Boussinesq models describe the phase-resolved hydrodynamics of unbroken waves and wave- induced currents in shallow coastal waters. Many enhanced versions of the Boussinesq equations are available in the literature, aiming to improve the representation of linear dispersion and non-linearity. This paper describes the numerical solution of the extended Boussinesq equations derived by Madsen and Sørensen (Coastal Eng. 1992; 15: 371–388) on Cartesian cut-cell grids, the aim being to model non-linear wave interaction with coastal structures. An explicit second-order MUSCL-Hancock Godunov-type finite volume scheme is used to solve the non-linear and weakly dispersive Boussinesq-type equations. Interface fluxes are evaluated using an HLLC approximate Riemann solver. A ghost-cell immersed boundary method is used to update flow information in the smallest cut cells and overcome the time step restriction that would otherwise apply. The model is validated for solitary wave reflection from a vertical wall, diffraction of a solitary wave by a truncated barrier, and solitary wave scattering and diffraction from a vertical circular cylinder. In all cases, the model gives satisfactory predictions in comparison with the published analytical solutions and experimental measurements.