ar X iv : 0 71 1 . 04 06 v 1 [ m at h . N A ] 2 N ov 2 00 7 GENERALIZED MONOTONE SCHEMES , DISCRETE PATHS OF EXTREMA , AND DISCRETE ENTROPY CONDITIONS

Solutions to conservation laws satisfy the monotonicity property: the number of local extrema is a non-increasing function of time, and local maximum/minimum values decrease/increase monotonically in time. This paper investigates this property from a numerical standpoint. We introduce a class of fully discrete in space and time, high order accurate, difference schemes, called generalized monotone schemes. Convergence toward the entropy solution is proven via a new technique of proof, assuming that the initial data has a finite number of extremum values only, and the flux-function is strictly convex. We define discrete paths of extrema by tracking local extremum values in the approximate solution. In the course of the analysis we establish the pointwise convergence of the trace of the solution along a path of extremum. As a corollary, we obtain a proof of convergence for a MUSCL-type scheme being second order accurate away from sonic points and extrema. Published as : Math. of Comput. 68 (1998), 1025–1055. 1 Ecole Polytechnique, Palaiseau, France. CURRENT ADDRESS : Laboratoire Jacques-Louis Lions, Centre National de la Recherche Scientifique, Université de Paris 6, 4, Place Jussieu, 75252 Paris, France. E-mail address: LeFloch@ann.jussieu.fr. P.G.L. was supported in parts by the Centre National de la Recherche Scientifique, and by the National Science Foundation under grants DMS-88-06731, DMS 94-01003 and DMS 95-02766, and a Faculty Early Career Development award (CAREER). 2 Temple University, Philadelphia, USA. CURRENT ADDRESS : Department of Mathematics, University of Maryland, College Park, MD 20742-4015, USA. J.G.L. was partially supported by DOE grant DE-FG02 88ER-25053. Mathematics Subject Classification: 35L65, 65M12.