Adaptive modeling of shallow fully nonlinear gravity waves

Denys Dutykh (1, 2, 3) , Didier Clamond (4) , Dimitrios Mitsotakis (5)

1 LAMA - Laboratoire de Mathématiques
2 INSMI-CNRS - Institut National des Sciences Mathématiques et de leurs Interactions - CNRS Mathématiques
3 USMB [Université de Savoie] [Université de Chambéry] - Université Savoie Mont Blanc
4 LJAD - Laboratoire Jean Alexandre Dieudonné
5 Victoria University of Wellington

This paper presents an extended version of the celebrated Serre-Green-Naghdi (SGN) system. This extension is based on the well-known Bona-Smith-Nwogu trick which aims to improve the linear dispersion properties. We show that in the fully nonlinear setting it results in modifying the vertical acceleration. Even if this technique is well-known, the effect of this modification on the nonlinear properties of the model is not clear. The first goal of this study is to shed some light on the properties of solitary waves, as the most important class of nonlinear permanent solutions. Then, we propose a simple adaptive strategy to choose the optimal value of the free parameter at every instance of time. This strategy is validated by comparing the model prediction with the reference solutions of the full Euler equations and its classical counterpart. Numerical simulations show that the new adaptive model provides a much better accuracy for the same computational complexity.

Mots clés: dispersion relation; Green-Naghdi model; adaptive model; solitary waves; Shallow water waves; Serre equations

Domaines: Mécanique des fluides [physics.class-ph]; Dynamique des Fluides [physics.flu-dyn]; Géophysique [physics.geo-ph]; Physique Atmosphérique et Océanique [physics.ao-ph]; Physique Numérique [physics.comp-ph]; Formation de Structures et Solitons [nlin.PS]; Systèmes Solubles et Intégrables [nlin.SI]; Analyse numérique [math.NA]; Equations aux dérivées partielles [math.AP]

Denys Dutykh, Didier Clamond, Dimitrios Mitsotakis. Adaptive modeling of shallow fully nonlinear gravity waves. RIMS Kokyuroku, 2015, Mathematical Analysis in Fluid and Gas Dynamics, 1947, pp.45-65. ⟨hal-01088440v2⟩