  Riemann problems at geometric discontinuities are a classic and fascinating
issue of hydraulics. In the present paper, the complete solution to the Riemann
problem of the one-dimensional Shallow water Equations at monotonic width
discontinuities is presented. This solution is based on the assumption that the
relationship between the states immediately to the left and to the right of the
discontinuity is a stationary weak solution of the one-dimensional
variable-width Shallow water Equations. It is demonstrated that the solution to
the Riemann problem always exists and it is unique, but there are cases where
three solutions are possible. The appearance of multiple solutions is connected
to a phenomenon, known as hydraulic hysteresis, observed for supercritical flow
in contracting channel. The analysis of a Finite Volume numerical scheme from
the literature (Cozzolino et al. 2018b) shows that the algorithm captures the
solution with supercritical flow through the width discontinuity when multiple
solutions are possible. Interestingly, the one-dimensional variable-width
Shallow water Equations are formally identical to the one-dimensional Porous
Shallow water Equations, implying that the exact solutions and the numerical
scheme discussed in the present paper are relevant for two-dimensional Porous
Shallow water numerical models aiming at urban flooding simulations. The exact
solution presented here may be used not only as a benchmark, but also as a
guide for the construction of new algorithms, and it can be even embedded in an
exact solver.
