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We present an extension to high-order of a first-order Lagrange-projectionlike 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 ona decomposition between acoustic and transport operators associated to animplicit-explicit time integration, thus relaxing the constraint of acousticwaves on the time step. We propose here to use a discontinuous Galerkin methodfor the space approximation. Considering the isentropic Euler equations, wederive conditions to keep positivity of the mean value of density and satisfyan entropy inequality for the numerical solution in each element of the mesh atany approximation order in space. These results allow to design limitingprocedures to restore these properties at nodal values within elements.Numerical experiments support the conclusions of the analysis and highlightstability and robustness of the present method, though it allows the use oflarge time steps.

A robust high-order Lagrange-projection like scheme with large time steps for the isentropic Euler equations