A numerical method for wave-structure interactions in the Boussinesq regime

This work studies waves interacting with partially immersed objects moving freely in the vertical direction, and waves described by the one-dimensional Boussinesq-Abbott system. This problem can be reduced to a transmission problem, with transmission conditions involving the resolution of coupled forced ODEs satisfied by the vertical displacement of the object and the average discharge of the fluid under the object. We propose a new extended formulation of the problem that provides the value of the surface elevation at the two contact points. This formulation simplifies the numerical computation of transmission conditions and the discretization of the fluid equations in cells near the object. Based on this formulation, we propose a second order-scheme involving a generalization of the Mac-Cormack scheme with nonlocal flux and a source term. Several explicit solutions are derived to validate this scheme. They can serve as benchmark for future codes.