In this paper we extend the DGiT multirate framework, developed in Connors and Sockwell (2022) for scalar transmission problems, to a solid–solid interaction (SSI) problem involving two coupled elastic solids and a coupled air–sea model with the rotating, thermal shallow water equations. In so doing we aim to demonstrate the broad applicability of the mathematical theory and governing principles established in Connors and Sockwell (2022) to coupled problems characterized by subproblems evolving at different temporal scales. Multirate time integration algorithms employing different time steps, optimized for the dynamics of each subproblem, can significantly improve simulation efficiency for such coupled problems. However, development of multirate algorithms is a highly non-trivial task due to the coupling, which can impact accuracy, stability or other desired properties such as preservation of system invariants. DGiT provides a general template for multirate time integration that can achieve these properties. To elucidate the manner in which DGiT accomplishes this task, we fully detail each step in the application of the framework to the SSI and air–sea coupled problems. Numerical examples illustrate key properties of the resulting multirate schemes for both problems.

中文翻译：

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一种保守的不连续伽辽金实时 (DGiT) 多速率时间积分框架，用于解决界面耦合问题，并应用于固-固相互作用和气-海模型

在本文中，我们将 Connors 和 Sockwell（2022）针对标量传输问题开发的 DGiT 多速率框架扩展到固-固相互作用 (SSI) 问题，该问题涉及两个耦合弹性固体和一个具有旋转、热力的耦合空气-海洋模型。浅水方程。这样做的目的是证明 Connors 和 Sockwell（2022）建立的数学理论和控制原则对于以不同时间尺度演变的子问题为特征的耦合问题的广泛适用性。采用不同时间步长的多速率时间积分算法，针对每个子问题的动态进行优化，可以显着提高此类耦合问题的仿真效率。然而，由于耦合，多速率算法的开发是一项非常重要的任务，这可能会影响准确性、稳定性或其他所需的属性，例如系统不变量的保留。 DGiT 提供了一个用于多速率时间积分的通用模板，可以实现这些属性。为了阐明 DGiT 完成这项任务的方式，我们详细介绍了将该框架应用于 SSI 和海空耦合问题的每一步。数值示例说明了针对这两个问题所得到的多速率方案的关键属性。