Kinetic entropy for the layer-averaged hydrostatic Navier-Stokes equations

TL;DR

This paper develops a kinetic scheme for layer-averaged hydrostatic Navier-Stokes equations, ensuring positivity, well-balancing, and energy stability, extending shallow water system results to more complex models.

Contribution

It introduces a vertically implicit/horizontally explicit finite volume kinetic scheme for layer-averaged hydrostatic Navier-Stokes equations, ensuring key physical properties.

Findings

Ensures positivity of water depth in simulations

Maintains well-balanced solutions for steady states

Satisfies a fully discrete energy inequality

Abstract

We are interested in the numerical approximation of the hydrostatic free surface incompressible Navier-Stokes equations. By using a layer-averaged version of the equations, we are able to extend previous results obtained for shallow water system. We derive a vertically implicit / horizontally explicit finite volume kinetic scheme that ensures the positivity of the approximated water depth, the well-balancing and a fully discrete energy inequality.