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 form ut + f(k(x, t), u)x = 0, where 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 Kružkov 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.