A numerical scheme for an improved Green–Naghdi model in the Camassa-Holm regime for the propagation of internal waves

Christian Bourdarias (1) , Stéphane Gerbi (1) , Ralph Lteif (2, 1)

1 LAMA - Laboratoire de Mathématiques
2 Université Libanaise, Faculté des Sciences Section (1)

In this paper, we introduce a new reformulation of the Green-Naghdi model in the Camassa-Holm regime for the propagation of internal waves over a flat topography derived by Duchêne, Israwi and Talhouk [SIAM J. Math. Anal., 47(1), 240—290]. These new Green–Naghdi systems are adapted to improve the frequency dispersion of the original model, they share the same order of precision as the standard one but have an appropriate structure which makes them much more suitable for the numerical resolution. We develop a second order splitting scheme where the hyperbolic part of the system is treated with a high-order finite volume scheme and the dispersive part is treated with a finite difference approach. Numerical simulations are then performed to validate the model.

Mots clés: Green–Naghdi model; Nonlinear shallow water; Dispersive waves; Nonlinear interactions; Improved dispersion; Splitting method; Hybrid method; Finite volume; Finite Difference; High order scheme; WENO reconstruction

Domaines: Equations aux dérivées partielles [math.AP]; Mécanique des fluides [physics.class-ph]; Analyse numérique [math.NA]; Océanographie

Christian Bourdarias, Stéphane Gerbi, Ralph Lteif. A numerical scheme for an improved Green–Naghdi model in the Camassa-Holm regime for the propagation of internal waves. 2017. ⟨hal-01450221⟩