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

Abstract

We consider the Saint-Venant system for shallow water flows, with non-flat bottom. It is a hyperbolic system of conservation laws that approximately describes various geophysical flows, such as rivers, coastal areas, 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 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 semi-discrete entropy inequality.

On some fast well-balanced first order solvers for nonconservative systems

“…The following modification of the numerical scheme (2.4) is then considered: The entropy-satisfying and positivity-preserving properties of the chosen pathconservative scheme may also be satisfied by its modification (3.19) if the generalized hydrostatic reconstruction satisfies some properties. We give hereafter two general results which are satisfied for the particular case of the hydrostatic reconstruction introduced in [1] for the shallow water system with source terms. In fact, their proofs follow closely those of the corresponding results given in [3].…”

“…In this paragraph we follow the strategy designed in [9] (which is a generalization of that introduced in [1] for the shallow water system) to modify any path-conservative numerical scheme (2.4) in order to be well-balanced for all the curves of a subset Γ 0 of Γ.…”

“…The hydrostatic reconstruction technique has been used in [1] to obtain first and second order schemes for the shallow water system which are well-balanced for vacuum and water at rest solutions. It has also been used in [9] to obtain high order Roe schemes which are also well-balanced for the vacuum and water at rest solutions.…”

“…However, they are not well-balanced in general. In order to overcome this drawback, we present some well-balanced versions of these methods based on two different strategies: the first one is based on a modification of the identity matrix appearing in the definition of the methods; the second one consists of using the generalized hydrostatic reconstruction introduced in [9] which is an extension of the technique introduced in [1] for the numerical treatment of the source term in the shallow-water system.…”

Uncertainty related to high resolution topographic data use for flood event modeling over urban areas: toward a sensitivity analysis approach

“…The structure of the code is made to facilitate the development of new evolutions. This software resolves the shallow water equations [Cunge et al, 1980] thanks to a well-balanced finite volume method based on the hydrostatic reconstruction (introduced in [Audusse et al, 2004,Bouchut, 2004). This numerical method has good properties: water mass conservation, well-balancedness (at least preservation of a lake at rest equilibrium) and positivity water height preservation.…”