  We present an extension to high-order of a first-order Lagrange-projection
like method for the approximation of the Euler equations introduced in Coquel
{\it et al.} (Math. Comput., 79 (2010), pp.~1493--1533). The method is based on
a decomposition between acoustic and transport operators associated to an
implicit-explicit time integration, thus relaxing the constraint of acoustic
waves on the time step. We propose here to use a discontinuous Galerkin method
for the space approximation. Considering the isentropic Euler equations, we
derive conditions to keep positivity of the mean value of density and satisfy
an entropy inequality for the numerical solution in each element of the mesh at
any approximation order in space. These results allow to design limiting
procedures to restore these properties at nodal values within elements.
Numerical experiments support the conclusions of the analysis and highlight
stability and robustness of the present method, though it allows the use of
large time steps.

Numerical Analysis