Extended multirate infinitesimal step methods: Derivation of order conditions

Multirate methods are specially designed for problems with multiple time scales. The multirate infinitesimal step method (MIS) was developed as a generalization of the so called split-explicit Runge–Kutta methods, where the integration of the fast part is conducted analytically. The MIS method was originally evolved for applications related to numerical weather prediction, i.e. the integration of the compressible Euler equation. In this work, an extension to MIS methods will be presented where an arbitrary Runge–Kutta method (RK) is applied for the integration of the fast component. Furthermore, the order convergence from the original MIS method will be reinvestigated including the derivation of conditions up to order four. Additionally will be presented how well-known methods such as recursive flux splitting multirate method, (Schlegel et al., 2012) partitioned Runge–Kutta method, (Jackiewicz and Vermiglio, 2000) and generalized additive Runge–Kutta method, (Sandu and Günther, 2015) are related to or can be cast as an extended MIS method. An exemplary MIS method of order four with five stages will show that the convergence behaviour not only depends on the applied method for the integration of the fast component. The method will further indicate that the used fast time step plays a significant role.