A Variational Approach to Multirate Integration for Constrained Systems

The simulation of systems with dynamics on strongly varying time scales is quite challenging and demanding with regard to possible numerical methods. A rather naive approach is to use the smallest necessary time step to guarantee a stable integration of the fast frequencies. However, this typically leads to unacceptable computational loads. Alternatively, multirate methods integrate the slow part of the system with a relatively large step size while the fast part is integrated with a small time step. In this work, a multirate integrator for constrained dynamical systems is derived in closed form via a discrete variational principle on a time grid consisting of macro and micro time nodes. Being based on a discrete version of Hamilton’s principle, the resulting variational multirate integrator is a symplectic and momentum preserving integration scheme and also exhibits good energy behaviour. Depending on the discrete approximations for the Lagrangian function, one obtains different integrators, e.g. purely implicit or purely explicit schemes, or methods that treat the fast and slow parts in different ways. The performance of the multirate integrator is demonstrated by means of several examples.