Symmetric parareal algorithms for Hamiltonian systems

¹ LSEC, Institute of Computational Mathematics and Scientific/Engineering Computing, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, China
daixy@lsec.cc.ac.cn
² École Nationale des Ponts et Chaussées, 6 et 8 avenue Blaise Pascal, 77455 Marne-La-Vallée Cedex 2, France
lebris@cermics.enpc.fr; legoll@lami.enpc.fr
³ INRIA Rocquencourt, MICMAC team-project, Domaine de Voluceau, B.P. 105, 78153 Le Chesnay Cedex, France
⁴ UPMC Univ. Paris 06, UMR 7598, Laboratoire J.-L. Lions, Boîte courrier 187, 75252 Paris Cedex 05, France
maday@ann.jussieu.fr
⁵ Division of Applied Mathematics, Brown University, Providence, RI, USA

Received: 24 November 2010
Revised: 14 February 2012

Abstract

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.