  We present an implicit-explicit finite volume scheme for the Euler equations.
We start from the non-dimensionalised Euler equations where we split the
pressure in a slow and a fast acoustic part. We use a Suliciu type relaxation
model which we split in an explicit part, solved using a Godunov-type scheme
based on an approximate Riemann solver, and an implicit part where we solve an
elliptic equation for the fast pressure. The relaxation source terms are
treated projecting the solution on the equilibrium manifold. The proposed
scheme is positivity preserving with respect to the density and internal energy
and asymptotic preserving towards the incompressible Euler equations. For this
first order scheme we give a second order extension which maintains the
positivity property. We perform numerical experiments in 1D and 2D to show the
applicability of the proposed splitting and give convergence results for the
second order extension.
