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.

翻译：模拟多时间尺度系统具有挑战性。快速动力学的稳定积分需要高精度近似，而对于慢变部分的模拟，较粗糙的近似已足够精确。考虑到数值方法的通用目标——高精度与低计算成本，一种流行的方法是对系统的慢变部分与快速部分进行差异化处理。将这种方法嵌入变分框架是本文工作的基石。通过建立连续与离散变分多速率动力学的对应关系，在由宏观与微观时间节点构成的时间网格上推导出积分器，这些积分器具有辛结构、动量守恒特性，并展现出良好的能量行为。对作用量离散近似方式的选择决定了格式的收敛阶，也决定了多速率系统不同部分的隐式或显式特性。利用变分误差分析理论证明了收敛阶。通过若干算例展示了多速率变分积分器的性能。