Multi-rate Runge-Kutta methods: stability analysis and applications

TL;DR

This paper introduces a new stability analysis for multi-rate Runge-Kutta methods, proposes an efficient implementation, and demonstrates significant efficiency improvements in physical modeling applications with multiple time scales.

Contribution

It provides a comprehensive stability analysis for all standard multi-rate Runge-Kutta methods and presents an efficient implementation within OpenModelica.

Findings

Enhanced stability assessment for multi-rate methods

Efficient implementation in OpenModelica

Demonstrated efficiency gains in physical simulations

Abstract

We present an approach for the efficient implementation of self-adjusting multi-rate Runge-Kutta methods and we introduce a novel stability analysis, that covers the multi-rate extensions of all standard Runge-Kutta methods and allows to assess the impact of different interpolation methods for the latent variables and of the use of an arbitrary number of sub-steps for the active variables. The stability analysis applies successfully to the model problem typically used in the literature for multi-rate methods. Furthermore,} we also propose a physically motivated model problem that can be used to assess stability to problems with purely imaginary eigenvalues and in situations closer to those arising in applications. Finally, we present an efficient implementation of multi-rate Runge-Kutta methods in the framework of the OpenModelica open-source modelling and simulation software. Results of several numerical experiments, performed with this implementation of the proposed methods, demonstrate the efficiency gains deriving from the use of the proposed multi-rate approach for physical modelling problems with multiple time scales.