Numerical stability for some equations of gas dynamics

Abstract

The isentropic gas dynamics equations in Eulerian coordinates are expressed by means of the density ρ \rho and the momentum q = ρ u q = \rho u , instead of the velocity u, in order to get domains bounded and invariant in the ( ρ , q ) (\rho ,q) -plane, for a wide class of pressure laws p ( ρ ) p(\rho ) and in the monodimensional case. A numerical scheme of the transport-projection type is proposed, which builds an approximate solution valued in such a domain. Since the characteristic speeds are bounded in this set, the stability condition is easily fulfilled and then estimates in the L ∞ {L^\infty } -norm are derived at any time step. Similar results are extended to the model involving friction and topographical terms, and for a simplified model of supersonic flows. The nonapplication of this study to the gas dynamics in Lagrangian coordinates is shown.