A second-order ADI difference scheme based on non-uniform meshes for the three-dimensional nonlocal evolution problem

This work constructs and analyzes a nonlocal evolution equation with a weakly singular kernel in three-dimensional space. In the temporal direction, the Crank-Nicolson (CN) method and product-integration (PI) rule are employed, from which the non-uniform meshes are used to eliminate the singular behaviour of the exact solution at t = 0. Then, a fully discrete scheme is obtained by the spatial discretization based on the finite difference method. Simultaneously, an alternating direction implicit (ADI) algorithm is designed to reduce the computational cost. The stability in L-2 norm and convergence are derived via the energy method, in which the convergence orders are O(k(2) + h(2)) with certain suitable assumptions, where k and h are corresponding space-time step sizes, respectively. Numerical results confirm the theoretical analysis.

Automated summary

This study constructs and analyzes a nonlocal evolution equation with a weakly singular kernel in three-dimensional space, using different numerical methods to ensure stability and convergence. The numerical results confirm the theoretical analysis.