Consistent semi-implicit staggered schemes for compressible flows Part II: Euler equations.

In this paper, we propose implicit and pressure correction schemes for the Euler equations, based on staggered space discretizations, namely the MAC finite volume scheme or the low-order (Rannacher-Turek or Crouzeix-Raviart) finite elements. Both schemes rely on the discretization of the internal energy balance equation, which offers two main advantages: first, we avoid the space discretization of the total energy, which involves cell-centered and face-centered variables; second, we obtain algorithms which boil down to usual schemes in the incompressible limit. However, since these schemes do 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. This latter step is performed under the guideline that, when adding the discrete kinetic equation to the discrete internal energy equation, the sum of the residual and corrective terms tends to zero with the mesh size and time step. 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