We present a hybrid solution strategy for the numerical solution of the two-dimensional (2D) partial differential equations of Green-Nagdhi (GN), which simulates fully nonlinear, weakly dispersive free surface waves. We re-write the standard form of the equations by splitting the original system in its elliptic and hyperbolic parts, through the definition of a new variable, accounting for the dispersive effects and having the role of a non-hydrostatic pressure gradient in the shallow water equations. We consider a two-step solution procedure. In the first step we compute a source term by inverting the elliptic coercive operator associated to the dispersive effects; then in a hyperbolic step we evolve the flow variables by using the non-linear shallow water equations, with all non-hydrostatic effects accounted by the source computed in the elliptic phase. The advantages of this procedure are firstly that the GN equations are used for propagation and shoaling, while locally reverting to the nonlinear shallow water equations to model energy dissipation in breaking regions. Secondly and from the numerical point of view, this strategy allows each step to be solved with an appropriate numerical method on arbitrary unstructured meshes. We propose a hybrid finite element (FE) finite volume (FV) scheme, where the elliptic part of the system is discretized by means of the continuous Galerkin FE method and the hyperbolic part is discretized using a third-order node-centred finite volume (FV) technique. The performance of the numerical model obtained is extensively validated against experimental measurements from a series of relevant benchmark problems.
Accurate simulations of water wave's propagation and non-linear wave transformations is of fundamental importance to marine and coastal engineering. Over the last decades, significant efforts in the development of depth averaged models have been made in order to provide the means of accurately predicting near-shore wave processes such as shoaling and runup, diffraction, refraction and harmonic interaction. One of the most applied depth averaged models is the Non-linear Shallow Water Equations (NSWE) which are able to model important aspects of wave propagation phenomena, the general characteristics of the ru-nup process, and the wave breaking with broken waves represented as shocks, but they are not appropriate for deeper waters and shoaling since they neglect all the dispersive effects that play a very important role. In order to take dispersive effects in to account we must keep the O(μ2) terms from the full water waves equations, which where neglected in the derivation of the NSWE. μ is the shallowness parameter defined as water depth to wavelength ratio h0/L. This leads to the Green-Naghdi (GN) equations (Green and Naghdi, 1976) known also as Serre equations. The range of validity of the model may vary as much as far the non-linearity parameter (defined as the ratio of wave amplitude to water depth A/h0) is concerned, but it requires the shallowness parameter μ to be small.