Integrators for highly oscillatory Hamiltonian systems: Anhomogenization approach

Claude Le Bris; Frédéric Legoll

doi:10.3934/dcdsb.2010.13.347

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2010, Volume 13, Issue 2: 347-373. Doi: 10.3934/dcdsb.2010.13.347

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Integrators for highly oscillatory Hamiltonian systems: An homogenization approach

Claude Le Bris^1, and
Frédéric Legoll^2,

1.
Université Paris-Est, CERMICS, Ecole Nationale des Ponts et Chaussées, 6 et 8 avenue Blaise Pascal, 77455 Marne-la-Valléauthore Cedex 2
2.
Université Paris-Est, Institut Navier, LAMI, Ecole Nationale des Ponts et Chaussées, 6 et 8 avenue Blaise Pascal, 77455 Marne-la-Vallée Cedex 2

Received: November 30, 2008

Revised: April 30, 2009

Published: August 2010

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Abstract

We introduce a systematic way to build symplectic schemes for the numerical integration of a large class of highly oscillatory Hamiltonian systems. The bottom line of our construction is to consider the Hamilton-Jacobi form of the Newton equations of motion, and to perform a two-scale expansion of the solution, for small times and high frequencies. The approximation obtained for the solution is then used as a generating function, from which the numerical scheme is derived. Several options for the derivation are presented. The various integrators obtained are tested and compared to several existing algorithms. The numerical results demonstrate their efficiency.

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Mathematics Subject Classification: Primary: 70K70, 37M15, 70H20, 65P10.

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