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A novel alternating-direction implicit spectral Galerkin method for a multi-term time-space fractional diffusion equation in three dimensions

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Abstract

In this paper, we develop an efficient spectral Galerkin method for the three-dimensional (3D) multi-term time-space fractional diffusion equation. Based on the L2-1σ formula for time stepping and the Legendre-Galerkin spectral method for space discretization, a fully discrete numerical scheme is constructed and the stability and convergence analyses are rigorously established. The results show that the fully discrete scheme is unconditionally stable and has second-order accuracy in time and optimal error estimation in space. In addition, we give the detailed implementation and apply the alternating-direction implicit (ADI) method to reduce the computational complexity. Furthermore, numerical experiments are presented to confirm the theoretical claims. As an application of the proposed method, the fractional Bloch-Torrey model is also solved.

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Funding

This work was funded by the National Natural Science Foundation of China (No.11501441 and 11772046), the Science Challenge Project (No. TZ2016002) and the Australian Research Council via the Discovery Projects (DP180103858 and DP190101889).

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Authors and Affiliations

  1. School of Science, Xi’an University of Architecture and Technology, Xi’an, 710055, Shaanxi, People’s Republic of China

    Ying Wang

  2. School of Mathematics and Statistics, Xi’an Jiaotong University, Xi’an, 710049, Shaanxi, People’s Republic of China

    Ying Wang & Liquan Mei

  3. School of Mathematical Sciences, Queensland University of Technology (QUT), Brisbane, QLD 4001, Australia

    Fawang Liu

  4. Faculty of Science, Engineering and Technology, Swinburne University of Technology, PO Box 218, Hawthorn, VIC 3122, Australia

    Vo V. Anh

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  1. Ying Wang
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Wang, Y., Liu, F., Mei, L. et al. A novel alternating-direction implicit spectral Galerkin method for a multi-term time-space fractional diffusion equation in three dimensions. Numer Algor 86, 1443–1474 (2021). https://doi.org/10.1007/s11075-020-00940-7

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