Multirate methods for ordinary differential equations

TL;DR

This survey reviews advanced multirate numerical methods for ordinary differential equations, highlighting techniques that adapt to different system activity levels to improve computational efficiency.

Contribution

It provides a comprehensive overview of current multirate schemes, including both basic and higher order methods, and discusses their theoretical foundations and practical implementations.

Findings

Multirate methods effectively exploit multiple time scales in ODEs.

Higher order multirate schemes generalize classical methods for improved accuracy.

The survey identifies key challenges and future directions in multirate method development.

Abstract

This survey provides an overview of state-of-the art multirate schemes, which exploit the different time scales in the dynamics of a differential equation model by adapting the computational costs to different activity levels of the system. We start the discussion with the straightforward approach based on interpolating and extrapolating the slow--fast coupling variables; the multirate Euler scheme, used as a base example, falls into this class. Next we discuss higher order multirate schemes that generalize classical singlerate linear multistep, Runge-Kutta, and extrapolation methods.