三维静力学纳维-斯托克斯系统的数值近似
Numerical approximation of the 3d hydrostatic Navier-Stokes system with free surface
AI总结：
本文提出了一种稳定且鲁棒的策略，用于近似具有自由表面的三维不可压缩欧拉和纳维-斯托克斯系统。该方法通过使用分片常数基函数的伽辽金型近似来获得水平速度垂直剖面的准确描述，从而在不需移动网格的情况下处理自由表面，并利用浅水类型系统的大量稳健和精确的数值技术。本文还展示了该模型具有动能解释，包括垂直交换项，并基于此结果提出了一种稳健的有限体积方案。
摘要：
在本文中，我们提出了一种稳定且鲁棒的策略，用于近似具有自由表面的三维不可压缩欧拉和纳维-斯托克斯系统。与纳维-斯托克斯系统的浅水近似相比，该方法的思路是使用分片常数基函数的伽辽金型近似来获得水平速度垂直剖面的准确描述。该策略具有多个优点。它允许将纳维-斯托克斯方程改写为具有源项的守恒律系统，易于处理自由表面，无需移动网格，并有可能利用大量为浅水类型系统开发的稳健和精确的数值技术。与一些作者之前的工作相比，本文研究了三维情况。我们展示了该模型具有动能解释，包括垂直交换项，并利用这一结果提出了一种稳健的有限体积方案用于其数值近似。所有离散方案的各个方面（通量、边界条件……）都被完全描述，讨论了所提出数值方案的稳定性性质（平衡性、水深的正性……）。我们通过一些数值学术示例（三维非稳态解析解）验证了模型和离散方案，并展示了离散模型再现现实 tsunami 波的能力。
英文摘要：
In this paper we propose a stable and robust strategy to approximate the 3d incompressible hydrostatic Euler and Navier-Stokes systems with free surface. Compared to shallow water approximation of the Navier-Stokes system, the idea is to use a Galerkin type approximation of the velocity field with piecewise constant basis functions in order to obtain an accurate description of the vertical profile of the horizontal velocity. Such a strategy has several advantages. It allows - to rewrite the Navier-Stokes equations under the form of a system of conservation lawswith source terms, - the easy handling of the free surface, which does not require moving meshes, - the possibility to take advantage of robust and accurate numerical techniques developed in extensive amount for Shallow Water type systems. Compared to previous works of some of the authors, the three dimensional case is studied in this paper. We show that the model admits a kinetic interpretation including the vertical exchanges terms, and we use this result to formulate a robust finite volume scheme for its numerical approximation. All the aspects of the discrete scheme (fluxes, boundary conditions,. . . ) are completely described and the stability properties of the proposed numerical scheme (well-balancing, positivity of the water depth,. . . ) are discussed. We validate the model and the discrete scheme with some numerical academic examples (3d non stationary analytical solutions) and illustratethe capability of the discrete model to reproduce realistic tsunami waves.
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