This paper is devoted to the theoretical and numerical investigation of the initial boundary value problem for a system of equations used for the description of waves in coastal areas, namely, the Boussinesq-Abbott system in the presence of topography. We propose a procedure that allows one to handle very general linear or nonlinear boundary conditions. It consists in reducing the problem to a system of conservation laws with nonlocal fluxes and coupled to an ODE. This reformulation is used to propose two hybrid finite volumes/finite differences schemes of first and second order respectively. The possibility to use many kinds of boundary conditions is used to investigate numerically the asymptotic stability of the boundary conditions, which is an issue of practical relevance in coastal oceanography since asymptotically stable boundary conditions would allow one to reconstruct a wave field from the knowledge of the boundary data only, even if the initial data is not known.

翻译：本文致力于研究描述近岸区域波浪运动的方程组初边值问题，具体针对含地形效应的Boussinesq-Abbott系统。我们提出了一套可处理非常一般的线性或非线性边界条件的程序，其核心思想是将原问题约化为具有非局部通量并与常微分方程耦合的守恒律系统。基于这一重构形式，我们分别构建了一阶和二阶两种混合有限体积/有限差分格式。利用多种边界条件的可实施性，本文数值研究了边界条件的渐近稳定性——这在近岸海洋学中具有实际应用价值，因为渐近稳定的边界条件允许仅通过边界数据重构波场，即使初始数据未知。