A Consistent Quasi-Second Order Staggered Scheme for the Two-Dimensional Shallow Water Equations
Abstract
A quasi-second order scheme is developed to obtain approximate solutions of the shallow water equationswith bathymetry. The scheme is based on a staggered finite volume scheme for the space discretization:the scalar unknowns are located in the discretisation cells while the vector unknowns are located on theedges (in 2D) or faces (in 3D) of the mesh. A MUSCL-like interpolation for the discrete convectionoperators in the water height and momentum equations is performed in order to improve the precisionof the scheme. The time discretization is performed either by a first order segregated forward Eulerscheme in time or by the second order Heun scheme. Both schemes are shown to preserve the waterheight positivity under a CFL condition and an important state equilibrium known as the lake at rest.Using some recent Lax-Wendroff type results for staggered grids, these schemes are shown to be Lax-consistent with the weak formulation of the continuous equations; besides, the forward Euler schemeis shown to be consistent with a weak entropy inequality. Numerical results confirm the efficiency andaccuracy of the schemes.
Keywords
Cite
@article{arxiv.2111.09726,
title = {A Consistent Quasi-Second Order Staggered Scheme for the Two-Dimensional Shallow Water Equations},
author = {R Herbin and J. -C Latché and Y Nasseri and N Therme},
journal= {arXiv preprint arXiv:2111.09726},
year = {2021}
}
Comments
This work is a revised version of the first part of V1 of the same manuscript. The second part of V1, namely the appendix, which concerns the Lax Wendroff theorem on general staggered grids, is now separate, published in SeMA Journal and uploaded as https://hal.archives-ouvertes.fr/hal-03168277/. IMA Journal of Numerical Analysis, OUP, In press