An alternating direction implicit scheme with graded time steps for a fractional evolution equation with a weakly singular kernel

This paper addresses the numerical solution of a fractional evolution equation with a weakly singular kernel. The temporal discretization is carried out by the Crank–Nicolson (CN) method based on graded time steps, while the spatial discretization is carried out using an alternating direction implicit (ADI) finite difference method. Stability and convergence rate are also discussed. The convergence rate is $O(k^2+h_x^2+h_y^2)$, where $k$ is a parameter of the maximal time steps and $h_x$, $h_y$ are the uniform grid steps in space. Three numerical examples are employed to demonstrate the errors and convergence behavior in practice.