Discontinuous-Galerkin Discretization of a New Class of Green-Naghdi Equations

Abstract

Abstract. 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 an expansion basis of arbitrary order. 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 handle broken waves is also described. The validity of the resulting model is assessed through extensive numerical validations.. . IntroductionDepth-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 [42] and the recent monograph [41]. Denoting by λ the typical horizontal scale of the flow and h 0 the typical depth, the shallow water regime usually corresponds to the configuration where µ := [52,54,57] 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. Removing the small amplitude assumption while still keeping all the O(µ) terms, we obtain the so-called Green-Naghdi equations (GN equations in the following) [34], also referred to as Serre equations [62] or fully non-linear Boussinesq equations [76]. A large number of numerical methods have been developed in the past few years for the BT equations. Let us mention for instance some Finite-Difference (FDM in the following) approaches [49,54,65,75] As far as flexibility is concerned, the use of discontinuous-Galerkin methods (dG methods in the following) would appear as a natural choice. Indeed, this class of method provides several appealing features, like compact discretization stencils and hp-adaptivity, flexibility with a natural handling of unstructured meshes, easy parallel computation and local conservation properties in the approximation of conservation laws. A general review of dG methods for convection dominated problems is performed in [14]. Concerning the approximation of more general problems, involving higher-order derivatives, several methods a...

On wave breaking for Boussinesq-type models

“…This criterion has been proven inadequate in some cases [39,44] since its use leads to less energy dissipation than needed. Several more sophisticated criteria have been developed based on physical or numerical arguments [11,39,6,21]. As pointed out in [24], this approach has a major limitation in the stability of the coupling which introduces spurious oscillations at the interface between the breaking and no-breaking region.…”

Hydromorpho: A coupled model for unsteady Stokes/Exner equations and numerical results with Feel++ library

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“…The complete model is composed of the equation of the fluid in two dimensional domain, the equation of the topography in one dimensional domain and the ALE equation in two dimensional domain. According to (18), the coupled model is written as follows :…”

“…It is deduced from the Navier Stokes equations, neglecting the vertical acceleration (see [23]). Many other free surface models have been developed to take into account non-hydrostatic effects with vertically averaged models: see [8,10,11,18,32]. Contrariwise, for this study we choose to conserve the z coordinate in our model, which raises the question of time-depending domain when the bathymetry changes with time: this coupling between fluid motion (including vertical effects) and domain evolution is at the core of this paper.…”

Application of a Combined Finite Element—Finite Volume Method to a 2D Non-hydrostatic Shallow Water Problem

“…However, when the hydrostatic assumption is no longer valid, what we call dispersive effects appear, then more complex models have to be used to represent these effects. Many free surface models are available to take into consideration this dispersive effect, see [11] for the classical Green-Naghdi (GN) model and [5,7,9,6] for other kinds of non hydrostatic models with bathymetry. In this approach, we propose a new method dealing with a formulation without high order terms, we treat the depth-averaged Euler system developed in [6] where the non-hydrostatic pressure is an unknown of the system.…”