Pre-tensorial alternative direction implicit method for nonlocal diffusion equations

In this paper, we propose a modified alternative direction implicit (ADI) method to solve the nonlocal diffusion equations. The nonlocal operator is discretized by the finite difference scheme, that is without holding the tensorial structure. The resulting discretized operator is split into two parts: The first part is dominant and linear with tensorial structure, that will be treated implicitly. The other part is non-tensorial that will be explicitly evaluated in the right-hand side of the discretized system. Furthermore, the ADI method is applied to solve the first part since it processes the tensorial structure. Theoretically, our pre-tensorial ADI method is confirmed to be unconditionally stable and convergent. Numerical experiments are conducted to demonstrate the accuracy and efficiency of the proposed method.