Multirate generalized additive Runge Kutta methods

TL;DR

This paper introduces a new class of multirate schemes derived from generalized additive Runge-Kutta methods, enabling different step sizes for various components while maintaining accuracy and stability.

Contribution

It develops a comprehensive multirate GARK framework that encompasses many existing schemes and extends their stability and accuracy properties.

Findings

Includes many well-known multirate schemes as special cases

Order conditions follow from GARK accuracy theory

Stability and monotonicity are inherited from base schemes

Abstract

This work constructs a new class of multirate schemes based on the recently developed generalized additive Runge-Kutta (GARK) methods (Sandu and Guenther, 2013). Multirate schemes use different step sizes for different components and for different partitions of the right-hand side based on the local activity levels. We show that the new multirate GARK family includes many well-known multirate schemes as special cases. The order conditions theory follows directly from the GARK accuracy theory. Nonlinear stability and monotonicity investigations show that these properties are inherited from the base schemes provided that additional coupling conditions hold.