Introduction

Here we consider the numerical energy behaviour for fixed positive hco in the interval of linear stability, for a class of nonlinear model problems with a single, constant high frequency uj. This problem class includes the Fermi-Pasta-Ulam model of alternating soft nonlinear and stiff linear springs. Although oscillations in the energy are of a size roughly proportional to (hio)2, the averages of the energy over finite time windows are preserved over very long time intervals. This is explained, and the deviation of the finite-time averages from the initial energy is explicitly determined in terms of certain modified energies which are well preserved over long times. The analysis uses only the symmetry of the method, not its symplecticness.