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<p>This paper proposes an highly efficient second-order backward difference formula (BDF2) alternating direction implicit (ADI) compact finite difference scheme, which is applicable to three-dimensional (3D) partial integro-differential equations (PIDEs) with multi-term weakly singular kernels. In terms of temporal discretization, the BDF2 and product integration (PI) rule on graded meshes are utilized to approximate the time first-order derivative term and Riemann-Liouville (R-L) fractional integrals, respectively, effectively capturing initial-layer singularities and constructing second-order temporal accurate formulas. In the spatial direction, the fourth-order compact difference scheme are adopted. By introducing the alternating direction implicit technique, the three-dimensional problem is decoupled into independent one-dimensional subproblems, significantly reducing computational complexity. By establishing a novel Gronwall-type inequality, we rigorously prove the stability and convergence of the proposed scheme via the discrete energy method. Finally, some numerical examples are provided. Through comparisons between results obtained by the proposed method and those from Zhou et al. (Comput. Appl. Math. <b>43</b>(8), 418 <CitationRef CitationID="CR1">2024</CitationRef>), we improve the temporal and spatial convergence, simultaneously.</p>

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The BDF2-ADI compact difference scheme on graded meshes for the three-dimensional PIDEs with multi-term weakly singular kernels

  • 期刊论文

  • 发表日期：

    2025年10月18日

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  • Tianyuan Liu,
  • Haixiang Zhang,
  • Xuehua Yang

摘要

This paper proposes an highly efficient second-order backward difference formula (BDF2) alternating direction implicit (ADI) compact finite difference scheme, which is applicable to three-dimensional (3D) partial integro-differential equations (PIDEs) with multi-term weakly singular kernels. In terms of temporal discretization, the BDF2 and product integration (PI) rule on graded meshes are utilized to approximate the time first-order derivative term and Riemann-Liouville (R-L) fractional integrals, respectively, effectively capturing initial-layer singularities and constructing second-order temporal accurate formulas. In the spatial direction, the fourth-order compact difference scheme are adopted. By introducing the alternating direction implicit technique, the three-dimensional problem is decoupled into independent one-dimensional subproblems, significantly reducing computational complexity. By establishing a novel Gronwall-type inequality, we rigorously prove the stability and convergence of the proposed scheme via the discrete energy method. Finally, some numerical examples are provided. Through comparisons between results obtained by the proposed method and those from Zhou et al. (Comput. Appl. Math. 43(8), 418 2024), we improve the temporal and spatial convergence, simultaneously.