|
|
| |
| CONVERGENCE OF THE LAX-FRIEDRICHS SCHEME AND STABILITY FOR CONSERVATION LAWS WITH A DISCONTINUOUS SPACE-TIME DEPENDENT FLUX |
| |
Citation： |
K. H. KARLSEN,J. D. TOWERS.CONVERGENCE OF THE LAX-FRIEDRICHS SCHEME AND STABILITY FOR CONSERVATION LAWS WITH A DISCONTINUOUS SPACE-TIME DEPENDENT FLUX[J].Chinese Annals of Mathematics B,2004,25(3):287~318 |
| Page view： 1166
Net amount： 1081 |
Authors： |
K. H. KARLSEN; J. D. TOWERS |
Foundation： |
Project supported by the BeMatA Program of the Research Council of Norway and the European
network HYKE, funded by the EC as contract HPRN-CT-2002-00282. |
|
|
| Abstract： |
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
\cite{KRT:L1stable}, the authors 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, 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. |
Keywords： |
Conservation law, Discontinuous coefficient, Nonconvex flux, Lax-Friedrichs difference scheme, Convergence, Compensated compactness,
Entropy condition, Uniqueness |
Classification： |
35L65, 35L45, 65M06, 65M12 |
|
|
Download PDF Full-Text
|
|
|
|
