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Fast direct solver for CN-ADI-FV scheme to two-dimensional Riesz space-fractional diffusion equations

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  • Qu, Wei
  • Li, Zhi

Abstract

Based on the Crank-Nicolson alternating direction implicit finite volume (CN-ADI-FV) discretization of two dimensional Riesz space-fractional diffusion equations (2 D RSFDEs) proposed in [1], a fast direct solver is proposed to obtain the numerical solutions. Linear systems result from the CN-ADI-FV discretization can be treated as two matrix equations with symmetric positive definite (SPD) Toeplitz coefficient matrices. Then numerical solutions, expressed as multiple matrix-vector multiplications involving the inverse of SPD Toeplitz matrices, can be effectively obtained by using Gohberg-Semencul formula (GSF), where the preconditioned conjugate gradient (PCG) method is applied as an acceleration technique. Moreover, under certain assumptions, the spectral of the preconditioned matrices of the linear system associated with the GSF are theoretically proved to be clustered around 1. Then the PCG method is verified to converge superlinearly and can be implemented with O(NlogN) operations, where N is the number of spatial unknowns in both x- and y- directions. Besides, the PCG method is used only twice in the whole solution process, and the complete solution process can be finished with O(MN2logN) operations and O(N) memory, where M is the number of temporal unknowns. Numerical experiments are given to demonstrate the efficiency of our approach.

Suggested Citation

  • Qu, Wei & Li, Zhi, 2021. "Fast direct solver for CN-ADI-FV scheme to two-dimensional Riesz space-fractional diffusion equations," Applied Mathematics and Computation, Elsevier, vol. 401(C).
    • Handle: RePEc:eee:apmaco:v:401:y:2021:i:c:s0096300321000813
    DOI: 10.1016/j.amc.2021.126033
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    References listed on IDEAS

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    1. Zhao, Yanmin & Bu, Weiping & Huang, Jianfei & Liu, Da-Yan & Tang, Yifa, 2015. "Finite element method for two-dimensional space-fractional advection–dispersion equations," Applied Mathematics and Computation, Elsevier, vol. 257(C), pages 553-565.
    2. Raberto, Marco & Scalas, Enrico & Mainardi, Francesco, 2002. "Waiting-times and returns in high-frequency financial data: an empirical study," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 314(1), pages 749-755.
    3. Xing, Zhiyong & Wen, Liping, 2019. "Numerical analysis and fast implementation of a fourth-order difference scheme for two-dimensional space-fractional diffusion equations," Applied Mathematics and Computation, Elsevier, vol. 346(C), pages 155-166.
    4. 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. Yang, Hong & Lao, Cheng-Xue & She, Zi-Hang, 2023. "Fast solution methods for Riesz space fractional diffusion equations with non-separable coefficients," Applied Mathematics and Computation, Elsevier, vol. 445(C).
    2. Guan, Chaoyue & Zhang, Jian, 2026. "Adaptive spectral solver for Riesz fractional reaction–diffusion equations via penalized minimum residual iteration," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 241(PB), pages 431-451.

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