  Attention of the researchers has increased towards a simplification of the
complete Shallow water Equations called the Local Inertia Approximation (LInA),
which is obtained by neglecting the advection term in the momentum conservation
equation. In the present paper it is demonstrated that a shock is always
developed at moving wetting-drying frontiers, and this justifies the study of
the Riemann problem on even and uneven beds. In particular, the general exact
solution for the Riemann problem on horizontal frictionless bed is given,
together with the exact solution of the non-breaking wave propagating on
horizontal bed with friction, while some example solution is given for the
Riemann problem on discontinuous bed. From this analysis, it follows that
drying of the wet bed is forbidden in the LInA model, and that there are
initial conditions for which the Riemann problem has no solution on smoothly
varying bed. In addition, propagation of the flood on discontinuous sloping bed
is impossible if the bed drops height have the same order of magnitude of the
moving-frontier shock height. Finally, it is found that the conservation of the
mechanical energy is violated. It is evident that all these findings pose a
severe limit to the application of the model. The numerical analysis has proven
that LInA numerical models may produce numerical solutions, which are
unreliable because of mere algorithmic nature, also in the case that the LInA
mathematical solutions do not exist. The applicability limits of the LInA model
are discouragingly severe, even if the bed elevation varies continuously. More
important, the non-existence of the LInA solution in the case of discontinuous
topography and the non-existence of receding fronts radically question the
viability of the LInA model in realistic cases. It is evident that classic SWE
models should be preferred in the majority of the practical applications.
