Stability of leap-frog constant-coefficients semi-implicit schemes for the fully elastic system of Euler equations. Case with orography
Bénard, P. ; Masek, J. ; Smolíková, P.
Année de publication
2005
The stability of constant-coefficients semi-implicit schemes for the <br>hydrostatic primitive equations and the fully elastic Euler equations in<br> the presence of explicitly treated thermal residuals has been <br>theoretically examined in the earlier literature, but only for the case <br>of a flat terrain. This paper extends these analyses to a case in which <br>an orography is present, in the shape of a uniform slope. It is shown, <br>with mass-based coordinates, that for the Euler equations, the presence <br>of a slope reduces furthermore the set of the prognostic variables that <br>can be used in the vertical momentum equation to maintain the robustness<br> of the scheme, compared to the case of a flat terrain. The situation <br>appears to be similar for systems cast in mass-based and height-based <br>vertical coordinates. Still for mass-based vertical coordinates, an <br>optimal prognostic variable is proposed and is shown to result in a <br>robustness similar to the one observed for the hydrostatic primitive <br>equations system. The prognostic variables that lead to robust <br>semi-implicit schemes share the property of having cumbersome evolution <br>equations, and an alternative time treatment of some terms is then <br>required to keep the evolution equation reasonably simple. This <br>treatment is shown not to modify substantially the stability of the time<br> scheme. This study finally indicates that with a pertinent choice for <br>the prognostic variables, mass-based vertical coordinates are equally <br>suitable as height-based coordinates for efficiently solving the <br>compressible Euler equations system</div>
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