A Fast Method for Solving Advection-dispersion Equations

In this paper, a fast numerical algorithm for Riesz-space fractional advection-dispersion equation is considered. Firstly, the spatial variables of the equation are discretized using a fractional central difference scheme, while the time variables are discretized via the Crank-Nicolson scheme. This yields a second-order finite difference scheme, which further results in a linear system after discretizing the equation. Secondly, since the coefficient matrix of this linear system has a Toeplitz structure, a \boldsymbol\tau preconditioner based on sine transform is proposed to accelerate the solution of the linear system. The convergence of the preconditioned matrix is analyzed theoretically, and we prove that the spectrum of the preconditioned matrix is clustered in the interval (1/2,3/2). Finally, the second-order accuracy of the algorithm and the effectiveness of the preconditioner are verified by numerical experiments.