Section: New Results

DG method for dispersive Green-Naghdi equations

Concerning the development of the WaveBox code, we have introduced in [2] the first available numerical code allowing to solve some fully nonlinear and weakly dispersive asymptotic shallow water models on unstructured meshes. More precisely, we introduce a discontinuous Finite Element formulation (discontinuous-Galerkin) 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 sup- plemented 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, with a use of a L-DG method. 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. To improve the efficiency of the resolution of the elliptic part of the formulation, we also investigate the use of very recent skeleton Hybrid-High-Order (HHO) methods. These methods allow to dramatically reduce the number of degrees of freedom (DOF), using only the DOF located on the mesh skeleton. To initiate the development of such methods for nonlinear and un-stationnary problems, a new discrete formulation was developed for the advective Cahn-Hilliard equations in [17]. Such an approach will be extended to more complex asymptotic shallow water models in a near future.