Energy balance numerical schemes for shallow water equations with discontinuous topography

The well-balanced property that ensures quiescent equilibrium when solving the shallow-water equations with varying topography is extended in this work to ensure numerically a constant level of energy in steady cases with velocity when necessary. This is done in the context of augmented solvers that consider in their definition the presence of a discontinuous bed. In order to guarantee a constant energy state a proper integral approach of the bed source term is presented. This approach is systematically assessed via a series of steady test cases and Riemann problems including the resonance regime.