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

Fractional diffusion equations model phenomena exhibiting anomalous diffusion that cannot be modeled accurately by second-order diffusion equations. Because of the non-local property of fractional differential operators, numerical methods for space-fractional diffusion equations generate dense or even full coefficient matrices with complicated structures. Traditionally, these methods were solved with Gaussian elimination, which requires computational work of 0 (N-3) per time step and 0 (N-2) of memory to store where N is the number of spatial grid points in the discretization. The significant computational work and memory requirement of these methods makes a numerical simulation of three-dimensional space-fractional diffusion equations computationally prohibitively expensive. 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 0 (N log N) per iteration and a memory requirement of 0 (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. (C) 2013 Elsevier Inc. All rights reserved.