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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 ( τ 2 + h 6 ) and O ( τ 2 + h 8 ) , 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. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we develop two classes of finite difference schemes for the reaction–subdiffusion equations by using a mixed spline function in space direction, forward and backward difference in time direction, respectively. It has been shown that some of the previous known difference schemes can be derived from our schemes if we suitably choose the spline parameters. By Fourier method, we prove that one class of difference scheme is unconditionally stable and convergent, the other is conditionally stable and convergent. Finally, some numerical results are provided to demonstrate the effectiveness of the proposed difference schemes. [Copyright &y& Elsevier]