High-order numerical methods for the Riesz space fractional advection-dispersion equations

TL;DR

This paper develops high-order finite difference methods for solving Riesz space fractional advection-dispersion equations, demonstrating stability, convergence, and improved accuracy through Richardson extrapolation with numerical validation.

Contribution

The paper introduces a stable, convergent finite difference scheme for RSFADE using weighted and shifted Grünwald operators, and enhances accuracy with Richardson extrapolation.

Findings

The scheme is unconditionally stable and convergent with second-order accuracy.

Richardson extrapolation improves convergence to fourth order.

Numerical examples confirm the effectiveness and theoretical predictions.

Abstract

In this paper, we propose high-order numerical methods for the Riesz space fractional advection-dispersion equations (RSFADE) on a {f}inite domain. The RSFADE is obtained from the standard advection-dispersion equation by replacing the first-order and second-order space derivative with the Riesz fractional derivatives of order $\alpha\in(0,1)$ and $\beta\in(1,2]$, respectively. Firstly, we utilize the weighted and shifted Gr\"unwald difference operators to approximate the Riesz fractional derivative and present the {f}inite difference method for the RSFADE. Specifically, we discuss the Crank-Nicolson scheme and solve it in matrix form. Secondly, we prove that the scheme is unconditionally stable and convergent with the accuracy of $\mathcal {O}(\tau^2+h^2)$. Thirdly, we use the Richardson extrapolation method (REM) to improve the convergence order which can be $\mathcal {O}(\tau^4+h^4)$. Finally, some numerical examples are given to show the effectiveness of the numerical method, and the results are excellent with the theoretical analysis.