We consider trigonometric and modified trigonometric integrators applied to oscillatory Hamiltonian differential equations with several constant high frequencies. Examples are the impulse method which coincides with the method of Deuflhard, the mollified impulse method, the St"ormer--Verlet method and an IMEX method.

For the exact solution of these equations, one has exact conservation of the total energy and long-time near-conservation of the oscillatory energy. For the considered numerical methods, we will show in the talk near-conservation of the total and oscillatory energies over long time scales that cover arbitrary negative powers of the step size.

A main issue in such long-time results are resonances between the frequencies and resonances between the frequencies and the time step-size. The presented results require non-resonance conditions between the time step-size and the frequencies, but in contrast to previous results they do not require any non-resonance conditions among the frequencies.

This is joint work with Ernst Hairer (Geneva) and Christian Lubich (Tübingen).

Slides