  We study the implicit upwind finite volume scheme for numerically
approximating the linear continuity equation in the low regularity
DiPerna-Lions setting. That is, we are concerned with advecting velocity fields
that are spatially Sobolev regular and data that are merely integrable. We
prove that on unstructured regular meshes the rate of convergence of
approximate solutions generated by the upwind scheme towards the unique
distributional solution of the continuous model is at least 1/2. The numerical
error is estimated in terms of logarithmic Kantorovich-Rubinstein distances and
provides thus a bound on the rate of weak convergence.
