多速率无穷小方法的自适应时间步长控制
Adaptive time step control for multirate infinitesimal methods
摘要：
多速率方法已经使用了几十年来时间演变的初始值问题，其中不同的组件在不同的时间尺度上演变，因此使用不同的步长为这些组件可以导致增加的计算效率。通常，这些方法基于实验或稳定性考虑来选择这些不同的步长。对于在单个时间尺度上演化的问题，努力控制局部时间误差的自适应方法被广泛用于以最小的计算工作量实现所需精度的数值结果，同时减轻对具有不同时间步长的手动实验的需要。然而，有一个显着的差距，在出版记录的自适应时间步长控制器的多速率方法的发展。本文将Gustafsson（1994）的单采样率控制器的工作推广到多采样率方法的设定。具体而言，我们开发的控制器的基础上多项式逼近的"快"和"慢"的时间尺度内的多速率无穷小（MRI）方法的主要误差函数。此外，我们还研究了各种方法，用于估计MRI方法中每个时间尺度所产生的误差。然后，我们数值评估所提出的多速率控制器和误差估计策略的多速率测试问题的范围内，比较其性能对估计的最佳性能。通过这项工作，我们结合了这些方法中最有性能的一组多速率自适应时间步长控制器，以最小的计算工作量鲁棒地实现所需的解决方案的精度。
英文摘要：
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. 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 multirate infinitesimal (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.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
