Frédéric Darboux

Olivier Delestre, Carine Lucas, Pierre-Antoine Ksinant et al.

Numerous codes are being developed to solve Shallow Water equations. Because there are used in hydraulic and environmental studies, their capability to simulate properly flow dynamics is critical to guarantee infrastructure and human safety. While validating these codes is an important issue, code validations are currently restricted because analytic solutions to the Shallow Water equations are rare and have been published on an individual basis over a period of more than five decades. This article aims at making analytic solutions to the Shallow Water equations easily available to code developers and users. It compiles a significant number of analytic solutions to the Shallow Water equations that are currently scattered through the literature of various scientific disciplines. The analytic solutions are described in a unified formalism to make a consistent set of test cases. These analytic solutions encompass a wide variety of flow conditions (supercritical, subcritical, shock, etc.), in 1 or 2 space dimensions, with or without rain and soil friction, for transitory flow or steady state. The corresponding source codes are made available to the community (http://www.univ-orleans.fr/mapmo/soft/SWASHES), so that users of Shallow Water-based models can easily find an adaptable benchmark library to validate their numerical methods.

Olivier Delestre, Stéphane Cordier, Frédéric Darboux et al.

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.

Olivier Delestre, Carine Lucas, Pierre-Antoine Ksinant et al.

Numerous codes are being developed to solve Shallow Water equations. Because they are used in hydraulics and environmental studies, their capability to simulate properly flow dynamics is essential to guarantee infrastructure and human safety. Hence, validating these codes and the associated numerical methods is an important issue. Analytic solutions would be excellent benchmarks for these issues. However, analytic solutions to Shallow Water equations are rare. Moreover, they have been published on an individual basis over a period of more than five decades, making them scattered through the literature. In this paper, a significant number of analytic solutions to the Shallow Water equations is described in a unified formalism. They encompass a wide variety of flow conditions (supercritical, subcritical, shock ...), in 1 or 2 space dimensions, with or without rain and soil friction, for transitory flow or steady state. An original feature is that the corresponding source codes are made freely available to the community (http://www.univ-orleans.fr/mapmo/soft/SWASHES), so that users of Shallow Water based models can easily find an adaptable benchmark library to validate their numerical methods.