Meshless spline-based DQ methods of high-dimensional space–time fractional advection–dispersion equations for fluid flow in heterogeneous porous media

Abstract

Fractional advection–dispersion equations (ADEs) appear to have great potential to predict various non-Fickian dispersion processes as the anomalous transport in surface and subsurface water. In this paper, a class of meshless spline-based differential quadrature (DQ) approaches is proposed to study the high-dimensional space–time fractional ADEs. These methods are established through the design of an efficient spline-based DQ approximation for the Riemann–Liouville fractional derivative in space with the cubic B-splines being trial functions. A class of high-order finite difference (FD) schemes is constructed to discretize the Caputo fractional derivative in time and then the stability, convergence are rigorously analyzed. The proposed DQ methods not only have advantages of high accuracy, strong flexibility and the ease of use, but also enjoy small computational cost. Illustrative experiments are carried out, and the simulation of a rotating non-Gaussian concentration hump in advection-dominated flow governed by the fractional derivatives is performed to validate its reliability and efficiency.

Keywords

• Fractional advection–dispersion equation
• Cubic B-spline
• Differential quadrature
• Convergence