Generalized Split-Explicit Runge–Kutta Methods for the Compressible Euler Equations

Abstract

he compressible Euler equations exhibit wave phenomena on different scales. A suitable spatial discretization results in partitioned ordinary differential equations where fast and slow modes are present. Generalized split-explicit methods for the time integration of these problems are presented. The methods combine explicit Runge?Kutta methods for the slow modes and with a free choice integrator for the fast modes. Order conditions for these methods are discussed.Construction principles to develop methods with enlarged stability area are presented. Among the generalized class several new methods are developed and compared to the well-established three-stage low-storage Runge?Kutta method (RK3). The new methods allow a 4 times larger macro step size. They require a smaller integration interval for the fast modes. Further, these methods satisfy the order conditions for order three even for nonlinear equations. Numerical tests on more complex problems than the model equation confirm the enhanced stability properties of these methods.