Multirate Runge–Kutta for Nonlinearly Partitioned Systems

Abstract

Abstract. 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 componentwise 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 the joint linear stability of MR-NPRK methods. We then introduce a class of second- and third-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. Last, we conduct numerical experiments to validate our proposed methods and show the benefits of multirating a nonlinear partition. Reproducibility of computational results. This paper has been awarded the “SIAM Reproducibility Badge: Code and data available” as a recognition that the authors have followed reproducibility principles valued by SISC and the scientific computing community. Code and data that allow readers to reproduce the results in this paper are available at https://github.com/buvoli/nprk-mr-intro/ and in the supplementary materials ( M174737_Supplementary_Materials_1.pdf [364KB], nprk-mr-intro-main.zip [130KB]). [Formula: see text]

The authors' abstract, as published at the source. SIAM Journal on Scientific Computing, 2026 · DOI ↗