Dispersive shallow water wave modelling. Part II: Numerical simulation on a globally flat space

In this paper we describe a numerical method to solve numerically the weakly dispersive fully nonlinear Serre–Green–Naghdi (SGN) celebrated model. Namely, our scheme is based on reliable finite volume methods, proven to be very efficient for the hyperbolic part of equations. The particularity of our study is that we develop an adaptive numerical model using moving grids. Moreover, we use a special form of the SGN equations where non-hydrostatic part of pressure is found by solving a nonlinear elliptic equation. Moreover, this form of governing equations allows to determine the natural form of boundary conditions to obtain a well-posed (numerical) problem.