A Fast Second-Order Finite Difference Method for Space-Fractional Diffusion Equations

Abstract

Fractional diffusion equations provide an adequate and accurate description of transport processes that exhibit anomalous diffusion that cannot be modeled accurately by classical second-order diffusion equations. However, numerical discretizations of fractional diffusion equations yield full coefficient matrices, which require a computational operation of $O(N^3)$ per time step and a memory of $O(N^2)$ for a problem of size $N$. In this paper we develop a fast second-order finite difference method for space-fractional diffusion equations, which only requires memory of $O(N)$ and computational work of $O(N log^2 N)$. Numerical experiments show the utility of the method.