  In this paper, we present convergence analysis of high-order finite element
based methods, in particular, we focus on a discontinuous Galerkin scheme using
summation-by-parts operators. To this end, it is crucial that structure
preserving properties, such as positivity preservation and entropy inequality
hold. We demonstrate how to ensure them and prove the convergence of our
multidimensional high-order DG scheme via dissipative weak solutions. In
numerical simulations, we verify our theoretical results.
