Multirate Runge-Kutta for Nonlinearly Partitioned Systems

Multirate integration is an increasingly relevant tool that enables scientists to simulate multiphysics systems. Existing multirate methods are designed for equations whose fast and slow variables can be linearly separated using additive or component-wise partitions. However, in realistic applications, this assumption is not always valid. Building on the recently developed class of nonlinearly partitioned Runge-Kutta (NPRK) methods, we develop a framework for multirate NPRK (MR-NPRK) that allows for arbitrary nonlinear splittings of the evolution operator. We discuss order conditions, formalize different types of coupling between timescales, and analyze joint linear stability of MR-NPRK methods. We then introduce a class of 2nd- and 3rd-order methods, referred to as ``implicitly-wrapped'' multirate methods, that combine a user-specified explicit method for integrating the fast timescale with several slow implicit stages. These methods are designed to be algorithmically simple with low memory costs and minimal operator evaluations. Lastly, we conduct numerical experiments to validate our proposed methods and show the benefits of multirating a nonlinear partition.