A cell-centred pressure-correction scheme for the compressible Euler equations

We propose a robust pressure-correction scheme for the numerical solution of the compressible Euler equations discretized by a collocated finite volume method. The scheme is based on an internal energy formulation, which ensures that the internal energy is positive. More generally, the scheme enjoys fundamental stability properties: without restriction on the time step, both the density and the internal energy are positive, the integral of the total energy over the computational domain is preserved thanks to an estimate on the discrete kinetic energy and a discrete entropy inequality is satisfied. These stability properties ensure the existence of a solution to the scheme. The internal energy balance features a corrective source term, which is needed for the scheme to compute the correct shock solutions; we are indeed able to prove a Lax-consistency-type convergence result, in the sense that, under some compactness assumptions, the limit of a converging sequence of approximate solutions obtained with space and time discretization steps tending to zero is an entropy weak solution of the Euler equations. Moreover, constant pressure and velocity are preserved through contact discontinuities. The obtained theoretical results and the scheme accuracy are verified by numerical experiments; a numerical stabilization is introduced in order to reduce the oscillations that appear for some tests. The qualitative behaviour of the scheme is assessed on one-dimensional and two-dimensional Riemann problems and compared with other schemes.