A framework is presented to design multirate time stepping algorithms for two dissipative models with coupling across a physical interface. The coupling takes the form of boundary conditions imposed on the interface, relating the solution variables for both models to each other. The multirate aspect arises when numerical time integration is performed with different time step sizes for the component models. In this paper, we seek to identify a unified approach to develop multirate algorithms for these coupled problems. This effort is pursued though the use of discontinuous-Galerkin time stepping methods, acting as a general unified framework, with different time step sizes. The subproblems are coupled across user-defined intervals of time, called {\it coupling windows}, using polynomials that are continuous on the window. The coupling method is shown to reproduce the correct interfacial energy dissipation, discrete conservation of fluxes, and asymptotic accuracy. In principle, methods of arbitrary order are possible. As a first step, herein we focus on the presentation and analysis of monolithic methods for advection-diffusion models coupled via generalized Robin-type conditions. The monolithic methods could be computed using a Schur-complement approach. We conclude with some discussion of future developments, such as different interface conditions and partitioned methods.
翻译:提出一个框架,用于设计两种分散式模型的多时间阶梯算法,同时将物理界面连接在一起。组合采用界面上的边界条件形式,将两种模型的解决方案变量相互联系起来。当对组件模型采用不同的时间级大小来进行数字时间整合时,就会出现多维的方面。在本文件中,我们寻求找到一种统一的方法来为这些相交的问题制定多级算法。通过使用不连续的-伽勒金时间阶梯法,作为通用的统一框架,以不同的时间级大小作为一般的统一框架,来进行这一努力。次级问题在用户确定的时间间隔之间相互结合,将两个模型的解决方案称为 huit 组合式窗口 。 组合法显示的是复制正确的间能量消散、离散的通量和调节准确性。 原则上, 任意秩序的方法是可能的。 作为第一步, 我们在此集中介绍并分析用于适应分解- 化模式的单一方法, 使用在窗口上连续的多位组合式组合式组合式组合模式, 将采用不同的方法, 并用普通化的组合式组合式组合式的组合式组合式组合式组合式模式, 。