In this talk, we propose an all regime Lagrange-Projection like numerical scheme for the gas dynamics
equations. By all regime, we mean that the numerical scheme is able to compute accurate
approximate solutions with an under-resolved discretization with respect to the Mach number
M, i.e. such that the ratio between the Mach number M and the mesh size or the time step is
small with respect to 1. The key idea is to decouple acoustic and transport phenomenon and
then alter the numerical flux in the acoustic approximation to obtain a uniform truncation error
in term of M. This modified scheme is conservative and endowed with good stability properties
with respect to the positivity of the density and the internal energy. A discrete entropy inequality
under a condition on the modification is obtained thanks to a reinterpretation of the modified
scheme in the Harten Lax and van Leer formalism. A natural extension to multi-dimensional
problems discretized over unstructured mesh is proposed. Then a simple and efficient semi
implicit scheme is also proposed. The resulting scheme is stable under a CFL condition driven
by the (slow) material waves and not by the (fast) acoustic waves and so verifies the all regime
property. Numerical evidences are proposed and show the ability of the scheme to deal with
tests where the flow regime may vary from low to high Mach values.
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