Boussinesq-type (BT) models play a major role in coastal engineering applications. Due to their asymptotic nature, these models have lent themselves to constant study and improvement based on the inclusion/removal of asymptotically small terms. This work is divided in two main axis. First, we study a hybrid approach combining a finite volume (FV) and a finite element (FE) method to solve a fully-nonlinear and weakly-dispersive depth averaged wave propagation model. The FV method is used to solve the underlying hyperbolic shallow water system, while a standard 𝑃1 finite element method is used to solve the elliptic system associated to the dispersive correction. We study the impact of several numerical aspects: the impact of the reconstruction used in the hyperbolic phase; the representation of the FV data in the FE method used in the elliptic phase and their impact on the theoretical accuracy of the method; the well-posedness of the overall method. For the first element we proposed a systematic implementation of an iterative reconstruction providing on arbitrary meshes up to third order solutions, full second order first derivatives, as well as a consistent approximation of the second derivatives. These properties are exploited to improve the assembly of the elliptic solver, showing dramatic improvement of the finale accuracy, if the FV representation is correctly accounted for. Concerning the elliptic step, the original problem is usually better suited for an approximation in (𝐻 𝑑𝑖𝑣) spaces.
However, it has been shown that perturbed problems involving similar operators with a small Laplace perturbation are well behaved in 𝐻1 . We show, based on both heuristic and strong numerical evidence, that numerical dissipation plays a major role in stabilizing the coupled method, and not only providing convergent results, but also providing the expected convergence rates.
Secondly, we go back to the fundamental question of whether full nonlinearity in weakly dispersive BT models is a necessity. We reconsider the tests first used in the literature to address this issue, as well as a number of more demanding issues. We also consider different families of weakly nonlinear BT models, with different shoaling characteristics, especially when nonlinear waves are involved. Our study allows us to point out that for many cases, it is quite hard to conclude whether full nonlinearity is really necessary. There are a few discriminant cases, which are unfortunately not those mostly used in the literature proposing new models or new numerical methods.
