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

2016/11/30 by Fethi Ben Belgacem, Belgacem, F. Ben, P-E Jabin +1 · 1 citation
Engineering · Mathematics · #Analysis of PDEs (math.AP) #Computational Fluid Dynamics and Aerodynamics #FOS: Mathematics #Gas Dynamics and Kinetic Theory #Navier-Stokes equation solutions #Numerical Analysis (math.NA)

paper · pdf · doi:10.48550/arxiv.1611.10271

openalex publication_date 2016/11/30 · openalex created_date 2022/10/06 · openalex updated_date 2026/08/01

Abstract

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

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