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

TL;DR

This paper proves that the hydrostatic reconstruction scheme for the Saint-Venant system satisfies a fully discrete entropy inequality with an error term that vanishes as the spatial step decreases, especially when using a kinetic solver.

Contribution

It demonstrates that the hydrostatic reconstruction scheme with a kinetic solver satisfies a fully discrete entropy inequality with an error term tending to zero, extending previous semi-discrete results.

Findings

The scheme satisfies a fully discrete entropy inequality with an error term.

The error term diminishes as the space step tends to zero.

Without the kinetic solver, the scheme does not satisfy the entropy inequality.

Abstract

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.