Multirate methods have been used for decades to temporally evolve initial-value problems in which different components evolve on distinct time scales, and thus use of different step sizes for these components can result in increased computational efficiency. Generally, such methods select these different step sizes based on experimentation or stability considerations. Meanwhile for problems that evolve on a single time scale, adaptivity approaches that strive to control local temporal error are widely used to achieve numerical results of a desired accuracy with minimal computational effort, while alleviating the need for manual experimentation with different time step sizes. However, there is a notable gap in the publication record on the development of adaptive time-step controllers for multirate methods. In this paper, we extend the single-rate controller work of Gustafsson (1994) to the multirate method setting. Specifically, we develop controllers based on polynomial approximations to the principal error functions for both the "fast" and "slow" time scales within infinitesimal multirate (MRI) methods. We additionally investigate a variety of approaches for estimating the errors arising from each time scale within MRI methods. We then numerically evaluate the proposed multirate controllers and error estimation strategies on a range of multirate test problems, comparing their performance against an estimated optimal performance. Through this work, we combine the most performant of these approaches to arrive at a set of multirate adaptive time step controllers that robustly achieve desired solution accuracy with minimal computational effort.

翻译：数十年来一直使用多种方法来临时演变初始值问题,其中不同组成部分在不同的时间尺度上演变,因此对这些组成部分使用不同的职级大小可以提高计算效率。一般而言,这些方法根据实验或稳定性因素选择不同的职级大小。与此同时,对于在单一时间尺度上演变的问题,努力控制当地时间误差的适应性方法被广泛用于在最小的计算努力下实现理想准确度的数字结果,同时减轻以不同时间级尺进行人工实验的需要。然而,在开发适应性时级控制器以用于多种方法方面,出版物记录中存在明显差距。在本文件中,我们将Gustfsson(1994年)的单级控制器工作扩展至多级方法设置。具体地说,我们在“快”和“慢”时间尺度上的主要误差函数上,广泛使用适应性方法,在极小的多级多级计算方法中,我们进一步调查了各种办法,用以估计在多种时间尺度方法中出现的误差。我们随后用数字评估了拟议的多级制级控制器的单级控制器工作,然后用最稳健的计算方法,我们用最稳健的性的工作估计了这些测试方法,我们用最稳性性的工作方法来比较了各种的多级的绩效评估。