  This work investigates the topical problem of balancing the shallow water
equations over the bottom steps of different heights. The current approaches in
the literature are essentially based on mathematical analysis of the hyperbolic
system of balance equations and take into account the relevant progress in
treating the non-conservative form of the governing system in the framework of
path-conservative schemes. An important problem under debate is the correct
position of the momentum balance closure when the bottom elevation is
discontinuous. Cases of technical interest are systematically analyzed,
consisting of backward-facing steps and forward-facing steps, tackled
supercritical and subcritical flows; critical (sonic) conditions are also
analyzed and discussed. The fundamental concept governing the problem and
supported by the present computations is that the energy-conserving approach is
the only approach that is consistent with the classical shallow water equations
formulated with geometrical source terms and that the momentum balance is
properly closed if a proper choice of a conventional depth on the bottom step
is performed. The depth on the step is shown to be included between the depths
just upstream and just downstream of the step. It is also shown that current
choices (as given in the literature) of the depth on (or in front of) the step
can lead to unphysical configurations, similar to some energy-increasing
solutions.
