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

We consider second-order differential systems where high-frequency oscillations are generated by a linear part. We present a frequency expansion of the solution, and we discuss two invariants of the system that determine the coefficients of the frequency expansion. These invariants are related to the total energy and the oscillatory harmonic energy of the original system. For the numerical solution we study a class of symmetric methods that discretize the linear part without error. We are interested in the case where the 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 frequency expansion where the coefficients are the solution of a modified system. This allows us to prove the near-conservation of the total and the oscillatory energy over very long time intervals.