  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 K-convergence of numerical solutions and approximate turbulent
solutions together with the Reynolds stress defect and the energy defect.
