  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.
