This article presents a semi-discrete, multilayer set of equations describing the three-dimensional motion of an incompressible fluid bounded below by topography and above by a moving free-surface. This system is a consistent discretisation of the incompressible Euler equations, valid without assumptions on the slopes of the interfaces. Expressed as a set of conservation laws for each layer, the formulation has a clear physical interpretation and makes a seamless link between the hydrostatic Saint-Venant equations, dispersive Boussinesq-style models and the incompressible Euler equations. The associated numerical scheme, based on an approximate vertical projection and multigrid-accelerated column relaxations, provides accurate and efficient solutions for all regimes. The same model can thus be applied to study metre-scale waves, even beyond breaking, with results closely matching those obtained using small-scale Euler/Navier–Stokes models, and coastal or global scale dispersive waves, with an accuracy and efficiency comparable to extended Boussinesq wave models. The implementation is adaptive, parallel and open source as part of the Basilisk framework and the documented source codes sufficient to reproduce all results and figures are provided.

本文提出了一个半离散的多层方程组，描述了不可压缩流体的三维运动，该三维运动的下面是地形，上面是运动的自由表面。该系统是不可压缩的Euler方程的一致离散，在不假设界面斜率的情况下有效。表示为每一层的一组守恒定律，该公式具有清晰的物理解释，并在静液压的Saint-Venant方程，弥散的Boussinesq型模型和不可压缩的Euler方程之间建立了无缝的联系。基于近似的垂直投影和多重网格加速的列松弛的关联数值方案为所有方案提供了准确而有效的解决方案。因此，可以将相同的模型应用于研究米级波浪，甚至不止是破裂，其结果与使用小规模Euler / Navier-Stokes模型以及沿海或全球性色散波获得的结果非常匹配，其准确性和效率可与扩展的Boussinesq波模型相媲美。作为Basilisk框架的一部分，该实现​​是自适应的，并行的和开放源代码，并且提供了足以复制所有结果和图形的文档化源代码。