Abstract
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.
Highlights
Fractional derivatives are almost as old as their more familiar integer-order counterparts [1, 2]
Having developed a numerical scheme and shown that it is consistent, in the following theorems, we show this method is solvable, unconditionally stable, and convergent
The following theorem shows that the storage of matrix A can be stored in O(K) memories, instead of O(K2) memories
Summary
Fractional derivatives are almost as old as their more familiar integer-order counterparts [1, 2]. Meerschaert and Tadjeran [14] developed finite difference approximations for one-dimensional fractional advection-dispersion equations with Dirichlet boundary conditions. Some authors have discussed numerical method for fractional equations with fractional derivative boundary conditions. Jia and Wang [15] developed fast implicit finite difference methods for two-sided space fractional diffusion equations with fractional derivative boundary conditions. As far as we know, the study on numerical method for one-dimensional advection-dispersion equations with fractional derivative boundary conditions is still limited. Implicit finite difference method for the advection-dispersion equations with fractional derivative boundary conditions (1)–(3) can be written as follows: pin+1 − pin Δt The implicit finite difference method for the advection-dispersion equations with fractional derivative boundary conditions defined by (7) is unconditionally stable.
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