The parareal in time algorithm allows for efficient parallel numerical simulations of
time-dependent problems. It is based on a decomposition of the time interval into
subintervals, and on a predictor-corrector strategy, where the propagations over each
subinterval for the corrector stage are concurrently performed on the different processors
that are available. In this article, we are concerned with the long time integration of
Hamiltonian systems. Geometric, structure-preserving integrators are preferably employed
for such systems because they show interesting numerical properties, in particular
excellent preservation of the total energy of the system. Using a symmetrization procedure
and/or a (possibly also symmetric) projection step, we introduce here several variants of
the original plain parareal in time algorithm [L. Baffico, et al. Phys. Rev. E
66 (2002) 057701; G. Bal and Y. Maday, A parareal time
discretization for nonlinear PDE’s with application to the pricing of an American put, in
Recent developments in domain decomposition methods, Lect.
Notes Comput. Sci. Eng. 23 (2002) 189–202; J.-L. Lions, Y. Maday
and G. Turinici, C. R. Acad. Sci. Paris, Série I 332 (2001)
661–668.] that are better adapted to the Hamiltonian context. These variants are
compatible with the geometric structure of the exact dynamics, and are easy to implement.
Numerical tests on several model systems illustrate the remarkable properties of the
proposed parareal integrators over long integration times. Some formal elements of
understanding are also provided.