We present an approach for the efficient implementation of self-adjusting multi-rate Runge-Kutta methods and we extend the previously available stability analyses of these methods to the case of an arbitrary number of sub-steps for the active components. We propose a physically motivated model problem that can be used to assess the stability of different multi-rate versions of standard Runge-Kutta methods and the impact of different interpolation methods for the latent variables. Finally, we present the results of several numerical experiments, performed with implementations of the proposed methods in the framework of the \textit{OpenModelica} open-source modelling and simulation software, which demonstrate the efficiency gains deriving from the use of the proposed multi-rate approach for physical modelling problems with multiple time scales.

翻译：本文提出了一种高效实现自适应多速率龙格-库塔方法的技术路径，并将该类方法已有的稳定性分析扩展到活跃分量采用任意数量子步的情形。我们构建了一个基于物理意义的模型问题，该问题可用于评估标准龙格-库塔方法不同多速率版本的稳定性，以及隐变量不同插值方法的影响。最后，我们展示了在开源建模与仿真软件\textit{OpenModelica}框架下，通过实现所提方法开展的多项数值实验的结果。这些结果证明了将所提出的多速率方法应用于具有多时间尺度物理建模问题时所能获得的效率提升。