Long-term analysis of numerical integrators for oscillatory Hamiltonian systems under minimal non-resonance conditions
D. Cohen, L. Gauckler, E. Hairer and Ch. Lubich
Abstract. 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. The results form numerical counterparts to the analytical result
of Gauckler, Hairer and Lubich [Commun. Math. Phys. 321 (2013), 803--815] and
Bambusi, Giorgilli, Paleari and Penati [Preprint (2014)], where long-time near-conservation
of the oscillatory energy along exact solutions is shown without any non-resonance condition.
Key Words. Oscillatory Hamiltonian systems, Modulated Fourier expansions,
Trigonometric integrators, St\"ormer-Verlet scheme, IMEX scheme,
Long-time energy conservation, Numerical resonances, Non-resonance condition.