In recent years, fractional differential equations have been extensively applied to model various complex dynamic systems. The studies on highly accurate and efficient numerical methods for fractional differential equations have become necessary. In this paper, an effective recurrence algorithm for computing both the fractional Riemann-Liouville and Caputo derivatives is proposed, and then spectral collocation methods based on the algorithm are investigated for solving fractional differential equations. By the recurrence method, the numerical stability with respect to $N$, the number of collocation points, can be improved remarkably in comparison with direct algorithm. Its robustness ensures that a highly accurate spectral collocation method can be applied widely to various fractional differential equations.