Implicit Multirate GARK Methods

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 treat different partitions the system timesteps a more targeted and efficient solution compared to monolithic single rate approaches. With implicit used across all partitions, must find balance between stability cost nonlinear stages. In order characterize this important trade-off, we explore coupling strategies, problems assessing linear stability, techniques efficiently implement Newton iterations stage equations. Unlike much existing analysis which is limited in scope particular methods, present general statements on describe fundamental limitations certain types schemes. New up fourth are derived, their accuracy efficiency properties verified numerical tests.