The paper presents a hybrid scheme for the solution of 2DH extended Boussinesq equations. The finite volume method is applied to the advective part of the equations, while dispersive and source terms are discretized by the finite difference technique. To validate the numerical model, a classical refraction-diffraction test is proposed. Special attention is devoted to verify the shock-capturing capabilities of the scheme: the model is applied to one- and two- dimensional runup test cases with good results, showing that no ad hoc treatment is required at the shoreline.
In the latest decades two different approaches have been developed in the study of nearshore hydrodynamics: one is based on the solution of Boussinesq-type equations, the other on the use of De Saint Venant or nonlinear shallow water equations (NSWE), a non-dispersive subset of the former. Boussinesq-type equations have proved to be a powerful and well-tested means for the simulation of wave propagation in coastal areas.
The standard Boussinesq equations for variable bathymetry were first derived by Peregrine (1967), retaining the lowest-order effects of nonlinearity and dispersion. Owing to the poor dispersive properties, standard equations could only be applied to shallow water areas, a severe restriction for most practical problems. Considerable effort has been made to improve the dispersive characteristics of standard equations, rearranging the dispersive terms or using different velocity variables (eg. Madsen et al., 1991; Nwogu, 1993).
These methods of derivation lead to sets of equations, referred to as extended or improved, whose dispersion relations are closer to the exact linear dispersion relation in intermediate water than standard equations. Recently new higher-order formulations have been derived, extending the range of applicability of Boussinesq-type equations to highly nonlinear and dispersive waves (Madsen et al., 2006) but the integration of these sets of equations is still computationally expensive.