Modified Shallow Water Equations for significantly varying seabeds

Denys Dutykh (1) , Didier Clamond (2)

1 LAMA - Laboratoire de Mathématiques
2 JAD - Laboratoire Jean Alexandre Dieudonné

In the present study, we propose a modified version of the Nonlinear Shallow Water Equations (Saint-Venant or NSWE) for irrotational surface waves in the case when the bottom undergoes some significant variations in space and time. The model is derived from a variational principle by choosing an appropriate shallow water ansatz and imposing some constraints. Our derivation procedure does not explicitly involve any small parameter and is straightforward. The novel system is a non-dispersive non-hydrostatic extension of the classical Saint-Venant equations. A key feature of the new model is that, like the classical NSWE, it is hyperbolic and thus similar numerical methods can be used. We also propose a finite volume discretisation of the obtained hyperbolic system. Several test-cases are presented to highlight the added value of the new model. Some implications to tsunami wave modelling are also discussed.

Mots clés: Shallow water; Saint-Venant equations; finite volumes; UNO scheme

Domaines: Mécanique des fluides [physics.class-ph]; Mécanique des fluides [physics.class-ph]; Dynamique des Fluides [physics.flu-dyn]; Physique Numérique [physics.comp-ph]; Analyse numérique [math.NA]; Equations aux dérivées partielles [math.AP]; Milieux fluides et réactifs; Océan, Atmosphère; Océanographie

Denys Dutykh, Didier Clamond. Modified Shallow Water Equations for significantly varying seabeds: Modified Shallow Water Equations. 2015. ⟨hal-00675209v4⟩