  Differential equations arising in many practical applications are
characterized by multiple time scales. Multirate time integration seeks to
solve them efficiently by discretizing each scale with a different, appropriate
time step, while ensuring the overall accuracy and stability of the numerical
solution. In a seminal paper Knoth and Wolke (APNUM, 1998) proposed a hybrid
solution approach: discretize the slow component with an explicit Runge-Kutta
method, and advance the fast component via a modified fast differential
equation. The idea led to the development of multirate infinitesimal step (MIS)
methods by Wensch et al. (BIT, 2009.)G\"{u}nther and Sandu (BIT, 2016)
explained MIS schemes as a particular case of multirate General-structure
Additive Runge-Kutta (MR-GARK) methods. The hybrid approach offers extreme
flexibility in the choice of the numerical solution process for the fast
component.
  This work constructs a family of multirate infinitesimal GARK schemes
(MRI-GARK) that extends the hybrid dynamics approachin multiple ways. Order
conditions theory and stability analyses are developed, and practical explicit
and implicit methods of up to order four are constructed. Numerical results
confirm the theoretical findings. We expect the new MRI-GARK family to be most
useful for systems of equations with widely disparate time scales, where the
fast process is dispersive, and where the influence of the fast component on
the slow dynamics is weak.
