Integrators for highly oscillatory Hamiltonian systems: An homogenization approach

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March 2010, 13(2): 347-373. doi: 10.3934/dcdsb.2010.13.347

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, France
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, France

Received December 2008 Revised May 2009 Published December 2009

Abstract
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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.

Keywords: Numerical schemes, two-scale expansion, highly oscillatory Hamiltonian systems, symplectic algorithms., generating function.

Mathematics Subject Classification: Primary: 70K70, 37M15, 70H20, 65P1.

Citation: Claude Le Bris, Frédéric Legoll. Integrators for highly oscillatory Hamiltonian systems: An homogenization approach. Discrete & Continuous Dynamical Systems - B, 2010, 13 (2) : 347-373. doi: 10.3934/dcdsb.2010.13.347

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