Spectral Deferred Correction Methods for Fractional Differential Equations

In this paper, we propose and analyze a spectral deferred correction method for the fractional differential equation of order alpha. The proposed method is based on a well-known finite difference method of (2-alpha)-order, see [Sun and Wu, Appl. Numer. Math., 56(2), 2006] and [Lin and Xu, J. Comput. Phys., 225(2), 2007], for prediction of the numerical solution, which is then corrected through a spectral deferred correction method. In order to derive the convergence rate of the prediction-correction iteration, we first derive an error estimate for the (2-alpha)-order finite difference method on some non-uniform meshes. Then the convergence rate of orders O (tau((2-alpha)(p+1)) and O (tau((2-alpha)+p)) of the overall scheme is demonstrated numerically for the uniform mesh and the Gauss-Lobatto mesh respectively, where tau is the maximal time step size and p is the number of correction steps. The performed numerical test confirms the efficiency of the proposed method.