  We study the implicit upwind finite volume scheme for numerically
approximating the advection-diffusion equation with a vector field 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 study the implicit upwind finite volume scheme for numerically
approximating the advection-diffusion equation with a vector field in the low
regularity DiPerna-Lions setting. We prove that on unstructured regular meshes
the rate of convergence of approximate solutions generated by the upwind scheme
towards the unique solution of the continuous model is at least one. The
numerical error is estimated in terms of logarithmic Kantorovich-Rubinstein
distances and provides thus a bound on the rate of weak convergence.
