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CN ADI fast algorithm on non-uniform meshes for the three-dimensional nonlocal evolution equation with multi-memory kernels in viscoelastic dynamics

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  • Zhou, Ziyi
  • Zhang, Haixiang
  • Yang, Xuehua

Abstract

This paper proposes a Crank-Nicolson alternating direction implicit (CN-ADI) finite difference scheme for solving the three-dimensional nonlocal evolution equation with multi-memory kernels in viscoelastic dynamic for the first time. Due to the weakly singular behavior of the exact solution near the initial time t=0, we use the non-uniform meshes to capture the rapid change of the solution at t=0. The Crank-Nicolson method and product-integration (PI) rule are proposed to approximate temporal derivative and the Riemann-Liouville (R-L) fractional integral term, respectively. The fully discrete scheme is obtained by the standard central finite difference method (FDM) in space. The stability in L2-norm and convergence of the CN-ADI difference scheme are strictly proved, where the convergence reached O(τ2+hx2+hy2+hz2). The ADI algorithm greatly reduces the computational cost of the three-dimensional problems in viscoelastic dynamics. At last, the results of numerical examples verify the correctness of the theoretical analysis and prove the effectiveness of the proposed method.

Suggested Citation

  • 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).
    • Handle: RePEc:eee:apmaco:v:474:y:2024:i:c:s0096300324001528
    DOI: 10.1016/j.amc.2024.128680
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    References listed on IDEAS

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    1. Alikhanov, Anatoly A. & Huang, Chengming, 2021. "A high-order L2 type difference scheme for the time-fractional diffusion equation," Applied Mathematics and Computation, Elsevier, vol. 411(C).
    2. Chang, Ailian & Sun, HongGuang & Zhang, Yong & Zheng, Chunmiao & Min, Fanlu, 2019. "Spatial fractional Darcy’s law to quantify fluid flow in natural reservoirs," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 519(C), pages 119-126.
    3. 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.
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