Recently developed concept of dissipative measure-valued solution for compressible flows is a suitable tool to describe oscillations and singularities possibly developed in solutions of multidimensional Euler equations. In this paper we study the convergence of the first-order finite volume method based on the exact Riemann solver for the complete compressible Euler equations. Specifically, we derive entropy inequality and prove the consistency of numerical method. Passing to the limit, we show the weak and strong convergence of numerical solutions and identify their limit. The numerical results presented for the spiral, Kelvin-Helmholtz and the Richtmyer-Meshkov problem are consistent with our theoretical analysis.
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