  We prove quantitative regularity estimates for the solutions to non-linear
continuity equations and their discretized numerical approximations on
Cartesian grids when advected by a rough force field. This allow us to recover
the known optimal regularity for linear transport equations but also to obtain
the convergence of a wide range of numerical schemes. Our proof is based on a
novel commutator estimates, quantifying and extending to the non-linear case
the classical commutator approach of the theory of renormalized solutions.
