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Numerical analysis and fast implementation of a fourth-order difference scheme for two-dimensional space-fractional diffusion equations

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  • Xing, Zhiyong
  • Wen, Liping

Abstract

In this paper, a fourth-order difference scheme (FODS) is proposed for solving the two-dimensional Riesz space-fractional diffusion equations with homogeneous Dirichlet boundary conditions. It is proved that the FODS is uniquely solvable, unconditionally stable, and convergent with order O(τ2+hx4+hy4) in the discrete L∞- norm, where τ is the time step size, and hx, hy are the space grid sizes in the x direction and the y direction, respectively. Based on the special structure and symmetric positive definiteness of the coefficient matrix, a fast method is developed for the implementation of the FODS. The fast method reduces the storage requirement of O(N2) and computational cost of O(N3) down to O(M+J) and O(Nlog N), where N=MJ,M and J are the numbers of the spatial grid points in the x direction and the y direction, respectively. Finally, several numerical results are shown to verify the theoretical results and the efficiency of the fast method.

Suggested Citation

  • 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.
    • Handle: RePEc:eee:apmaco:v:346:y:2019:i:c:p:155-166
    DOI: 10.1016/j.amc.2018.10.057
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    References listed on IDEAS

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    1. Chen, S. & Liu, F. & Jiang, X. & Turner, I. & Anh, V., 2015. "A fast semi-implicit difference method for a nonlinear two-sided space-fractional diffusion equation with variable diffusivity coefficients," Applied Mathematics and Computation, Elsevier, vol. 257(C), pages 591-601.
    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.
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    Cited by:

    1. 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).
    2. Xiaoyong Yang & Zhendong Luo, 2022. "An Unchanged Basis Function and Preserving Accuracy Crank–Nicolson Finite Element Reduced-Dimension Method for Symmetric Tempered Fractional Diffusion Equation," Mathematics, MDPI, vol. 10(19), pages 1-13, October.
    3. Xing, Zhiyong & Wen, Liping & Wang, Wansheng, 2021. "An explicit fourth-order energy-preserving difference scheme for the Riesz space-fractional Sine–Gordon equations," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 181(C), pages 624-641.
    4. Zhao, Jingjun & Li, Yu & Xu, Yang, 2019. "An explicit fourth-order energy-preserving scheme for Riesz space fractional nonlinear wave equations," Applied Mathematics and Computation, Elsevier, vol. 351(C), pages 124-138.
    5. Almushaira, Mustafa, 2023. "An efficient fourth-order accurate conservative scheme for Riesz space fractional Schrödinger equation with wave operator," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 210(C), pages 424-447.

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