A discontinuous Galerkin method for a new class of Green-Naghdi equations on simplicial unstructured meshes

In this paper, we introduce a discontinuous Finite Element formulation on simplicial unstructured meshes for the study of free surface flows based on the fully nonlinear and weakly dispersive Green-Naghdi equations. Working with a new class of asymptotically equivalent equations, which have a simplified analytical structure, we consider a decoupling strategy: we approximate the solutions of the classical shallow water equations supplemented with a source term globally accounting for the non-hydrostatic effects and we show that this source term can be computed through the resolution of scalar elliptic second-order sub-problems. The assets of the proposed discrete formulation are: (i) the handling of arbitrary unstructured simplicial meshes, (ii) an arbitrary order of approximation in space, (iii) the exact preservation of the motionless steady states, (iv) the preservation of the water height positivity, (v) a simple way to enhance any numerical code based on the nonlinear shallow water equations. The resulting numerical model is validated through several benchmarks involving nonlinear wave transformations and run-up over complex topographies

Combined Hybridizable Discontinuous Galerkin (HDG) and Runge-Kutta Discontinuous Galerkin (RK-DG) formulations for Green-Naghdi equations on unstructured meshes

In this paper, we introduce some new high-order discrete formulations on general unstructured meshes, especially designed for the study of irrotational free surface flows based on partial differential equations belonging to the family of fully nonlinear and weakly dispersive shallow water equations. Working with a recent family of optimized asymptotically equivalent equations, we benefit from the simplified analytical structure of the linear dispersive operators to conveniently reformulate the models as the classical nonlin-ear shallow water equations supplemented with several algebraic source terms, which globally account for the non-hydrostatic effects through the introduction of auxiliary coupling variables. High-order discrete approximations of the main flow variables are obtained with a RK-DG method, while the trace of the auxiliary variables are approximated on the mesh skeleton through the resolution of second-order linear elliptic sub-problems with high-order HDG formulations. The combined use of hybrid unknowns and local post-processing significantly helps to reduce the number of globally coupled unknowns in comparison with previous approaches. The proposed formulation is then extended to a more complex family of three parameters enhanced Green-Naghdi equations. The resulting numerical models are validated through several benchmarks involving nonlinear waves transformations and propagation over varying topographies, showing good convergence properties and very good agreements with several sets of experimental data

Un modèle d'équilibre d'ordre élevé pour les équations de Saint-Venant et extension au cas dispersif pour la propagation de la houle en milieu littoral

Nous rappelons ici brièvement les méthodes numériques introduites dans la cadre du modèle récent SURF_WB introduit par Marche et col. pour l'étude de la propagation des vagues en zone de surf. Une extension vers un schéma équilibre d'ordre élevé est ensuite présentée, en vue de l'étude des macro-structures de vorticité générée dans la zone de surf, suivie de validations simples. Enfin, nous proposons une extension vers un modèle de type Boussinesq afin d'être en mesure de simuler l'ensemble des processus concernant la propagation de la houle en milieu littoral

Contributions to the numerical approximation of shallow water asymptotics

HD

Numerical approximation of a new class of 2D dispersive Green-Naghdi equations

International audienc

Numerical approximation of a new class of 2D dispersive Green-Naghdi equations

International audienc

Derivation of a new two-dimensional viscous shallow water model with varying topography, bottom friction and capillary effects

International audienc

Theoretical and numerical study of shallow water models (applications to nearshore hydrodynamics)

Ce travail est consacré à l'étude théorique et numérique de modèles de type Saint-Venant. La première partie est consacrée à la dérivation formelle d'un nouveau modèle de Saint-Venant visqueux avec termes de friction et de tension de surface additionnels. Nous réalisons ensuite une analyse mathématique de ce modèle en montrant des résultats d'existence de solutions en un sens faible.\\ Dans la deuxième partie, nous développons un nouveau modèle numérique de type équilibre'' particulièrement adapté à la simulation de problème impliquant des mouvements de la ligne d'eau et de fortes variations de topographie, dans un contexte bi-dimensionnel.\\ La troisième partie est consacré à l'application de notre modèle numérique à quelques problèmes spécifiques à l'hydrodynamique littorale et c\^otière. En particulier, nous nous intéressons dans un premier temps à la simulation de la propagation en zone c\^otière d'ondes longues de type tsunami. Dans un deuxième temps nous étudions les interactions houle/courant dans la Zone de Surf Interne.BORDEAUX1-BU Sciences-Talence (335222101) / SudocSudocFranceF

A new class of fully nonlinear and weakly dispersive Green-Naghdi models for efficient 2D simulations

International audienceWe introduce a new class of two-dimensional fully nonlinear and weakly dispersive Green-Naghdi equations over varying topography. These new Green-Naghdi systems share the same order of precision as the standard one but have a mathematical structure which makes them much more suitable for the numerical resolution, in particular in the demanding case of two dimensional surfaces. For these new models, we develop a high order, well balanced, and robust numerical code relying on an hybrid finite volume and finite difference splitting approach. The hyperbolic part of the equations is handled with a high-order finite volume scheme allowing for breaking waves and dry areas. The dispersive part is treated with a finite difference approach. Higher order accuracy in space and time is achieved through WENO reconstruction methods and through a SSP-RK time stepping. Particular effort is made to ensure positivity of the water depth. Numerical validations are then performed, involving one and two dimensional cases and showing the ability of the resulting numerical model to handle waves propagation and transformation, wetting and drying; some simple treatments of wave breaking are also included. The resulting numerical code is particularly efficient from a computational point of view and very robust; it can therefore be used to handle complex two dimensional configurations