Coupled Multirate Infinitesimal GARK Schemes for Stiff Systems with Multiple Time Scales
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Differential equations derived from many real-world applications are dominated by multiple time scales. Multirate time integration strategies are able to efficiently and accurately propagate these equations in time. Multirate infinitesimal step (MIS) methods allow arbitrary accuracy in the integration of the fast dynamics of a system. Günther and Sandu portrayed MIS schemes as members of the larger class of multirate General-structure Additive Runge-Kutta (MR-GARK) methods. In this work we extend the derivation of multirate infinitesimal GARK schemes to include coupled implicit stages that involve both the fast and slow components of the multirate system. Although the coupled stages computed over large timesteps may not provide the desired accuracy for the fast components of the system, they capture the slow dynamics and can be used to steer one or more MIS integrations with the goal of resolving the fast system more accurately. In this work. two approaches are discussed for this strategy and theoretical analysis is provided for the accuracy and stability implications of each approach. Various methods of up to order four are derived and numerically tested.
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