  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.
