New numerical methods for the Riesz space fractional partial differential equations

In this paper, we consider the numerical solution of the Riesz space fractional diffusion equation and advection–dispersion equation. First, a system of ordinary differential equations is obtained from the above equations with respect to the space variable by using the improved matrix transform method. Furthermore, we use the (2,2)(2,2) Pade approximation to compute the exponential matrix in the analytic solution of the ordinary differential equation, and get two difference schemes. Second, using the matrix analysis method, we prove that the two difference schemes are unconditionally stable. Finally, some numerical results are given, which demonstrate the effectiveness of the two difference schemes.