Alternating direction implicit-Euler method for the two-dimensional fractional evolution equation

In this paper, we present a fast and efficient numerical method to solve a two-dimensional fractional evolution equation on a finite domain. This numerical method combines the alternating direction implicit (ADI) approach with the second-order difference quotient in space, the backward Euler in time and order one convolution quadrature approximating the integral term. By using the discrete energy method, we prove that the ADI scheme is unconditionally stable and the numerical solution converges to the exact one with order O(k+hx2+hy2), where k   is the temporal grid size and hx,hyhx,hy are spatial grid sizes in the x and y directions, respectively. Two numerical examples with known exact solution are also presented, and the behavior of the error is analyzed to verify the order of convergence of the ADI-Euler method.