Long-term analysis of the Störmer–Verlet method for Hamiltonian systems with a solution-dependent high frequency

Abstract

The long-time behaviour of the Stormer---Verlet---leapfrog method is studied when this method is applied to highly oscillatory Hamiltonian systems with a slowly varying, solution-dependent high frequency. Using the technique of modulated Fourier expansions with state-dependent frequencies, which is newly developed here, the following results are proved: the considered Hamiltonian systems have the action as an adiabatic invariant over long times that cover arbitrary negative powers of the small parameter. The Stormer---Verlet method approximately conserves a modified action and a modified total energy over a long time interval that covers a negative integer power of the small parameter. This power depends on the size of the product of the stepsize with the high frequency.