Symplectic schemes for highly oscillatory Hamiltonian systems with varying fast frequencies

Matthew Dobson; Claude Le Bris; Frédéric Legoll

doi:10.1016/j.crma.2010.08.005

Numerical Analysis

Symplectic schemes for highly oscillatory Hamiltonian systems with varying fast frequencies
[Intégrateurs symplectiques pour des systèmes Hamiltoniens hautement oscillants avec fréquences rapides variables]

Présenté par : Olivier Pironneau

Matthew Dobson ¹ ; Claude Le Bris ^1,² ; Frédéric Legoll ^3,²

¹ CERMICS, École des Ponts ParisTech, 77455 Marne-La-Vallée cedex 2, France
² INRIA Rocquencourt, MICMAC project, domaine de Voluceau, B.P. 105, 78153 Le Chesnay cedex, France
³ Institut Navier, LAMI, École des Ponts ParisTech, 77455 Marne-La-Vallée cedex 2, France

Comptes Rendus. Mathématique, Volume 348 (2010) no. 17-18, pp. 1033-1038.

Résumé
Abstract

Nous dérivons des intégrateurs symplectiques pour une classe de systèmes Hamiltoniens hautement oscillants. L'approche est basée sur un développement à deux échelles de la solution de l'équation de Hamilton–Jacobi associée à la dynamique originale. Cette Note présente une extension des techniques précédemment introduites dans Le Bris et Legoll (2007, 2010) [10,11] au cas où les fréquences rapides du système ne sont plus constantes, mais dépendent des variables lentes.

We derive symplectic integrators for a class of highly oscillatory Hamiltonian systems. Our approach is based upon a two-scale expansion of the solution to the Hamilton–Jacobi equation associated to the original dynamics. This Note presents an extension of the approach previously introduced in Le Bris and Legoll (2007, 2010) [10,11] to the case where the fast frequencies of the system, instead of being constant, explicitly depend on the slow degrees of freedom.

Reçu le : 2010-05-15
Accepté le : 2010-08-17
Publié le : 2010-09-01

DOI : 10.1016/j.crma.2010.08.005

Affiliations des auteurs :

Matthew Dobson ¹ ; Claude Le Bris ^{1, 2} ; Frédéric Legoll ^{3, 2}

¹ CERMICS, École des Ponts ParisTech, 77455 Marne-La-Vallée cedex 2, France
² INRIA Rocquencourt, MICMAC project, domaine de Voluceau, B.P. 105, 78153 Le Chesnay cedex, France
³ Institut Navier, LAMI, École des Ponts ParisTech, 77455 Marne-La-Vallée cedex 2, France

@article{CRMATH_2010__348_17-18_1033_0,
     author = {Matthew Dobson and Claude Le Bris and Fr\'ed\'eric Legoll},
     title = {Symplectic schemes for highly oscillatory {Hamiltonian} systems with varying fast frequencies},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {1033--1038},
     publisher = {Elsevier},
     volume = {348},
     number = {17-18},
     year = {2010},
     doi = {10.1016/j.crma.2010.08.005},
     language = {en},
}

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Matthew Dobson; Claude Le Bris; Frédéric Legoll. Symplectic schemes for highly oscillatory Hamiltonian systems with varying fast frequencies. Comptes Rendus. Mathématique, Volume 348 (2010) no. 17-18, pp. 1033-1038. doi : 10.1016/j.crma.2010.08.005. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2010.08.005/

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Cité par Sources :

^☆ This work was supported in part by the NSF Mathematical Sciences Postdoctoral Research Fellowship, by the INRIA under the grant “Action de Recherche Collaborative” HYBRID, and by the Agence Nationale de la Recherche, under grant ANR-09-BLAN-0216-01 (MEGAS).

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