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

In this paper, we propose high-order numerical methods for the Riesz space fractional advection-dispersion equations (RSFADE) on a finite 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 α∈(0,1) and β∈(1,2], respectively. Firstly, we utilize the weighted and shifted Grünwald difference operators to approximate the Riesz fractional derivative and present the finite 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 𝒪(τ^2+h^2). Thirdly, we use the Richardson extrapolation method (REM) to improve the convergence order which can be 𝒪(τ^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.