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.

中文翻译：

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Riesz 空间分数阶对流-弥散方程的高阶数值方法

在本文中，我们针对 {f} 有限域上的 Riesz 空间分数阶对流-弥散方程 (RSFADE) 提出了高阶数值方法。RSFADE 是通过将一阶和二阶空间导数替换为 $\alpha\in(0,1)$ 和 $\beta\in(1, 2]$，分别。首先，我们利用加权和移位 Gr\"unwald 差分算子来逼近 Riesz 分数阶导数，并提出 RSFADE 的{f}有限差分方法。具体来说，我们讨论了 Crank-Nicolson 方案并求解其矩阵形式。其次，我们证明该方案是无条件稳定和收敛的，精度为$\mathcal {O}(\tau^2+h^2)$。第三，我们使用理查森外推法 (REM) 来改进收敛阶数，它可以是 $\mathcal {O}(\tau^4+h^4)$。最后，给出了一些数值算例，证明了该数值方法的有效性，与理论分析结果相吻合。