Strongly nonlinear and weakly dispersive water waves can be described quite accurately by a set of nonlinear and dispersive partial differential equations known to as the Serre-Green-Naghdi equations (SGN). We solve numerically the SGN system using stable, accurate and efficient fully discrete numerical schemes based on Galerkin/finite element methods.
After reviewing the properties of the SGN system we present the convergence properties of the numerical scheme. A detailed study of the dynamics of the solitary waves of the SGN system over variable bottom topographies and wall-boundary conditions is also presented. In the presence of surface tension additional weakly singular traveling waves are presented.