A finite-volume method for the one-dimensional shallow-water equations including topographic source terms is presented. Exploiting an original idea by Leroux, the system of partial-differential equations is completed by a trivial equation for the bathymetry. By applying a change of variable, the system is given a celerity-speed formulation, and linearized. As a result, an approximate Riemann solver preserving the positivity of the celerity can be constructed, permitting wetting and drying flow simulations to be performed. Finally, the simulation of numerical test cases is presented.