Stability Properties of the Constant Coefficients Semi-Implicit Time Schemes Solving Nonfiltering Approximation of the Fully Compressible Equations

Abstract

Abstract A set of control parameters is introduced in the fully elastic nonhydrostatic Euler equations formulated in the mass-based vertical coordinate of Laprise. Contrary to the classical approach, the hydrostatic limit is represented by a subspace of control parameters, instead of a single point. By finding a suitable path from the fully compressible equations to the hydrostatic subspace, we are able to construct a blended system with acoustic modes slowed down and gravity modes nearly unaffected. Numerical stability of the discretized system is thus improved, and the solution remains essentially the fully compressible one. Alternatively, control parameters can be used to redefine the linear model of the constant coefficients semi-implicit time scheme, increasing the numerical stability of the fully compressible system. With a careful choice of the control parameters in both, the linear model used in the semi-implicit temporal scheme, and in the full model, the blended system does not deteriorate the compressible solution while its semi-implicit temporal discretization is more stable. We illustrate the potential of the method in several simple examples and in real case studies using the ALADIN system.