A one-dimensional spatial fractional avdvection dispersion equation is considered in this paper.The finite difference scheme with the shifted Grünwald formula is firstly applied to discrete the spatial derivative of the equation.Then the temporal discretization by the boundary value methods can transfer the equation to a large linear system.The GMRES iterative method is employed to solving the resulting linear system and a preconditioner is constructed to accelerate the convergence.Numerical examples demonstrate the superiority of this algorithm.