A Consistent Quasi-Second Order Staggered Scheme for the Two-Dimensional Shallow Water Equations

TL;DR

This paper introduces a quasi-second order staggered finite volume scheme for 2D shallow water equations with bathymetry, improving accuracy while preserving key physical properties like water height positivity.

Contribution

It develops a novel staggered scheme with MUSCL-like interpolation and analyzes its consistency, positivity preservation, and accuracy for shallow water equations.

Findings

Scheme is Lax-consistent with the weak formulation.

Positivity of water height is preserved under CFL condition.

Numerical results demonstrate efficiency and accuracy.

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.