  This work focuses on the construction of a new class of fourth-order accurate
methods for multirate time evolution of systems of ordinary differential
equations. We base our work on the Recursive Flux Splitting Multirate (RFSMR)
version of the Multirate Infinitesimal Step (MIS) methods and use recent
theoretical developments for Generalized Additive Runge-Kutta methods to
propose our higher-order Relaxed Multirate Infinitesimal Step extensions. The
resulting framework supports a range of attractive properties for multirate
methods, including telescopic extensions, subcycling, embeddings for temporal
error estimation, and support for changes to the fast/slow time-scale
separation between steps, without requiring any sacrifices in linear stability.
In addition to providing rigorous theoretical developments for these new
methods, we provide numerical tests demonstrating convergence and efficiency on
a suite of multirate test problems.
