MARCHE , Fabien

HDR

International audience

International audience

We 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...

We consider in this work the discontinuous Galerkin discretization of the nonlinear Shallow Water equations on unstructured triangulations. In the recent years, several improvements have been made in the quality of the discontinuous Galerkin approximations for the Shallow Water equations. In this paper, we first perform a review of the recent...

We study here the propagation of long waves in the presence of vorticity. In the irrotational framework, the Green-Naghdi equations (also called Serre or fully nonlinear Boussinesq equations) are the standard model for the propagation of such waves. These equations couple the surface elevation to the vertically averaged horizontal velocity and...

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...

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...

We consider a particular viscous shallow water model with topography and friction laws, formally derived by asymptotic expansion from the three-dimensional free surface Navier-Stokes equations. Emphasize is put on the numerical study: the viscous system is regarded as an hyperbolic system with source terms and discretized using a second order...

We introduce a robust numerical strategy for the numerical simulation of several free-boundary problems arising in the study of nonlinear wave-structure interactions in shallow-water flows. We investigate two types of boundary-evolution equations: (i) a kinematic-type equation, associated with the interaction of waves with a moving lateral...

In this work, a numerical method is introduced for the study of nonlinear interactions between freesurface shallow-water flows and a partly immersed floating object. At the continuous level, the fluid's evolution is modeled with the nonlinear hyperbolic shallow-water equations. The description of the flow beneath the object reduces to an...

We propose a novel Hybrid High-Order method for the Cahn--Hilliard problem with convection. The proposed method is valid in two and three space dimensions, and it supports arbitrary approximation orders on general meshes containing polyhedral elements and nonmatching interfaces. An extensive numerical validation is presented, which shows...

In this work, we develop a fully implicit Hybrid High-Order algorithm for the Cahn–Hilliard problem in mixed form. The space discretization hinges on local reconstruction operators from hybrid polynomial unknowns at elements and faces. The proposed method has several assets: (i) It supports fairly general meshes possibly containing polygonal...