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

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. Introduction Depth-averaged equations are widely used in coastal engineering for the simulation of nonlinear waves propagation and transformations in nearshore areas. The full description of surface water waves in an incompressible, homogeneous, inviscid fluid, is provided by the free surface Euler (or water waves) equations but this problem remains mathematically and numerically challenging. As a consequence, the use of depth averaged equations helps to reduce the three-dimensional problem to a two-dimensional problem, while keeping a good level of accuracy in many configurations. Many Boussinesq-like models are used nowadays and a detailed review can be found in [44] and the recent monograph [43]. Denoting by L the typical horizontal scale of the flow and h0 the typical depth, the shallow water regime usually corresponds to the configuration where μ := h20 L2 ≪ 1. If approximations of order O(μ ) of the free surface Euler equations are furnished by the Boussinesq-type (BT equations in the following) equations, see [54, 56, 60] for instance, an additional smallness amplitude assumption on the typical wave amplitude a is classically performed: ε := a h0 = O(μ). This assumption often appears as too restrictive for many applications in coastal oceanography.