Finite Volume Implementation of Non-Dispersive, Non-Hydrostatic Shallow Water Equations

Vincent Guinot (1) , Didier Clamond (2) , Denys Dutykh (3, 4, 5, 6)

1 HSM - Hydrosciences Montpellier
2 LJAD - Laboratoire Jean Alexandre Dieudonné
3 School of Mathematical Sciences [Dublin]
4 LAMA - Laboratoire de Mathématiques
5 USMB [Université de Savoie] [Université de Chambéry] - Université Savoie Mont Blanc
6 INSMI-CNRS - Institut National des Sciences Mathématiques et de leurs Interactions - CNRS Mathématiques

A shock-capturing, finite volume implementation of recently proposed non-hydrostatic two-dimensional shallow water equations, is proposed. The discretization of the equations in conservation form implies the modification of the time derivative of the conserved variable, in the form of a mass/inertia matrix, and extra terms in the flux functions. The effect of this matrix is to slow down wave propagation in the presence of significant bottom slopes. The proposed model is first derived in conservation form using mass and momentum balance principles. Its finite volume implementation is then presented. The additional terms to the shallow water equations can be discretized very easily via a simple time-stepping procedure. Two application examples are presented. These examples seem to indicate that the proposed model does not exhibit strong differences with the classical hydrostatic shallow water model under steady-state conditions, but that its behavior is significantly different when transients are involved.

Mots clés: Bottom acceleration; Shallow water model; Non-hydrostatic pressure distribution

Domaines: Mécanique des fluides [physics.class-ph]; Mécanique des fluides [physics.class-ph]; Analyse numérique [math.NA]; Dynamique des Fluides [physics.flu-dyn]; Physique Numérique [physics.comp-ph]; Géophysique [physics.geo-ph]; Géophysique [physics.geo-ph]; Physique Atmosphérique et Océanique [physics.ao-ph]

Vincent Guinot, Didier Clamond, Denys Dutykh. Finite Volume Implementation of Non-Dispersive, Non-Hydrostatic Shallow Water Equations. Philippe Gourbesville, Jean Cunge, Guy Caignaert. Advances in Hydroinformatics, Springer Singapore, pp.197-206, 2014, Springer Hydrogeology 2014, 978-981-4451-42-0. ⟨10.1007/978-981-4451-42-0_17⟩. ⟨hal-00907108⟩