Variational multirate integrators

TL;DR

This paper introduces variational multirate integrators designed for systems with multiple time scales, achieving high accuracy and stability while preserving geometric properties through a variational framework.

Contribution

It develops a novel variational multirate integration method that is symplectic, momentum preserving, and adaptable for different time scales, with proven convergence.

Findings

Integrators are symplectic and momentum preserving.

The scheme exhibits good energy behavior.

Convergence order depends on discrete approximations.

Abstract

The simulation of systems that act on multiple time scales is challenging. A stable integration of the fast dynamics requires a highly accurate approximation whereas for the simulation of the slow part, a coarser approximation is accurate enough. With regard to the general goals of any numerical method, high accuracy and low computational costs, a popular approach is to treat the slow and the fast part of a system differently. Embedding this approach in a variational framework is the keystone of this work. By paralleling continuous and discrete variational multirate dynamics, integrators are derived on a time grid consisting of macro and micro time nodes that are symplectic, momentum preserving and also exhibit good energy behaviour. The choice of the discrete approximations for the action determines the convergence order of the scheme as well as its implicit or explicit nature for the different parts of the multirate system. The convergence order is proven using the theory of variational error analysis. The performance of the multirate variational integrators is demonstrated by means of several examples.