Convergence of the Lax-Friedrichs scheme and stability for conservation laws with a discontinuous space-time dependent flux

The authors give the first convergence proof for the Lax-Friedrichs finite difference scheme for non-convex genuinely nonlinear scalar conservation laws of the form u(t) + 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, here the 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], the authors propose a definition of entropy solution that extends the classical Kruzkov definition to the situation where k(x,t) is piecewise Lipschitz continuous in the (x,t) plane, and prove the 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. It is shown 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.