An analysis of the error of the upwind scheme for transport equation with
discontinuous coefficients is provided. We consider here a velocity field that
is bounded and one-sided Lipschitz continuous. In this framework, solutions are
defined in the sense of measures along the lines of Poupaud and Rascle's work.
We study the convergence order of the upwind scheme in the Wasserstein
distances. More precisely, we prove that in this setting the convergence order
is 1/2. We also show the optimality of this result. In the appendix, we show
that this result also applies to other "diffusive" "first order" schemes and to
a forward semi-Lagrangian scheme
