Stability of Leap-Frog Constant-Coefficients Semi-Implicit Schemes for the Fully Elastic System of Euler Equations. Case with Orography.
Résumé
The stability of constant-coefficients semi-implicit schemes for the hydrostatic primitive equations and the fully elastic Euler equations in presence of explicitly treated thermal residuals has been theoretically examined in the earlier literature, but only for the case of a flat terrain. This paper extends these analyses to the case where an orography is present, in the shape of a uniform slope. It is shown, with mass-based coordinates, that for the Euler equations, the presence of a slope reduces furthermore the set of the prognostic variables which can be used in the vertical momentum equation to maintain the robustness of the scheme, compared to the case of a flat terrain. The situation appears to be similar for systems cast in mass-based and height-based vertical coordinates. Still for mass-based vertical coordinates, an optimal prognostic variable is proposed, and shown to result in a robustness similar to the one observed for the hydrostatic primitive equations system. This study finally indicates that with a pertinent choice for the prognostic variables, mass-based vertical coordinates are equally suitable as height-based coordinates for efficiently solving the compressible Euler equations system.