Convergence of numerical approximations to non-linear continuity equations with rough force fields
Provides theoretical guarantees for numerical methods in non-linear transport problems with rough coefficients, which is relevant for computational fluid dynamics and PDE solvers.
The paper proves quantitative regularity estimates for solutions to non-linear continuity equations with rough force fields, enabling convergence proofs for a wide range of numerical schemes. The results recover optimal regularity for linear transport equations and extend commutator estimates to the non-linear case.
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.