Conservative modified Serre-Green-Naghdi equations with improved dispersion characteristics

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

1 JAD - Laboratoire Jean Alexandre Dieudonné
2 LAMA - Laboratoire de Mathématiques
3 USMB [Université de Savoie] [Université de Chambéry] - Université Savoie Mont Blanc
4 INSMI-CNRS - Institut National des Sciences Mathématiques et de leurs Interactions - CNRS Mathématiques
5 School of Mathematics, Statistics and Computer Science [Wellington]

For surface gravity waves propagating in shallow water, we propose a variant of the fully nonlinear Serre-Green-Naghdi equations involving a free parameter that can be chosen to improve the dispersion properties. The novelty here consists in the fact that the new model conserves the energy, contrary to other modified Serre's equations found in the literature. Numerical comparisons with the Euler equations show that the new model is substantially more accurate than the classical Serre equations, specially for long time simulations and for large amplitudes.

Mots clés: energy conservation; improved dispersion; Shallow water waves

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

Didier Clamond, Denys Dutykh, Dimitrios Mitsotakis. Conservative modified Serre-Green-Naghdi equations with improved dispersion characteristics: Improved Serre-Green-Naghdi equations. Communications in Nonlinear Science and Numerical Simulation, 2017, 45, pp.245-257. ⟨10.1016/j.cnsns.2016.10.009⟩. ⟨hal-01232370v3⟩