Convergence of numerical approximations to non-linear continuity equations with rough force fields

TL;DR

This paper establishes regularity estimates and convergence results for numerical schemes solving non-linear continuity equations with rough force fields, extending classical linear transport theory to the non-linear setting.

Contribution

It introduces a novel commutator estimate framework that extends the classical approach to the non-linear case, enabling convergence analysis of various numerical schemes.

Findings

Proves optimal regularity for solutions to non-linear continuity equations.

Demonstrates convergence of multiple numerical schemes under rough force fields.

Extends classical linear transport results to non-linear equations.

Abstract

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.