High-order algorithms for Riesz derivative and their applications (II)

In this paper, we firstly develop two high-order approximate formulas for the Riesz fractional derivative. Secondly, we propose a temporal second order numerical method for a fractional reaction-dispersion equation, where we discretize the Riesz fractional derivative by using two numerical schemes. We prove that the numerical methods for a spatial Riesz fractional reaction dispersion equation are both unconditionally stable and convergent, and the orders of convergence are O(τ2+h6)O(τ2+h6) and O(τ2+h8)O(τ2+h8), in which τ and h are spatial and temporal step sizes, respectively. Finally, we test our numerical schemes and observe that the numerical results are in good agreement with the theoretical analysis.