A stable explicitly solvable numerical method for the Riesz fractional advection-dispersion equations

In this paper, we present a finite difference scheme for solving the Riesz fractional advection-dispersion equations (RFADEs). The scheme is obtained by using asymmetric discretization technique and modify the shifted Grünwald approximation to fractional derivative. By calculating the unknowns in differential nodal-point sequences at the odd and even time-levels, the discrete solution of the scheme can be obtained explicitly. The computational cost for the scheme at each time step can be O(Klog K) by using the fast matrix-vector multiplication with the help of Toeplitz structure, where K is the number of unknowns. We prove that the scheme is solvable and unconditionally stable. We derive the error estimates in discrete l2-norm, which is optimal in some cases. Numerical examples are presented to verify our theoretical results.