We investigate here the ability of a Green–Naghdi model to reproduce strongly nonlinear and dispersive wave propagation. We test in particular the behavior of the new hybrid finite-volume and finite-difference splitting approach recently developed by the authors and collaborators on the challenging benchmark of waves propagating over a submerged bar. Such a configuration requires a model with very good dispersive properties, because of the high-order harmonics generated by topography-induced nonlinear interactions. We thus depart from the aforementioned work and choose to use a new Green–Naghdi system with improved frequency dispersion characteristics. The absence of dry areas also allows us to improve the treatment of the hyperbolic part of the equations. This leads to very satisfying results for the demanding benchmarks under consideration.