Long-term analysis of numerical integrators for oscillatory Hamiltonian   systems under minimal non-resonance conditions
  For trigonometric and modified trigonometric integrators applied to oscillatory Hamiltonian differential equations with one or several constant high frequencies, near-conservation of the total and oscillatory energies are shown over time scales that cover arbitrary negative powers of the step size. This requires non-resonance conditions between the step size and the frequencies, but in contrast to previous results the results do not require any non-resonance conditions among the frequencies. The proof uses modulated Fourier expansions with appropriately modified frequencies. 
David Cohen, Ludwig Gauckler, Ernst Hairer, Christian Lubich
math.NA