In this work we present two new families of multirate time step adaptivity controllers, that are designed to work with embedded multirate infinitesimal (MRI) time integration methods for adapting time steps when solving problems with multiple time scales. We compare these controllers against competing approaches on two benchmark problems, showing that the proposed methods offer dramatically improved performance and flexibility. The combination of embedded MRI methods and the proposed controllers enable adaptive simulations of problems with a potentially arbitrary number of time scales, achieving high accuracy while maintaining low computational cost. Additionally, we introduce a new set of embeddings for the family of explicit multirate exponential Runge--Kutta (MERK) methods of orders 2 through 5, resulting in the first-ever fifth-order embedded MRI method. Finally, we compare the performance of a wide range of embedded MRI methods on our benchmark problems to provide guidance on how to select an appropriate MRI method and multirate controller.

翻译：本文提出了两个新型多速率时间步长自适应控制器系列，专为配合嵌入式多速率无穷小（MRI）时间积分方法而设计，用于在求解多时间尺度问题时自适应调整时间步长。我们在两个基准问题上将这些控制器与现有方法进行对比，结果表明所提方法在性能和灵活性方面均有显著提升。嵌入式MRI方法与所提控制器的结合，使得对具有任意数量时间尺度问题的自适应模拟成为可能，在保持高精度的同时实现了较低的计算成本。此外，我们为2至5阶显式多速率指数龙格-库塔（MERK）方法系列引入了一套新的嵌入式构造，从而首次实现了五阶嵌入式MRI方法。最后，我们在基准问题上对比了多种嵌入式MRI方法的性能，为如何选择合适的MRI方法和多速率控制器提供了指导。