  Many complex applications require the solution of initial-value problems
where some components change fast, while others vary slowly. Multirate schemes
apply different step sizes to resolve different components of the system,
according to their dynamics, in order to achieve increased computational
efficiency. The stiff components of the system, fast or slow, are best
discretized with implicit base methods in order to ensure numerical stability.
To this end, linearly implicit methods are particularly attractive as they
solve only linear systems of equations at each step.
  This paper develops the Multirate GARK-ROS/ROW (MR-GARK-ROS/ROW) framework
for linearly-implicit multirate time integration. The order conditions theory
considers both exact and approximative Jacobians. The effectiveness of implicit
multirate methods depends on the coupling between the slow and fast
computations; an array of efficient coupling strategies and the resulting
numerical schemes are analyzed. Multirate infinitesimal step linearly-implicit
methods, that allow arbitrarily small micro-steps and offer extreme
computational flexibility, are constructed. The new unifying framework includes
existing multirate Rosenbrock(-W) methods as particular cases, and opens the
possibility to develop new classes of highly effective linearly implicit
multirate integrators.
