A fast characteristic finite difference method for fractional advection–diffusion equations

Abstract

Fractional advection–diffusion equations provide an adequate and accurate description of the movement of solute in an aquifer. However, there are major obstacles that restrict their applications. From a modeling viewpoint, one of the major limitations in the application of fractional advection–diffusion equations to hydrology is the poor predictability of model parameters [27]. From a computational view point, one of the major limitations in numerical solution of fractional advection–diffusion equations in multiple space dimensions is that they generate full coefficient matrices in their numerical approximations, which require O( N 3) of computational cost and O( N 2) storage for a problem of size N. This paper presents a preliminary step towards the efficient numerical solution of fractional advection–diffusion equations. In this paper we develop a fast characteristic finite difference method for the efficient solution of space-fractional transient advection–diffusion equations in one space dimension. This method generates more accurate solutions than standard implicit methods even if much larger time steps and spatial meshes are used, leading to a discrete system with a greatly reduced size. Furthermore, we explore the structure of the coefficient matrix to come up with an efficient iterative solver which requires only O( N) account of storage and roughly O( N log N) account of computational cost. Our preliminary numerical example runs for some simple one dimensional model problems seem to indicate the following observations: to achieve the same accuracy, the new method uses no more than one thousandth of CPU and about one thousandth of the storage used by the standard method. This demonstrates the strong potential of the method.