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Symmetric parareal algorithms for Hamiltoniansystems

Published online by Cambridge University Press:  04 March 2013

Xiaoying Dai ,
Claude Le Bris ,
Frédéric Legoll  and
Yvon Maday
Xiaoying Dai
Affiliation:
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 UPMC Univ. Paris 06, UMR 7598, Laboratoire J.-L. Lions, Boîte courrier 187, 75252 Paris Cedex 05, France; maday@ann.jussieu.fr
Claude Le Bris
Affiliation:
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 INRIA Rocquencourt, MICMAC team-project, Domaine de Voluceau, B.P. 105, 78153 Le Chesnay Cedex, France
Frédéric Legoll
Affiliation:
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 INRIA Rocquencourt, MICMAC team-project, Domaine de Voluceau, B.P. 105, 78153 Le Chesnay Cedex, France
Yvon Maday
Affiliation:
UPMC Univ. Paris 06, UMR 7598, Laboratoire J.-L. Lions, Boîte courrier 187, 75252 Paris Cedex 05, France; maday@ann.jussieu.fr Division of Applied Mathematics, Brown University, Providence, RI, USA
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Abstract

The parareal in time algorithm allows for efficient parallel numerical simulations oftime-dependent problems. It is based on a decomposition of the time interval intosubintervals, and on a predictor-corrector strategy, where the propagations over eachsubinterval for the corrector stage are concurrently performed on the different processorsthat are available. In this article, we are concerned with the long time integration ofHamiltonian systems. Geometric, structure-preserving integrators are preferably employedfor such systems because they show interesting numerical properties, in particularexcellent preservation of the total energy of the system. Using a symmetrization procedureand/or a (possibly also symmetric) projection step, we introduce here several variants ofthe original plain parareal in time algorithm [L. Baffico, et al. Phys. Rev. E66 (2002) 057701; G. Bal and Y. Maday, A parareal timediscretization for nonlinear PDE’s with application to the pricing of an American put, inRecent developments in domain decomposition methods, Lect.Notes Comput. Sci. Eng. 23 (2002) 189–202; J.-L. Lions, Y. Madayand G. Turinici, C. R. Acad. Sci. Paris, Série I 332 (2001)661–668.] that are better adapted to the Hamiltonian context. These variants arecompatible with the geometric structure of the exact dynamics, and are easy to implement.Numerical tests on several model systems illustrate the remarkable properties of theproposed parareal integrators over long integration times. Some formal elements ofunderstanding are also provided.

Keywords

Parallel integratorsHamiltonian dynamicslong-time integrationsymmetric algorithmssymmetric projectiongeometric integration
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 47 , Issue 3 , May 2013 , pp. 717 - 742
Copyright
© EDP Sciences, SMAI, 2013

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