Parareal and Spectral Deferred Corrections

A new class of iterative time parallel methods for initial value ordinary differential equations are developed. Methods based on a parallel variation of spectral deferred corrections (SDC) are compared and contrasted with the parareal method. It is shown that there is a strong similarity between the serial step in the parareal algorithm and the correction step in the SDC method. This observation is used to construct a hybrid strategy combining features of both the parareal and SDC methods which can significantly reduce the computational cost of each iteration compared to parareal. A numerical example is presented to compare the effectiveness of the hybrid strategies.