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 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 epsilon, in the case that the solution remains bounded as epsilon --> 0. We show that in general one must require that k(2)/epsilon 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(epsilon). The same holds for higher-order collocation at Gaussian points. The encountered restrictions on k are typically still better than the corresponding ones for explicit schemes.