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Convergence of the kinetic hydrostatic reconstruction scheme for the Saint Venant system with topography


Authors: François Bouchut and Xavier Lhébrard
Journal: Math. Comp. 90 (2021), 1119-1153
MSC (2020): Primary 65M12, 76M12, 35L65
DOI: https://doi.org/10.1090/mcom/3600
Published electronically: December 28, 2020
MathSciNet review: 4232219
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Abstract: We prove the convergence of the hydrostatic reconstruction scheme with kinetic numerical flux for the Saint Venant system with continuous topography with locally integrable derivative. We use a recently derived fully discrete sharp entropy inequality with dissipation, that enables us to establish an estimate in the inverse of the square root of the space increment $\Delta x$ of the $L^2$ norm of the gradient of approximate solutions. By DiPerna’s method we conclude the strong convergence towards bounded weak entropy solutions.


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Additional Information

François Bouchut
Affiliation: Université Paris-Est, Laboratoire d’Analyse et de Mathématiques Appliquées (UMR 8050), CNRS, UPEM, UPEC, F-77454, Marne-la-Vallée, France
MR Author ID: 314037
ORCID: 0000-0002-2545-1655

Xavier Lhébrard
Affiliation: Université Paris-Est, Laboratoire d’Analyse et de Mathématiques Appliquées (UMR 8050), CNRS, UPEM, UPEC, F-77454, Marne-la-Vallée, France

Keywords: Saint Venant system with topography, well-balanced scheme, hydrostatic reconstruction, convergence, entropy inequality, kinetic function
Received by editor(s): September 6, 2019
Received by editor(s) in revised form: September 4, 2020
Published electronically: December 28, 2020
Article copyright: © Copyright 2020 American Mathematical Society