Convergence of Numerical Approximations to Non-linear Continuity Equations with Rough Force Fields

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 allows us to not only 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 novel commutator estimates, quantifying and extending to the non-linear case the classical commutator approach of the theory of renormalized solutions.