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.

翻译：本文旨在研究在垂向上可自由运动的半浸没物体与波浪的相互作用，且波浪传播由一维Boussinesq-Abbott系统描述。该问题可简化为该Boussinesq系统的传输问题，其中物体左右两侧域分量间的传输条件通过求解物体垂向位移及物体下方流体区域平均流量所满足的耦合受迫常微分方程组（ODEs）随时间变化而确定。我们提出了一种新的扩展公式，在该公式中，这些ODEs由接触点处水面高程迹线所满足的两个附加受迫ODEs加以补充。该扩展公式的优势在于：强迫项易于数值计算，且接触点处的水面高程可自然获得。基于此公式，我们提出了一种二阶格式，该格式涉及具有非局部通量和源项的广义MacCormack格式，并与求解ODEs的二阶Heun格式相耦合。为验证该格式，我们推导了该波-结构相互作用问题的若干显式解，这些解可作为未来数值计算的标准算例。作为副产品，我们的方法为Boussinesq-Abbott系统在数值域入口处的波浪生成提供了一种二阶格式。