Discontinuous-Galerkin discretization of a new class of Green-Naghdi equations

Arnaud Duran (1, 2) , Fabien Marche (1, 2)

1 I3M - Institut de Mathématiques et de Modélisation de Montpellier
2 LEMON - Littoral, Environment: MOdels and Numerics

We describe in this work a discontinuous-Galerkin Finite-Element method to approximate the solutions of a new family of 1d Green-Naghdi models. These new models are shown to be more computationally efficient, while being asymptotically equivalent to the initial formulation with regard to the shallowness parameter. Using the free surface instead of the water height as a conservative variable, the models are recasted under a pre-balanced formulation and discretized using a nodal expansion basis. Independently from the polynomial degree in the approximation space, the preservation of the motionless steady-states is automatically ensured, and the water height positivity is enforced. A simple numerical procedure devoted to stabilize the computations in the vicinity of broken waves is also described. The validity of the resulting model is assessed through extensive numerical validations.

Mots clés: discontinuous-Galerkin; shallow water; dispersive

Domaines: Analyse numérique [math.NA]

Arnaud Duran, Fabien Marche. Discontinuous-Galerkin discretization of a new class of Green-Naghdi equations. Communications in Computational Physics, 2015, 17 (3), pp.721-760. ⟨10.4208/cicp.150414.101014a⟩. ⟨hal-00980826v2⟩