标题： 多速率Runge-Kutta用于非线性划分系统 显示英文标题

标题： Multirate Runge-Kutta for Nonlinearly Partitioned Systems

摘要： 多速率积分是一种日益重要的工具，使科学家能够模拟多物理系统。 现有的多速率方法适用于可以通过加性或分量划分线性分离快变和慢变变量的方程。 然而，在实际应用中，这一假设并不总是成立。 基于最近开发的一类非线性划分Runge-Kutta（NPRK）方法，我们构建了一个多速率NPRK（MR-NPRK）框架，允许演化算子进行任意非线性分裂。 我们讨论了阶条件，形式化了不同时间尺度之间的耦合类型，并分析了MR-NPRK方法的联合线性稳定性。 然后，我们引入了一类2阶和3阶方法，称为“隐式包裹”多速率方法，它结合了用户指定的用于积分快时间尺度的显式方法和多个慢隐式阶段。 这些方法旨在在算法上简单，具有低内存成本和最少的操作符评估。 最后，我们进行了数值实验以验证我们提出的方法，并展示了对非线性划分进行多速率化的优点。 显示英文摘要

摘要： 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.