A Multirate Discontinuous-Galerkin-in-Time Framework for Interface-Coupled Problems

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.