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

Kenneth H. Karlsen and John. D. Towers

Abstract: 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 \cite{KRT:L1stable}, we propose a definition of entropy solution that extends the classical Kru\v{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.

Paper:
Available as Postscript (384 Kbytes) or gzipped PostScript (128 Kbytes; uncompress using gunzip).
Author(s):
Kenneth H. Karlsen, <kennethk@math.uio.no>
John. D. Towers
Publishing information:
To appear in Chinese Ann. Math. Ser. B.