  A semi-implicit in time, entropy stable finite volume scheme for the
compressible barotropic Euler system is designed and analyzed and its weak
convergence to a dissipative measure-valued (DMV) solution [E. Feireisl et al.,
Dissipative measure-valued solutions to the compressible Navier-Stokes system,
Calc. Var. Partial Differential Equations, 2016] of the Euler system is shown.
The entropy stability is achieved by introducing a shifted velocity in the
convective fluxes of the mass and momentum balances, provided some CFL-like
condition is satisfied to ensure stability. A consistency analysis is performed
in the spirit of the Lax's equivalence theorem under some physically reasonable
boundedness assumptions. The concept of K-convergence [E. Feireisl et al.,
K-convergence as a new tool in numerical analysis, IMA J. Numer. Anal., 2020]
is used in order to obtain some strong convergence results, which are then
illustrated via rigorous numerical case studies. The convergence of the scheme
to a DMV solution, a weak solution and a strong solution of the Euler system
using the weak-strong uniqueness principle and relative entropy are presented.
