A fast finite difference method for three-dimensional time-dependent space-fractional diffusion equations and its efficient implementation

In this paper we develop an efficient and faithful solution method for the implicit finite difference discretization of time-dependent space-fractional diffusion equations in three space dimensions, by carefully analyzing the structure of the coefficient matrix of the finite difference method and delicately decomposing the coefficient matrix into a combination of sparse and structured dense matrices. The fast method has a computational work count of O(NlogN) per iteration and a memory requirement of O(N), while retaining the same accuracy as the underlying finite difference method solved with Gaussian elimination. Numerical experiments of a three-dimensional space-fractional diffusion equation show the utility of the fast method.