We will present a class of numerical methods (based on the trigonometric methods) for such Hamiltonian problems. We will then present a frequency expansion of the numerical solution: the modulated Fourier expansion. The system that determines the coefficients of this expansion has two formal invariants which are related to the total energy and the oscillatory energy of the original system. This allows us to prove the near-conservation of the total and the oscillatory energy for the numerical schemes over very long time intervals.
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