Explicit Two-Step Peer Methods for the Compressible Euler Equations

In atmospheric models the highest-frequency modes are often not the physical modes of interest. On the other hand severe stability constrains for the numerical integrator arise from those meteorologically irrelevant modes. A common strategy to avoid this problem is a splitting approach: The differential equation is split into two parts. The slow part is integrated with one numerical method and a time step size restricted by the CFL number of the low-frequency modes. For the integration of the high-frequency modes a simpler method is used together with smaller time steps so that these small time step sizes satisfy the CFL condition dictated by the high-frequency modes. The disadvantage of the commonly used split-explicit methods is the fact that the high-frequency modes still constrain the maximal time step size if no additional damping term such as divergence damping is used. For split-explicit Runge-Kutta methods there is the constraint cs∆t/∆x < π where cs is the speed of sound. In contrast the CFL number of advection e.g. for RK3 is √ 3 which means that in case of maximal wind speeds below of 190ms−1 the acoustic modes constrain the maximal time step size. We present a new methodology to describe time-splitting methods. The basic principle of the presented approach is the assumption that one part, the fast part, of the split differential equation can be solved analytically so that stability and order investigations can be made for the underlying method which solves the slow part. With this methodology we consider common split-explicit Runge-Kutta methods like RK3 of Wicker and Skamarock, a new class of generalized split-explicit ’Runge-Kutta’ methods developed by Wensch and Knoth and we present a new class of splitexplicit methods which use peer methods as underlying method for the solution of split differential equations.