An efficient ADI difference scheme for the nonlocal evolution equation with multi-term weakly singular kernels in three dimensions

Abstract

The paper constructs a fast efficient numerical scheme for the nonlocal evolution equation with three weakly singular kernels in three-dimensional space. In the temporal direction, We apply the backward Euler (BE) alternating direction implicit (ADI) method for the time derivative, simultaneously the first-order convolution quadrature formula is employed to deal with Riemann-Liouville (R-L) fractional integral term. In order to obtain a completely discrete implicit difference scheme, we use the standard central finite difference method (FDM) in space. The stability and convergence of the BE ADI difference scheme are proved rigorously with the convergence order $O(\tau +h^2)$ in which h and τ are corresponding on the step size of space and time respectively. The ADI algorithm greatly reduces the computational cost of the three-dimensional problem. At last, several numerical results are given to verify that the numerical results are in agreement with our theoretical analysis.

Keywords:

• Nonlocal evolution equation
• finite difference method
• alternating direction implicit
• weakly singular kernel
• stability and convergence

2020 AMS Subject Classifications:

• 45K05
• 65M06

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

The work was supported by National Natural Science Foundation of China Mathematics Tianyuan Foundation (12226337, 12226340, 12126321), Scientific Research Fund of Hunan Provincial Education Department (21B0550, 22C0323), Hunan Provincial Natural Science Foundation of China (2022JJ50083).