A Fast Implicit Finite Difference Method for Fractional Advection-Dispersion Equations with Fractional Derivative Boundary Conditions

Abstract EN

Fractional advection-dispersion equations, as generalizations of classical integer-order advection-dispersion equations, are used to model the transport of passive tracers carried by fluid flow in a porous medium.

In this paper, we develop an implicit finite difference method for fractional advection-dispersion equations with fractional derivative boundary conditions.

First-order consistency, solvability, unconditional stability, and first-order convergence of the method are proven.

Then, we present a fast iterative method for the implicit finite difference scheme, which only requires storage of O(K) and computational cost of O(KlogK).

Traditionally, the Gaussian elimination method requires storage of O(K2) and computational cost of O(K3).

Finally, the accuracy and efficiency of the method are checked with a numerical example.