Analysis of the SBP-SAT Stabilization for Finite Element Methods Part II: Entropy Stability

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.