The midpoint scheme and variants for Hamiltonian systems: Advantages and pitfalls

The (implicit) midpoint scheme, like higher-order Gauss-collocation schemes , is algebraically stable and symplectic, and it preserves quadratic integr al invariants. It may appear particularly suitable for the numerical soluti on of highly oscillatory Hamiltonian systems, such as those arising in mole cular dynamics or structural mechanics, because there is no stability restr iction when it is applied to a simple harmonic oscillator. Although it is w ell known that the midpoint scheme may also exhibit instabilities in variou s stiff situations, one might still hope for good results when resonance-ty pe instabilities are avoided. In this paper we investigate the suitability of the midpoint scheme for hig hly oscillatory, frictionless mechanical systems, where the step size k is much larger than the system's small parameter epsilon, in the case that the solution remains bounded as epsilon --> 0. We show that in general one mus t require that k(2)/epsilon be small enough or else even the errors in slow ly varying quantities like the energy may grow undesirably (especially when fast and slow modes are tightly coupled) or, worse, the computation may yi eld misleading information. In some cases this may already happen when k = O(epsilon). The same holds for higher-order collocation at Gaussian points. The encountered restrictions on k are typically still better than the corr esponding ones for explicit schemes.