Multirate Runge-Kutta for Nonlinearly Partitioned Systems
Tommaso Buvoli, Brian K. Tran, Ben S. Southworth
TL;DR
The paper tackles multirate time integration for nonlinear splittings by introducing multirate NPRK (MR-NPRK), which allows arbitrary nonlinear partitions $F(u,v)$ rather than linear additive or component splits. It develops order and stability theory for MR-NPRK, including joint stability analyses, and proposes IMEX MR-NPRK methods with an implicit-wrapping construction that uses a user-specified explicit method for the fast scale while placing implicit stages around the explicit ones. The authors derive second- and third-order MR-NPRK methods, establish sparsity patterns that reduce computational cost, and provide practical implementation guidance to achieve low memory usage. Numerical experiments on a Burgers-type nonlinear diffusion problem validate the methods and show improved efficiency and stability when leveraging nonlinear partitioning compared with single-rate NPRK and explicit schemes, highlighting potential benefits for complex multiphysics applications such as radiation hydrodynamics.
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 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.