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Fast Iterative Method with a Second-Order Implicit Difference Scheme for Time-Space Fractional Convection–Diffusion Equation

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Abstract

In this paper we intend to establish fast numerical approaches to solve a class of initial-boundary problem of time-space fractional convection–diffusion equations. We present a new unconditionally stable implicit difference method, which is derived from the weighted and shifted Grünwald formula, and converges with the second-order accuracy in both time and space variables. Then, we show that the discretizations lead to Toeplitz-like systems of linear equations that can be efficiently solved by Krylov subspace solvers with suitable circulant preconditioners. Each time level of these methods reduces the memory requirement of the proposed implicit difference scheme from \({\mathcal {O}}(N^2)\) to \({\mathcal {O}}(N)\) and the computational complexity from \({\mathcal {O}}(N^3)\) to \({\mathcal {O}}(N\log N)\) in each iterative step, where N is the number of grid nodes. Extensive numerical examples are reported to support our theoretical findings and show the utility of these methods over traditional direct solvers of the implicit difference method, in terms of computational cost and memory requirements.

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Notes

  1. In this case, it should mention that we only need to solve three nonsymmetric Toeplitz systems, i.e., equations with the form like (3.5) and in Step 4 of Algorithm 3, for implementing the whole for loop.

  2. For the sake of clarity, here we do not list the number of iterations required for solving those three linear systems one by one.

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Acknowledgements

The authors would like to thank the Prof. Zhi-Zhong Sun and Dr. Zhao-Peng Hao for their insightful discussions about the convergence analysis of the proposed implicit difference scheme. The authors are also grateful to the anonymous referees for their useful suggestions and comments that improved the presentation of this paper. The work of Gu and Huang is supported by 973 Program (2013CB329404), NSFC (61370147 and 61402082), the Fundamental Research Funds for the Central Universities (ZYGX2014J084). Ji’s work is supported by the Fundamental Research Funds for the Central Universities and the Research and Innovation Project for College Graduates of Jiangsu Province (Grant No. KYLX15_0106). The work of the last author has been partially implemented with the financial support of the Russian Presidential grant for young scientists MK-3360.2015.1.

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Authors and Affiliations

  1. School of Mathematical Sciences, University of Electronic Science and Technology of China, Chengdu, 611731, Sichuan, People’s Republic of China

    Xian-Ming Gu & Ting-Zhu Huang

  2. Institute of Mathematics and Computing Science, University of Groningen, Nijenborgh 9, P.O. Box 407, 9700 AK, Groningen, The Netherlands

    Xian-Ming Gu

  3. Department of Mathematics, Southeast University, Nanjing, 210096, Jiangsu, People’s Republic of China

    Cui-Cui Ji

  4. School of Science and Technology, Nottingham Trent University, Clifton Campus, Nottingham, NG11 8NS, UK

    Bruno Carpentieri

  5. Institute of Applied Mathematics and Automation, Russian Academy of Sciences, ul. Shortanova 89 a, Nalchik, Russia, 360000

    Anatoly A. Alikhanov

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  1. Xian-Ming Gu
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Correspondence to Ting-Zhu Huang.

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Gu, XM., Huang, TZ., Ji, CC. et al. Fast Iterative Method with a Second-Order Implicit Difference Scheme for Time-Space Fractional Convection–Diffusion Equation. J Sci Comput 72, 957–985 (2017). https://doi.org/10.1007/s10915-017-0388-9

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