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

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- by Philippe G. LeFloch and Jian-Guo Liu PDF
- Math. Comp.
**68**(1999), 1025-1055 Request permission

## 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
- MR Author ID: 233036
- ORCID: 0000-0002-9911-4045
- Email: jliu@math.temple.edu
- 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.
- © Copyright 1999 American Mathematical Society
- Journal: Math. Comp.
**68**(1999), 1025-1055 - MSC (1991): Primary 35L65, 65M12
- DOI: https://doi.org/10.1090/S0025-5718-99-01062-5
- MathSciNet review: 1627801