  We propose and analyze a new asymptotic preserving (AP) finite volume scheme
for the multidimensional compressible barotropic Euler equations to simulate
low Mach number flows. The proposed scheme uses a stabilized upwind numerical
flux, with the stabilization term being proportional to the stiff pressure
gradient, and we prove its conditional energy stability and consistency.
Utilizing the concept of dissipative measure-valued (DMV) solutions, we
rigorously illustrate the AP properties of the scheme for well-prepared initial
data. In particular, we prove that the numerical solutions will converge weakly
to a DMV solution of the compressible Euler equations as the mesh parameter
vanishes, while the Mach number is fixed. The DMV solutions then converge to a
classical solution of the incompressible Euler system as the Mach number goes
to zero. Conversely, we show that if the mesh parameter is kept fixed, we
obtain an energy stable and consistent finite-volume scheme approximating the
incompressible Euler equations as the Mach number goes to zero. The numerical
solutions generated by this scheme then converge weakly to a DMV solution of
the incompressible Euler system as the mesh parameter vanishes. Invoking the
weak-strong uniqueness principle, we conclude that the DMV solution and
classical solution of the incompressible Euler system coincide, proving the AP
property of the scheme. We also present an extensive numerical case study in
order to illustrate the theoretical convergences, wherein we utilize the
techniques of K-convergence.
