Mixed order and multirate variational integrators for the simulation of dynamics on different time scales

Mechanical systems with dynamics on varying time scales, e.g. including highly oscillatory motion, impose challenging questions for numerical integration schemes. High resolution is required to guarantee a stable integration of the fast frequencies. However, for the simulation of the slow dynamics, integration with a lower resolution is accurate enough - and computationally cheaper, especially for costly function evaluations. Two approaches are presented, a mixed order Galerkin variational integrator and a multirate variational integrator, and analysed with respect to the preservation of invariants, computational costs, accuracy and linear stability.