Considered herein is a class of Boussinesq systems of Bona-Smith type that describe water waves in bounded two-dimensional domains with slip-wall boundary conditions and variable bottom topography. Such boundary conditions are necessary in situations involving water waves in channels, ports, and generally in basins with solid boundaries. We prove that, given appropriate initial conditions, the corresponding initial-boundary value problems have unique solutions locally in time, which is a fundamental property of deterministic mathematical modeling. Moreover, we demonstrate that the systems under consideration adhere to three basic conservation laws for water waves: mass, vorticity, and energy conservation. The theoretical analysis of these specific Boussinesq systems leads to a conservative mixed finite element formulation. Using explicit, relaxation Runge-Kutta methods for the discretization in time, we devise a fully discrete scheme for the numerical solution of initial-boundary value problems with slip-wall conditions, preserving mass, vorticity, and energy. Finally, we present a series of challenging numerical experiments to assess the applicability of the new numerical model.

翻译：本文研究一类描述具有滑壁边界条件和可变底部地形的二维有限域中水波的Bona-Smith型Boussinesq系统。此类边界条件在涉及水道、港口以及一般具有固体边界的流域中的水波问题时是必要的。我们证明，在给定适当初始条件下，相应的初边值问题在时间上具有局部唯一解，这是确定性数学建模的基本性质。此外，我们证明了所考虑的系统遵循水波的三个基本守恒定律：质量、涡量和能量守恒。对这些特定Boussinesq系统的理论分析导出了一个保守的混合有限元公式。采用显式松弛Runge-Kutta方法进行时间离散，我们设计了一个完全离散格式，用于数值求解具有滑壁条件的初边值问题，并保持质量、涡量和能量守恒。最后，我们通过一系列具有挑战性的数值实验来评估新数值模型的适用性。