We consider the numerical solution of second-order differential systems, where high-frequency oscillations are generated by a linear part. We apply the numerical methods with step sizes whose product with the high frequency is in the range of linear stability, but otherwise it is not assumed to be small. For Hamiltonian systems, we study the long-time conservation of the total energy and of the oscillatory energy. The proofs are based on a frequency expansion of the exact and numerical solutions. Numerical experiments with a Fermi-Pasta-Ulam chain are presented.

This work is done in collaboration with Christian Lubich. It is documented in two papers that are available from http://www.unige.ch/math/folks/hairer/