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A class of shifted high-order numerical methods for the fractional mobile/immobile transport equations

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  • Yin, Baoli
  • Liu, Yang
  • Li, Hong

Abstract

In this article, we apply the generalized BDF2-θ to the fractional mobile/immobile transport equations for its temporal discretization and the finite element method in the spatial direction. To derive the stability estimates and obtain the optimal error convergence rate, some properties of the convolution weights are proved based on which we show the scheme is unconditionally stable with an error of O(τ2+hr+1), where τ and h represent the temporal and spatial mesh size, respectively. We conduct exhaustive numerical tests to further confirm our theoretical analysis, and to overcome the initial singularity of the time fractional derivative we adopt the generalized BDF2-θ with starting parts in accordance with the framework of the shifted convolution quadrature (SCQ).

Suggested Citation

  • Yin, Baoli & Liu, Yang & Li, Hong, 2020. "A class of shifted high-order numerical methods for the fractional mobile/immobile transport equations," Applied Mathematics and Computation, Elsevier, vol. 368(C).
    • Handle: RePEc:eee:apmaco:v:368:y:2020:i:c:s009630031930791x
    DOI: 10.1016/j.amc.2019.124799
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    References listed on IDEAS

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    5. Feng, L.B. & Zhuang, P. & Liu, F. & Turner, I., 2015. "Stability and convergence of a new finite volume method for a two-sided space-fractional diffusion equation," Applied Mathematics and Computation, Elsevier, vol. 257(C), pages 52-65.
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    Cited by:

    1. Hao, Zhaopeng & Lin, Guang & Zhang, Zhongqiang, 2020. "Error estimates of a spectral Petrov–Galerkin method for two-sided fractional reaction–diffusion equations," Applied Mathematics and Computation, Elsevier, vol. 374(C).
    2. Yaxin Hou & Cao Wen & Hong Li & Yang Liu & Zhichao Fang & Yining Yang, 2020. "Some Second-Order σ Schemes Combined with an H 1 -Galerkin MFE Method for a Nonlinear Distributed-Order Sub-Diffusion Equation," Mathematics, MDPI, vol. 8(2), pages 1-19, February.
    3. Wang, Xiuping & Gao, Fuzheng & Liu, Yang & Sun, Zhengjia, 2020. "A Weak Galerkin Finite Element Method for High Dimensional Time-fractional Diffusion Equation," Applied Mathematics and Computation, Elsevier, vol. 386(C).
    4. Jie Zhao & Zhichao Fang & Hong Li & Yang Liu, 2020. "A Crank–Nicolson Finite Volume Element Method for Time Fractional Sobolev Equations on Triangular Grids," Mathematics, MDPI, vol. 8(9), pages 1-17, September.
    5. Niu, Yuxuan & Liu, Yang & Li, Hong & Liu, Fawang, 2023. "Fast high-order compact difference scheme for the nonlinear distributed-order fractional Sobolev model appearing in porous media," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 203(C), pages 387-407.

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