Consistent explicit staggered schemes for compressible flows Part II: the Euler equation

In this paper, we build and analyze the stability and consistency of an explicit scheme for the Euler equations. This scheme is based on staggered space discretizations, with an upwinding performed with respect to the material velocity only. The pressure gradient is defined as the transpose of the natural velocity divergence, and is thus centered. The energy equation which is solved is the internal energy balance, which offers two main advantages: first, we avoid the space discretization of the total energy, the expression of which involves cell-centered and face-centered variables; second, the discretization ensures by construction the positivity of the internal energy, under a CFL condition. However, since this scheme does not use the original (total) energy conservative equation, in order to obtain correct weak solutions (in particular, with shocks satisfying the Rankine-Hugoniot conditions), we need to introduce corrective terms in the internal energy balance. These corrective terms are found by deriving a discrete kinetic energy balance, observing that this relation contains residual terms which do not tend to zero (at least, under reasonable stability assumptions) and, finally, compensating them in the discrete internal energy balance. It is then shown in the 1D case, that, if the scheme converges, the limit is indeed a weak solution. Finally, we present numerical results which confort this theory