We present an overview of recent developments on the convergence analysis of numerical methods for inviscid multidimensional compressible flows that preserve underlying physical structures. We introduce the concept of generalized solutions, the so-called dissipative solutions, and explain their relationship to other commonly used solution concepts. In numerical experiments we apply \(\mathcal {K}-\) convergence of numerical solutions and approximate turbulent solutions together with the Reynolds stress defect and the energy defect.