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Error analysis of fast L1 ADI finite difference/compact difference schemes for the fractional telegraph equation in three dimensions

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  • Qiao, Leijie
  • Qiu, Wenlin
  • Xu, Da

Abstract

This article proposes the fast L1 alternating direction implicit (ADI) finite difference and compact difference schemes to solve the fractional telegraph equation in three-dimensional space. The fully-discrete fast L1 ADI finite difference scheme can be established via the fast L1 formula for the approximation of mixed Caputo fractional derivatives and the central difference formula for the approximation of the spatial derivative term, then from which an ADI algorithm is designed to reduce three-dimensional problems to a series of one-dimensional problems. We add the corresponding compact operators in all directions of the space to get the fully-discrete L1 ADI compact difference scheme. Then the convergence in L2 and H1 norms of two ADI schemes is derived via energy method. Eventually, numerical experiments are carried out to verify the theoretical estimates.

Suggested Citation

  • Qiao, Leijie & Qiu, Wenlin & Xu, Da, 2023. "Error analysis of fast L1 ADI finite difference/compact difference schemes for the fractional telegraph equation in three dimensions," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 205(C), pages 205-231.
    • Handle: RePEc:eee:matcom:v:205:y:2023:i:c:p:205-231
    DOI: 10.1016/j.matcom.2022.10.001
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    References listed on IDEAS

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    1. Hosseininia, M. & Heydari, M.H., 2019. "Meshfree moving least squares method for nonlinear variable-order time fractional 2D telegraph equation involving Mittag–Leffler non-singular kernel," Chaos, Solitons & Fractals, Elsevier, vol. 127(C), pages 389-399.
    2. Qiu, Wenlin & Chen, Hongbin & Zheng, Xuan, 2019. "An implicit difference scheme and algorithm implementation for the one-dimensional time-fractional Burgers equations," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 166(C), pages 298-314.
    3. Saffarian, Marziyeh & Mohebbi, Akbar, 2021. "Numerical solution of two and three dimensional time fractional damped nonlinear Klein–Gordon equation using ADI spectral element method," Applied Mathematics and Computation, Elsevier, vol. 405(C).
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    Cited by:

    1. Zhou, Ziyi & Zhang, Haixiang & Yang, Xuehua, 2024. "CN ADI fast algorithm on non-uniform meshes for the three-dimensional nonlocal evolution equation with multi-memory kernels in viscoelastic dynamics," Applied Mathematics and Computation, Elsevier, vol. 474(C).
    2. Luo, Man & Qiu, Wenlin & Nikan, Omid & Avazzadeh, Zakieh, 2023. "Second-order accurate, robust and efficient ADI Galerkin technique for the three-dimensional nonlocal heat model arising in viscoelasticity," Applied Mathematics and Computation, Elsevier, vol. 440(C).

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    More about this item

    Keywords

    Three-dimensional fractional telegraph equation; ADI difference/compact difference methods; Fast L1 algorithm; Convergence analysis; Numerical examples;
    All these keywords.

    JEL classification:

    • L1 - Industrial Organization - - Market Structure, Firm Strategy, and Market Performance

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