Discontinuous-Galerkin Discretization of a New Class of Green-Naghdi Equations (Communications in Computational Physics, 2018)
We describe in this work a discontinuous-Galerkin Finite-Element methodto approximate the solutions of a new family of 1d Green-Naghdi models. These newmodels are shown to be more computationally efficient, while being asymptoticallyequivalent to the initial formulation with regard to the shallowness parameter. Usingthe free surface instead of the water height as a conservative variable, the models arerecasted under a pre-balanced formulation and discretized using a nodal expansion basis.Independently from the polynomial degree in the approximation space, the preservationof the motionless steady-states is automatically ensured, and the water heightpositivity is enforced. A simple numerical procedure devoted to stabilizing the computationsin the vicinity of broken waves is also described. The validity of the resultingmodel is assessed through extensive numerical validations.