Analysis for parareal algorithms applied to Hamiltonian differential equations

Long-time integrations are an important issue in the numerical solution of Hamiltonian systems. They are time consuming and it is natural to consider the use of parallel architectures for reasons of efficiency. In this context the parareal algorithm has been proposed by several authors.The present work is a theoretical study of the parareal algorithm when it is applied to Hamiltonian differential equations. The idea of backward error analysis is employed to get insight into the long-time behavior of numerical approximations. One of the main results is that convergence of the parareal iterations restricts the length of the time window. For nearly integrable systems its length is bounded by the square root of the inverse of the accuracy of the coarse integrator. The theoretical bounds are confirmed by numerical experiments.