Analysis and efficient implementation of alternating direction implicit finite volume method for Riesz space-fractional diffusion equations in two space dimensions

In this article, we develop a Crank-Nicolson alternating direction implicit finite volume method for time-dependent Riesz space-fractional diffusion equation in two space dimensions. Norm-based stability and convergence analysis are given to show that the developed method is unconditionally stable and of second-order accuracy both in space and time. Furthermore, we develop a lossless matrix-free fast conjugate gradient method for the implementation of the numerical scheme, which only hasO(N)memory requirement andO(NlogN)computational complexity per iteration withNbeing the total number of spatial unknowns. Several numerical experiments are presented to demonstrate the effectiveness and efficiency of the proposed scheme for large-scale modeling and simulations.