Long-time energy conservation of numerical methods for oscillatory differential equations

We consider second-order differential systems where high-frequency oscillat ions are generated by a linear part. We present a frequency expansion of th e solution, and we discuss two invariants of the system that determine the coefficients of the frequency expansion. These invariants are related to th e total energy and the oscillatory harmonic energy of the original system. For the numerical solution we study a class of symmetric methods that discr etize the linear part without error. We are interested in the case where th e product of the step size with the highest frequency can be large. In the sense of backward error analysis we represent the numerical solution by a f requency expansion where the coefficients are the solution of a modified sy stem. This allows us to prove the near-conservation of the total and the os cillatory energy over very long time intervals.