Convergence of the Lax-Friedrichs Scheme and Stability for Conservation Laws With a Discontinuous Space-Time Dependent Flux

We give the first convergence proof for the Lax-Friedrichs finite difference scheme for non-convex genuinely nonlinear scalar conservation laws of the formνₜ+f(k(x,t),u)ₓ=0, the coefficient $k(x,t)$ is allowed to be discontinuous along curves in the $(x,t)$ plane. In contrast to most of the existing literature on problems with discontinuous coefficients, our convergence proof is not based on the singular mapping approach, but rather on the div-curl lemma (but not the Young measure) and a Lax type entropy estimate that is robust with respect to the regularity of $k(x,t)$. Following [14], we propose a definition of entropy solution that extends the classical Kru\{z}kov definition to the situation where $k(x,t)$ is piecewise Lipschitz continuous in the $(x,t)$ plane. We prove stability (uniqueness) of such entropy solutions, provided that the flux function satisfies a so-called crossing condition, and that strong traces of the solution exist along the curves where $k(x,t)$ is discontinuous. We show that a convergent subsequence of approximations produced by the Lax-Friedrichs scheme converges to such an entropy solution, implying that the entire computed sequence converges.