  The goal of this work is to study waves interacting with partially immersed
objects allowed to move freely in the vertical direction, and in a regime in
which the propagation of the waves is described by the one dimensional
Boussinesq-Abbott system. The problem can be reduced to a transmission problem
for this Boussinesq system, in which the transmission conditions between the
components of the domain at the left and at the right of the object are
determined through the resolution of coupled forced ODEs in time satisfied by
the vertical displacement of the object and the average discharge in the
portion of the fluid located under the object. We propose a new extended
formulation in which these ODEs are complemented by two other forced ODEs
satisfied by the trace of the surface elevation at the contact points. The
interest of this new extended formulation is that the forcing terms are easy to
compute numerically and that the surface elevation at the contact points is
furnished for free. Based on this formulation, we propose a second order scheme
that involves a generalization of the MacCormack scheme with nonlocal flux and
a source term, which is coupled to a second order Heun scheme for the ODEs. In
order to validate this scheme, several explicit solutions for this
wave-structure interaction problem are derived and can serve as benchmark for
future codes. As a byproduct, our method provides a second order scheme for the
generation of waves at the entrance of the numerical domain for the
Boussinesq-Abbott system.
