  In the research community, there exists the strong belief that a continuous
Galerkin scheme is notoriously unstable and additional stabilization terms have
to be added to guarantee stability. In the first part of the series [6], the
application of simultaneous approximation terms for linear problems is
investigated where the boundary conditions are imposed weakly. By applying this
technique, the authors demonstrate that a pure continuous Galerkin scheme is
indeed linear stable if the boundary conditions are done in the correct way. In
this work, we extend this investigation to the non-linear case and focusing on
entropy conservation. Switching to entropy variables, we will provide an
estimation on the boundary operators also for non-linear problems to guarantee
conservation. In numerical simulations, we verify our theoretical analysis.
