Kinetic entropy inequality and hydrostatic reconstruction scheme for the Saint-Venant system

• Main Authors: Audusse, Emmanuel, Bouchut, François, Bristeau, Marie-Odile, Sainte-Marie, Jacques
• Format: Journal Article
• Language: English
• Published: 12-09-2014
• Subjects: Mathematics - Numerical Analysis
• DOI: 10.48550/arxiv.1409.3825

A lot of well-balanced schemes have been proposed for discretizing the
classical Saint-Venant system for shallow water flows with non-flat bottom.
Among them, the hydrostatic reconstruction scheme is a simple and efficient
one. It involves the knowledge of an arbitrary solver for the homogeneous
problem (for example Godunov, Roe, kinetic,...). If this solver is entropy
satisfying, then the hydrostatic reconstruction scheme satisfies a
semi-discrete entropy inequality. In this paper we prove that, when used with
the classical kinetic solver, the hydrostatic reconstruction scheme also
satisfies a fully discrete entropy inequality, but with an error term. This
error term tends to zero strongly when the space step tends to zero, including
solutions with shocks. We prove also that the hydrostatic reconstruction scheme
does not satisfy the entropy inequality without error term.