Numerical stability for some equations of gas dynamics

Abstract

The isentropic gas dynamics equations in Eulerian coordinates are expressed by means of the density p and the momentum q = pu, instead of the velocity u, in order to get domains bounded and invariant in the (p. <?)-plane, for a wide class of pressure laws p(p) 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°°-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.

Asymptotic problems and numerical schemes for traffic flows with unilateral constraints describing the formation of jams

“…Then, we project the obtained piecewise constant solution to obtain the updated numerical unknown, constant on the cells C j . This projection step does not preserve the invariant region, since the latter is non-convex (for the role of the convexity of the invariant domain we refer the reader to [24] for gas dynamics equations, and more generally to the textbook [9, Prop. 2.11]).…”

High-Order Schemes and Entropy Condition for Nonlinear Hyperbolic Systems of Conservation Laws

“…We can only prove such a result in the scalar case, in some particular cases (e.g., Euler isentropic equations) and for some particular entropy (in the last case, L°°s tability is also proved, see Le Roux [8]). The problems are similar for any other Godunov type scheme.…”

Stability of numerical schemes solving quasilinear wave equations

“…Conclusion. The technique used to build the schemes (24), (27) and (29), (27) is suitable for other first order hyperbolic systems such as the shallow water model or some equations of gas dynamics, as stated in the introduction; see [5], [6], [7]. The stability of the Glimm scheme (see [1]) may be deduced from Theorem 1.…”

“…Such a property, to have bounded convex invariant sets in the phase plane, is also true for other important applications and implies the stability of the same numerical schemes in the L°°-norm. This is proved for the Shallow Water model, in [5] and [6], and for the isentropic gas dynamics equations, with a wide class of pressure laws, and for some supersonic models in [7]. When source terms appear, the increase of these sets may be estimated rather easily, and the stablity condition does not become too restrictive for computations with large values of the time.…”