A Fast and Stable Well-Balanced Scheme with Hydrostatic Reconstruction for Shallow Water Flows

TLDR

The Saint‑Venant system models shallow water flows over nonflat bottoms and is a hyperbolic conservation law used for rivers, coastal areas, oceans, and granular flows; conservative finite volume methods handle shocks but are inaccurate near steady states, which well‑balanced schemes can overcome. The authors propose a general strategy that uses a local hydrostatic reconstruction to convert any homogeneous numerical flux into a well‑balanced scheme. The method applies the reconstruction to the flux, yielding a scheme that preserves nonnegativity of the water height and satisfies a semidiscrete entropy inequality. When the underlying solver meets classical stability criteria, the resulting scheme is simple, fast, and well‑balanced.

Abstract

We consider the Saint-Venant system for shallow water flows, with nonflat bottom. It is a hyperbolic system of conservation laws that approximately describes various geophysical flows, such as rivers, coastal areas, and oceans when completed with a Coriolis term, or granular flows when completed with friction. Numerical approximate solutions to this system may be generated using conservative finite volume methods, which are known to properly handle shocks and contact discontinuities. However, in general these schemes are known to be quite inaccurate for near steady states, as the structure of their numerical truncation errors is generally not compatible with exact physical steady state conditions. This difficulty can be overcome by using the so-called well-balanced schemes. We describe a general strategy, based on a local hydrostatic reconstruction, that allows us to derive a well-balanced scheme from any given numerical flux for the homogeneous problem. Whenever the initial solver satisfies some classical stability properties, it yields a simple and fast well-balanced scheme that preserves the nonnegativity of the water height and satisfies a semidiscrete entropy inequality.