A Fast Finite Difference Method for Two-Dimensional Space-Fractional Diffusion Equations

Abstract

Fractional diffusion equations model phenomena exhibiting anomalous diffusion that cannot be modeled accurately by second-order diffusion equations. Because of the nonlocal property of fractional differential operators, numerical methods for space-fractional diffusion equations generate complicated dense or full coefficient matrices. Consequently, these numerical methods were traditionally solved by Gaussian elimination, which requires computational work of $O(N^3)$ per time step and $O(N^2)$ of memory, where $N$ is the number of spatial grid points in the discretization. The significant computational work and memory requirement of the numerical methods impose a serious challenge for the numerical simulation of two- and especially three-dimensional space-fractional diffusion equations. We develop a fast and yet accurate solution method for the implicit finite difference discretization of space-fractional diffusion equations in two 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 $O(N \log N)$ 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 show that the fast method has a significant reduction of CPU time, from two months and eight days as consumed by the traditional method to less than 40 minutes, with less than one ten-thousandth of the memory required by the traditional method, in the context of a two-dimensional space-fractional diffusion equation with 512 $\times$ 512 = 262,144 spatial nodes and 512 time steps. This demonstrates the utility of the method.