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<p>In this paper, a Crank–Nicolson (C–N) alternating direction implicit (ADI) compact difference scheme with second-order accuracy in time and fourth-order accuracy in space is constructed for the three-dimensional (3D) nonlocal evolution problem with a weakly singular kernel. The C–N method and trapezoidal convolution quadrature rules are used to approximate the time derivative and tempered fractional integral term, respectively. In order to obtain a fully discrete method, compact difference method is used to space. Combining with corresponding ADI algorithms, the overall computational cost is reduced significantly. Energy approaches are used to demonstrate the ADI compact difference scheme’s stability and convergence. In the end, numerical examples demonstrate our theoretical statements, the CPU time is extremely little.</p>

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A Crank–Nicolson ADI compact difference scheme for the three-dimensional nonlocal evolution problem with a weakly singular kernel

  • 期刊论文

  • 发表日期：

    2025年2月8日

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  • Kai Liu,
  • Zhangming He,
  • Haixiang Zhang,
  • Xuehua Yang

摘要

In this paper, a Crank–Nicolson (C–N) alternating direction implicit (ADI) compact difference scheme with second-order accuracy in time and fourth-order accuracy in space is constructed for the three-dimensional (3D) nonlocal evolution problem with a weakly singular kernel. The C–N method and trapezoidal convolution quadrature rules are used to approximate the time derivative and tempered fractional integral term, respectively. In order to obtain a fully discrete method, compact difference method is used to space. Combining with corresponding ADI algorithms, the overall computational cost is reduced significantly. Energy approaches are used to demonstrate the ADI compact difference scheme’s stability and convergence. In the end, numerical examples demonstrate our theoretical statements, the CPU time is extremely little.