Before revision:  The main ingredients of our analysis are energy-based modeling concepts like the port-Hamiltonian framework and the theory on the Legendre transform, which allow a convenient and general line of argumentation. Moreover, the case of the barotropic Euler equations is examined in more detail and a well-posedness result is proven for their approximation in our framework .
After revision:  The main ingredients of our analysis are energy-based modeling concepts like the port-Hamiltonian formalism, theory of Legendre transformation), which provide a convenient and general line of reasoning. Under mild assumptions on the approximation, local conservation of mass, an energy bound, and the inheritance of the port-Hamiltonian structure can be shown. Our approach is not limited to conventional space discretization but also covers complexity reduction of the case of the barotropic Euler equations is examined in more detail and a well-posedness result is proven for their approximation in our framework .