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.