On Some Difficulties in Integrating Highly Oscillatory Hamiltonian Systems

. The numerical integration of highly oscillatory Hamiltonian systems, such as those arising in molecular dynamics or Hamiltonian partial differential equations, is a challenging task. Various methods have been suggested to overcome the step-size restrictions of explicit methods such as the Verlet method. Among these are multiple-time-stepping, constrained dynamics, and implicit methods. In this paper, we investigate the suitability of time-reversible, semi-implicit methods. Here semi-implicit means that only the highly oscillatory part is integrated by an implicit method such as the midpoint method or an energy-conserving variant of it. The hope is that such methods will allow one to use a step-size k which is much larger than the period " of the fast oscillations. However, our results are not encouraging. Even in the absence of resonancetype instabilities, we show that in general one must require that k 2 =" be small enough. Otherwise the method might become unstable and/or it migh..