Fast finite volume methods for space-fractional diffusion equations

Abstract

Fractional diffusion equations model phenomena exhibiting anomalous diffusion that is characterized by a heavy tail or an inverse power law decay, which cannot be modeled accurately by second-order diffusion equations that is well known to model Brownian motions that are characterized by an exponential decay. However, fractional differential equations introduce new mathematical and numerical difficulties that have not been encountered in the context of traditional second-order differential equations. For instance, because of the nonlocal property of fractional differential operators, the corresponding numerical methods have full coefficient matrices which require storage of $O(N^2)$ and computational cost of $O(N^3)$ where $N$ is the number of grid points.  &nbspWe develop a fast locally conservative finite volume method for a time-dependent variable-coefficient conservative space-fractional diffusion equation. This method requires only a computational cost of $O(N \log N)$ at each iteration and a storage of $O(N)$. Numerical experiments are presented to investigate the performance of the method and to show the strong potential of these methods.