On some difficulties in integrating highly oscillatory Hamiltonian systems. (English) Zbl 0971.65113

Ascher, Uri M.; Reich, Sebastian

Deuflhard, Peter (ed.) et al., Computational molecular dynamics: challenges, methods, ideas. Proceedings of the 2nd international symposium on Algorithms for Macromolecular modelling, Berlin, Germany, May 21-24, 1997. Berlin: Springer. Lect. Notes Comput. Sci. Eng. 4, 281-296 (1999).

Summary: 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 \(\varepsilon\) of the fast oscillations.
However, our results are not encouraging. Even in the absence of resonance-type instabilities, we show that in general one must require that \(k^2/\varepsilon\) be small enough. Otherwise the method might become unstable and/or it might lead to a wrong approximation of the slowly varying solution components. The latter situation might, in some cases, even require that \(k/\varepsilon\) be small in order to avoid this danger. While certain (semi-implicit) energy conserving methods prove to be robust for some model problems, they may also yield deceptively-looking, wrong solutions for other simple model problems, in circumstances where the corresponding constrained dynamics formulation may not be easily derived and used.
For the entire collection see [Zbl 0904.00046].

MSC:

65P10	Numerical methods for Hamiltonian systems including symplectic integrators
37M15	Discretization methods and integrators (symplectic, variational, geometric, etc.) for dynamical systems

Keywords:

highly oscillatory Hamiltonian systems; molecular dynamics; Verlet method; time-reversible, semi-implicit methods