Linear Stability Analysis of Runge–Kutta-Based Partial Time-Splitting Schemes for the Euler Equations

For atmospheric simulation models with resolutions from about 10 km to the subkilometer cloud-resolving scale, the complete nonhydrostatic compressible Euler equations are often used. An important integration technique for them is the time-splitting (or split explicit) method. This article presents a comprehensive numerical stability analysis of Runge–Kutta (RK)-based partial time-splitting schemes. To this purpose a linearized two-dimensional (2D) compressible Euler system containing advection (as the slow process), sound, and gravity wave terms (as fast processes) is considered. These processes are the most important ones in limiting stability. First, the detailed stability properties are discussed with regard to several off-centering weights for each fast process described by horizontally explicit, vertically implicit schemes. Then the stability properties of the temporally and spatially discretized three-stage RK scheme for the complete 2D Euler equations and their stabilization (e.g., by divergence damping) are discussed. The main goal is to find optimal values for all of the occurring numerical parameters to guarantee stability in operational model applications. Furthermore, formal orders of temporal truncation errors for the time-splitting schemes are calculated. With the same methodology, two alternatives to the three-stage RK method, a so-called RK3-TVD method, and a new four-stage, second-order RK method are inspected.