Lagrange multiplier of the incompressibility constraint and p can be expressed, for free surface flows, as a function of the water depth of the fluid. Therefore, the hydrostatic assumption implies that the resulting model, even though it describes an incompressible fluid, has common features with models arising in compressible fluid mechanics.In geophysical problems, the hydrostatic assumption coupled with a shallow water type description of the flow is often used. Unfortunately, these models do not represent phenomena containing dispersive effects for which the non-hydrostatic contribution cannot be neglected. And more complex models have to be considered to take into account this kind of phenomena, together with numerical methods able to discretize the high order derivative terms coming from the dispersive effects. Many shallow water type dispersive models have been proposed such as KdV, Boussinesq, Green-Naghdi, see [21,14,6,30,31,27,19,26,2,3,13]. The modeling of the non-hydrostatic effects for shallow water flows does not raise insuperable difficulties but their discretization is more tricky. Numerical techniques for the approximaion of these models have been recently proposed [15,11,28].The model studied in the present paper has been derived and studied in [12]. Its numerical approximation based on a projection-correction strategy [16] is described in [1]. In [1], the discretization of the elliptic part arising from the non-hydrostatic terms is carried out in a finite difference framework. It is worth noticing that the numerical scheme given in [1] is endowed with robustness and stability properties such as positivity, well-balancing, discrete entropy and wet/dry interfaces treatment.The main contents of this paper is the derivation and validation of the correction step in a variational framework. Since the derivation in a 2d context of the model proposed in [12] does not raise difficulty, the results depicted in this paper pave the way for a discretization of the 2d model over an unstructured mesh, and we will often maintain general notations as far as possible.Notice that the non-hydrostatic model we consider slightly differs from the wellknown Green-Naghdi model [21] but the numerical approximation proposed in this paper can also be used for the numerical approximation of the Green-Naghdi system.Let Ω ⊂ R, be a 1d domain (an interval) and Γ = Γ in ∪ Γ out its boundary (see figure 1). The non-hydrostatic model derived in [12,1] reads
We propose a numerical method for a family of two-dimensional dispersive shallow water systems with topography. The considered models consist in shallow water approximations-without the hydrostatic assumptionof the incompressible Euler system with free surface. Hence, the studied models appear as extensions of the classical shallow water system enriched with dispersive terms. The model formulation motivates us to use a predictioncorrection scheme for its numerical approximation. The prediction part leads to solving a classical shallow water system with topography while the correction part leads to solving an elliptic-type problem. The numerical approximation of the considered dispersive models in the two-dimensional case over unstructured meshes is described, it requires to combine finite volume and finite element techniques. A special emphasis is given to the formulation and the numerical resolution of the correction step (variational formulation, inf-sup condition, boundary conditions,.. .). The numerical procedure is confronted with analytical and experimental test cases. Finally, an application to a real tsunami case is given.
We propose a numerical method for a two-dimensional non-hydrostatic shallow water system with topography [6]. We use a prediction-correction scheme initially introduced by Chorin-Temam [13], and which has been applied previously to the one dimensional problem in [1]. The prediction part leads to solving a shallow water system for which we use finite volume methods [3], while the correction part leads to solving a mixed problem in velocity/pressure using a finite element method. We present an application of the method with a comparison between a hydrostatic and a non-hydrostatic model.
Abstract. We propose to couple the Exner equation with the Stokes equations to model the bedload sediment in geophysical flows . This work is a preliminary study to directly model the hydrodynamic flow by the unsteady Stokes equation instead of the classical shallow water equation. We focus in this proceeding on the coupling applying fluid structure interaction approach to morphodynamical behavior. In other words, we follow the approach of fluid interaction models replacing the structure equation by the Exner equation. The aim of this work is to validate the proposed procedure. These equations are solved by finite element method using the library FEEL++. Résumé. Nous proposons de coupler l'équation d'Exner avec leséquations de Stokes afin de modéliserle transport des sédiments par charriage. Ce travail est uneétude préliminaire pour modéliser le flux hydrodynamique par leséquations instationnaires de Stokesà la place du choix classique deséquations de Saint-Venant. Ici, nous nous concentrons sur l'utilisation d'une approche interaction fluide structure pour le couplage, c'est-à-dire remplacer l'équation de structure par l'équation d'Exner. Le but de ce travail est de valider la procédure proposée. Ceséquations sont résolues par la méthode desélements finis en utilisant la bibliothèque FEEL++.
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