A limitation of the hydrostatic reconstruction technique for Shallow Water equations

TL;DR

This paper identifies a specific limitation of the hydrostatic reconstruction technique used in well-balanced schemes for Shallow Water equations, showing it can behave abnormally under certain conditions, which impacts its stability.

Contribution

The paper reveals a previously unrecognized limitation of the hydrostatic reconstruction method, highlighting conditions that lead to abnormal behavior and potential instability.

Findings

Hydrostatic reconstruction can behave abnormally with certain slopes and mesh sizes.

The abnormal behavior affects the stability of the numerical scheme.

The issue is linked to specific combinations of water height and grid parameters.

Abstract

Because of their capability to preserve steady-states, well-balanced schemes for Shallow Water equations are becoming popular. Among them, the hydrostatic reconstruction proposed in Audusse et al. (2004), coupled with a positive numerical flux, allows to verify important mathematical and physical properties like the positivity of the water height and, thus, to avoid unstabilities when dealing with dry zones. In this note, we prove that this method exhibits an abnormal behavior for some combinations of slope, mesh size and water height.