A Class of Staggered Schemes for the Compressible Euler Equations

We present a class of numerical schemes for the solution of the Euler equations; these schemes are based on staggered discretizations and work either on structured meshes or on general simplicial or tetrahedral/hexahedral meshes. The time discretization is performed by fractional-step algorithms, either based on semi-implicit pressure correction techniques or segregated in such a way that only explicit steps are involved (referred to hereafter as "explicit" variants). These schemes solve the internal energy balance, with corrective terms to ensure the correct capture of shocks, and, more generally, the consistency in the Lax-Wendroff sense. To keep the density, the internal energy and the pressure positive, positivity-preserving convection operators for the mass and internal energy balance equations are designed, using upwinding with respect of the material velocity only. The construction of the fluxes thus does not need any Riemann or approximate Riemann solver, and yields particularly efficient algorithms. The stability is obtained without restriction on the time step for the pressure correction time-stepping and under a CFL-like condition for explicit variants: the preservation of the integral of the total energy over the computational domain and the positivity of the density and of the internal energy are ensured, and entropy estimates are derived.