An Averaging Technique for Highly Oscillatory Hamiltonian Problems

In this paper, we are concerned with the numerical solution of highly oscillatory Hamiltonian systems with a stiff linear part. We construct an averaged system whose solution remains close to the exact one over bounded time intervals, possesses the same adiabatic and Hamiltonian invariants as the original system, and is nonstiff. We then investigate its numerical approximation through a method which combines a symplectic integration scheme and an acceleration technique for the evaluation of time averages developed in [E. Cances et al., Numer. Math., 100 (2005), pp. 211-232]. Eventually, we demonstrate the efficiency of our approach on two test problems with one or several frequencies.

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