Long-time Integration of Non-Stiff and Oscillatory Hamiltonian Systems

For the long-time integration of Hamiltonian systems (e.g., planetary motion, molecular dynamics simulation) much insight can be gained with 'backward error analysis'. For example, it explains why symplectic integrators nearly conserve the energy, and why they have at most a linear error growth for integrable systems. This theory breaks down in the presence of high oscillations, when the product of the step size with the highest frequency is not small. In the situation, where the high oscillations originate from a linear part in the differential equation, the theory of 'modulated Fourier expansion' yields much information on the long-time behavior of analytic and numerical solutions, in particular for large step sizes. After a short review on backward error analysis, the main ideas and results of modulated Fourier expansions are presented. They are then applied to get new insight into the distribution of the modal energy spectrum of the Fermi-Pasta-Ulam problem.