Formulations of the bed slope term in the shallow water equations: implication for energy balance

For an incompressible, isothermal, continuous and frictionless flow, the exact
resolution of the equation of conservation of mechanical energy is equivalent to the exact
resolution of the equation of momentum conservation. However, the discrete solution of
the equation of momentum conservation does not verify the equation of energy
conservation. This is particularly the case as a result of the presence of a non-conservative
bed slope term in the equation of momentum conservation.
In this research, the influence of (i) the formulation and (ii) the discretization
scheme of the non-conservative bed slope term on the energy conservation is analysed in
the case of a 1D horizontal and frictionless channel with a topographic step.
The results for two different formulations of the bed slope term are firstly
compared: the standard formulation of the bed slope source term and a divergence
formulation of the bed slope term. Then, different discretization schemes of the
divergence formulation of the bed slope term are analyzed to determine the one
minimizing the numerical variation of energy.