  This work considers multirate generalized-structure additively partitioned
Runge-Kutta (MrGARK) methods for solving stiff systems of ordinary differential
equations (ODEs) with multiple time scales. These methods treat different
partitions of the system with different timesteps for a more targeted and
efficient solution compared to monolithic single rate approaches. With implicit
methods used across all partitions, methods must find a balance between
stability and the cost of solving nonlinear equations for the stages. In order
to characterize this important trade-off, we explore multirate coupling
strategies, problems for assessing linear stability, and techniques to
efficiently implement Newton iterations for stage equations. Unlike much of the
existing multirate stability analysis which is limited in scope to particular
methods, we present general statements on stability and describe fundamental
limitations for certain types of multirate schemes. New implicit multirate
methods up to fourth order are derived, and their accuracy and efficiency
properties are verified with numerical tests.
