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Mathematics of Computation
Mathematics of Computation
ISSN 1088-6842(online) ISSN 0025-5718(print)

 

Generalized monotone schemes,
discrete paths of extrema,
and discrete entropy conditions


Authors: Philippe G. LeFloch and Jian-Guo Liu
Journal: Math. Comp. 68 (1999), 1025-1055
MSC (1991): Primary 35L65, 65M12
Published electronically: February 13, 1999
MathSciNet review: 1627801
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Abstract: Solutions of 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 that is second order accurate away from sonic points and extrema.


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Additional Information

Philippe G. LeFloch
Affiliation: Centre de Mathématiques Appliquées and Centre National de la Recherche Scientifique, URA 756, Ecole Polytechnique, 91128 Palaiseau, France
Email: lefloch@cmapx.polytechnique.fr

Jian-Guo Liu
Affiliation: Department of Mathematics, Temple University, Philadelphia, Pennsylvania 19122
Email: jliu@math.temple.edu

DOI: http://dx.doi.org/10.1090/S0025-5718-99-01062-5
PII: S 0025-5718(99)01062-5
Keywords: Conservation law, entropy solution, extremum path, monotone scheme, high order accuracy, MUSCL scheme
Received by editor(s): May 5, 1997
Received by editor(s) in revised form: November 10, 1997
Published electronically: February 13, 1999
Additional Notes: The first author 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) from NSF. The second author was partially supported by DOE grant DE-FG02 88ER-25053.
Article copyright: © Copyright 1999 American Mathematical Society