  Traditional time discretization methods use a single timestep for the entire
system of interest and can perform poorly when the dynamics of the system
exhibits a wide range of time scales. Multirate infinitesimal step (MIS)
methods (Knoth and Wolke, 1998) offer an elegant and flexible approach to
efficiently integrate such systems. The slow components are discretized by a
Runge-Kutta method, and the fast components are resolved by solving modified
fast differential equations. Sandu (2018) developed the Multirate Infinitesimal
General-structure Additive Runge-Kutta (MRI-GARK) family of methods that
includes traditional MIS schemes as a subset. The MRI-GARK framework allowed
the construction of the first fourth order MIS schemes. This framework also
enabled the introduction of implicit methods, which are decoupled in the sense
that any implicitness lies entirely within the fast or slow integrations. It
was shown by Sandu that the stability of decoupled implicit MRI-GARK methods
has limitations when both the fast and slow components are stiff and interact
strongly. This work extends the MRI-GARK framework by introducing coupled
implicit methods to solve stiff multiscale systems. The coupled approach has
the potential to considerably improve the overall stability of the scheme, at
the price of requiring implicit stage calculations over the entire system. Two
coupling strategies are considered. The first computes coupled Runge-Kutta
stages before solving a single differential equation to refine the fast
solution. The second alternates between computing coupled Runge-Kutta stages
and solving fast differential equations. We derive order conditions and perform
the stability analysis for both strategies. The new coupled methods offer
improved stability compared to the decoupled MRI-GARK schemes. The theoretical
properties of the new methods are validated with numerical experiments.
