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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 discouraging. 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.
The (implicit) midpoint scheme, like higher order Gauss-collocation schemes, is algebraically stable and symplectic, and it preserves quadratic integral invariants. It may appear particularly suitable for the numerical solution of highly oscillatory Hamiltonian systems, such as those arising in molecular dynamics or structural mechanics, because there is no stability restriction when it is applied to a simple harmonic oscillator. Although it is well-known that the midpoint scheme may also exhibit instabilities in various stiff situations, one might still hope for good results when resonance-type instabilities are avoided. In this paper we investigate the suitability of the midpoint scheme for highly oscillatory, frictionless mechanical systems, where the step-size $k$ is much larger than the system's small parameter $\varepsilon$, in case that the solution remains bounded as $\varepsilon \rightarrow 0$. We show that in general one must require that $k^2/\varepsilon$ be small enough, or else, even the errors in slowly varying quantities like the energy may grow undesirably (especially when fast and slow modes are tightly coupled) or, worse, the computation may yield misleading information. In some cases this may already happen when $k = O(\varepsilon )$. The same holds for higher order collocation at Gaussian points. The encountered restrictions on $k$ are still better than the corresponding ones for explicit schemes.