A high-order compact ADI finite difference scheme on uniform meshes for a weakly singular integro-differential equation in three space dimensions

We propose a new compact alternating direction implicit (ADI) finite difference scheme on uniform meshes for a weakly singular integro-differential equation in three space dimensions. Compared with the compact non-ADI scheme, the proposed compact ADI scheme is easy to implement and significantly reduces the computational cost of solving the resulting linear system, while maintaining the same order of the local truncation error. We prove that the new compact ADI scheme is unconditionally stable, and has second-order convergence in time and fourth-order convergence in space for weakly singular solutions. Numerical results are given to confirm the theoretical analysis result and demonstrate the computational efficiency of the proposed compact ADI scheme.