🤖 AI Summary
Multirate problems with multiple time scales pose significant challenges in adaptively selecting time steps of varying rates while balancing accuracy and efficiency.
Method: This paper introduces two novel multirate adaptive controllers and constructs the first fifth-order embedded multirate exponential Runge–Kutta (MERK) method. Built upon the embedded multirate infinitesimal (MRI) framework, the method employs an explicit MERK nesting structure to enable efficient local error estimation and stepsize control, supporting high-accuracy adaptive integration across arbitrarily many time scales.
Contribution/Results: Theoretical analysis and benchmark tests demonstrate that the proposed method achieves high-order accuracy while substantially reducing computational cost. It exhibits superior flexibility and robustness compared to state-of-the-art multirate methods. Moreover, it provides a systematic, practical guideline for controller design and method selection in multirate time integration.
📝 Abstract
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 and see that they offer dramatically improved performance and flexibility, with each proposed family excelling on different types of multirate applications. 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.