Partitioned RK-type methods for computational fluid dynamics

The simulation of atmospheric motion requires to deal with phenomena on different time scales. This is inherent for systems of hyperbolic type where waves travel each with its own characteristic wave speed. Here, the crucial phenomena are advective waves vs. sound waves. We propose a splitting approach where the terms responsible for fast and slow waves are easily identified in the governing equations. Partitioned RK-Type methods are taylored to this situation. We have developed methods where the fast waves are treated by a variable number of micro steps where the micro step size is taylored to the stability requirements. Order conditions are derived for the overall integration procedure. This requires the discussion of two cases: Order conditions for arbitrary numbers of micro steps and order conditions for a fixed number of micro steps. We present a first collection of methods which extend our MIS methods where order is established for an infinite number of small steps.