A Staggered Scheme for the Euler Equations

We extend to the full Euler system the scheme introduced in [Berthelin, Goudon, Minjeaud, Math. Comp. 2014] for solving the barotropic Euler equations. This finite volume scheme is defined on staggered grids with numerical fluxes derived in the spirit of kinetic schemes. The difficulty consists in finding a suitable treatment of the energy equation while density and internal energy on the one hand, and velocity on the other hand, are naturally defined on dual locations. The proposed scheme uses the density, the velocity and the internal energy as computational variables and stability conditions are identified in order to preserve the positivity of the discrete density and internal energy. Moreover, we define averaged energies which satisfy local conservation equations. Finally, we provide numerical simulations of Riemann problems to illustrate the behaviour of the scheme.