A new class of entropy stable fluctuations for the discontinuous Galerkin method with application to the Saint-Venant-Exner model

In this work we consider entropy stable discontinuous Galerkin methods applied to nonconservative hyperbolic systems. We introduce a new class of entropy conservative fluctuations that allow us to construct entropy conservative schemes without any system-specific derivations. We also demonstrate that a loss of entropy symmetrization for nonconservative systems restricts the design of entropy stable fluctuations and propose a novel blending procedure to construct entropy stable dissipation terms from general numerical viscosity matrices. The resulting methodology is applied to develop a high-order, entropy stable, and well-balanced approximation for the Saint-Venant-Exner system. Numerical tests are presented to verify the theoretical findings and demonstrate the performance and robustness of the proposed scheme.