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The parareal in time algorithm allows to efficiently use parallel computingfor the simulation of time-dependent problems. It is based on a decompositionof the time interval into subintervals, and on a predictor-corrector strategy,where the propagations over each subinterval for the corrector stage areconcurrently performed on the processors. In this article, we are concernedwith the long time integration of Hamiltonian systems. Geometric,structure-preserving integrators are preferably employed for such systemsbecause they show interesting numerical properties, in particular excellentpreservation of the total energy of the system. Using a symmetrizationprocedure and/or a (possibly also symmetric) projection step, we introduce hereseveral variants of the original plain parareal in time algorithm [Lions, Madayand Turinici 2001, Baffico, Bernard, Maday, Turinici and Zerah 2002, Bal andMaday 2002] that are better adapted to the Hamiltonian context. These variantsare compatible with the geometric structure of the exact dynamics, and are easyto implement. Numerical tests on several model systems illustrate theremarkable properties of the proposed parareal integrators over longintegration times. Some formal elements of understanding are also provided.

Symmetric parareal algorithms for Hamiltonian systems