alexander Ostermann

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  1. Yong-Liang Zhao
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Zhao, YL., Gu, XM. & Ostermann, A. A Preconditioning Technique for an All-at-once System from Volterra Subdiffusion Equations with Graded Time Steps. J Sci Comput 88, 11 (2021). https://doi.org/10.1007/s10915-021-01527-7

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