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

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 ( ) and computational cost of ( log ). Traditionally, the Gaussian elimination method requires storage of ( 2 ) and computational cost of ( 3 ). Finally, the accuracy and efficiency of the method are checked with a numerical example.

Finite difference scheme for time‐space fractional diffusion equation with fractional boundary conditions

“…It is known that the weighted and shifted Grünwald-Letnikov operator can be used to approximate the Riemann-Liouville fractional derivative to the second-order accuracy. However, when the operator is used to approximate the space fractional derivative in the fractional boundary condition, because of the shift of the operator, there will appear points u(x ′ , t), where x ′ > L. In previous studies, [20][21][22][23] there are not similar case, as the space fractional derivative in the boundary condition are approximated by the standard Grünwald-Letnikov operator to first-order accuracy. Therefore, the difficulties in the paper are how to deal with u(x ′ , t) and the corresponding proof of stabilty and convergence.…”

“…The boundary condition , enforcing a no‐flux condition at the point y =0 in the state space, can be considered as the appropriate fractional analogue of a reflecting boundary condition in the traditional diffusion equation. Wang et al, developed a fast finite difference method for space‐fractional diffusion equations with fractional boundary conditions. Liu et al, developed an implicit finite difference method for fractional advection‐dispersion equations with fractional boundary conditions.…”

Alternating-direction implicit finite difference methods for a new two-dimensional two-sided space-fractional diffusion equation

“…are similar to the definitions of the x direction. As we cannot easily get the explicit analytical solutions of the fractional equations, so many researchers resort to their numerical solutions [4][5][6][7][8][9][10].…”

Augmented upwind numerical schemes for a fractional advection-dispersion equation in fractured groundwater systems

“…The development and analysis of numerical solutions for the fractional advectiondispersion equation has remained relevant, to continually pursue improved numerical solution methods [32,34,31,17,2,3,7,13,18]. Upwind finite difference numerical methods have been applied to fractional partial differential equations [38,36,37], and recently [1] presented augmented upwind schemes for the advectiondispersion equation with local operators, where an upwind Crank-Nicolson and weighted upwind-downwind finite difference schemes were developed.…”