A key difficulty in the analysis and numerical approximation of the shallow water
equations is the nonconservative product of measures due to the gravitational force acting on a sloped
bottom. Solutions may be nonunique, and numerical schemes are not only consistent discretizations
of the shallow water equations, but they also determine how to model the physics. Our derivation is
based on a subcell reconstruction using infinitesimal singular layers at the cell boundaries, as inspired
by S. Noelle, Y. Xing, and C.-W. Shu [J. Comput. Phys., 226 (2007), pp. 29-58]. One key step
is to separate the singular measures. Another aspect is the reconstruction of the solution variables
in the singular layers. We study three reconstructions. The first leads to the well-known scheme of
Audusse et al., [SIAM J. Sci. Comput., 25 (2004), pp. 2050-2065], which introduces the hydrostatic
reconstruction. The second is a modification proposed in [T. Morales de Luna, M. J. Castro D´ıaz,
and C. Par´es, Appl. Math. Comput., 219 (2013), pp. 9012-9032], which analyzes whether a wave
has enough energy to overcome a step. The third is our new scheme, which borrows its structure
from the wet-dry front. For a number of cases discussed in recent years, where water runs down a
hill, Audusse’s scheme converges slowly or fails. Morales’ scheme gives a visible improvement. Both
schemes are clearly outperformed by our new scheme.