Schémas numériques explicites à mailles décalées pour le calcul d'écoulements compressibles

Résumé

We develop and analyse explicit in time schemes for the computation of compressible flows, based on staggered in space. Upwinding is performed equation by equation only with respect to the velocity. The pressure gradient is built as the transpose of the natural divergence. For the barotropic Euler equations, the velocity convection is built to obtain a discrete kinetic energy balance, with residual terms which are non-negative under a CFL condition. We then show that, in 1D, if a sequence of discrete solutions converges to some limit, then this limit is the weak entropy solution. For the full Euler equations, we choose to solve the internal energy balance since a discretization of the total energy is rather unnatural on staggered meshes. Under CFL-like conditions, the density and internal energy are kept positive, and the total energy cannot grow. To obtain correct weak solutions with shocks satisfying the Rankine-Hugoniot conditions, we establish a kinetic energy identity at the discrete level, then choose the source term of the internal energy equation to recover the total energy balance at the limit. More precisely speaking, we prove that in 1D, if we assume the L∞ and BV-stability and the convergence of the scheme, passing to the limit in the discrete kinetic and internal energy equations, we show that the limit of the sequence of solutions is a weak solution. Finally, we consider the computation of radial flows, governed by Euler equations in axisymetrical (2D) or spherical (3D) coordinates, and obtain similar results to the previous sections. In all chapters, we show numerical tests to illustrate for theoretical results.

mots clés

Titre traduit

Explicit staggered schemes for compressible flows
Résumé

We develop and analyse explicit in time schemes for the computation of compressible flows, based on staggered in space. Upwinding is performed equation by equation only with respect to the velocity. The pressure gradient is built as the transpose of the natural divergence. For the barotropic Euler equations, the velocity convection is built to obtain a discrete kinetic energy balance, with residual terms which are non-negative under a CFL condition. We then show that, in 1D, if a sequence of discrete solutions converges to some limit, then this limit is the weak entropy solution. For the full Euler equations, we choose to solve the internal energy balance since a discretization of the total energy is rather unnatural on staggered meshes. Under CFL-like conditions, the density and internal energy are kept positive, and the total energy cannot grow. To obtain correct weak solutions with shocks satisfying the Rankine-Hugoniot conditions, we establish a kinetic energy identity at the discrete level, then choose the source term of the internal energy equation to recover the total energy balance at the limit. More precisely speaking, we prove that in 1D, if we assume the L∞ and BV-stability and the convergence of the scheme, passing to the limit in the discrete kinetic and internal energy equations, we show that the limit of the sequence of solutions is a weak solution. Finally, we consider the computation of radial flows, governed by Euler equations in axisymetrical (2D) or spherical (3D) coordinates, and obtain similar results to the previous sections. In all chapters, we show numerical tests to illustrate for theoretical results.