Stability analysis of a finite element approximation for the Navier-Stokes equation with free surface

G. Barrenechea, E. Audusse, A. Decoene, Pierrick Quemar
{"title":"Stability analysis of a finite element approximation for the Navier-Stokes equation with free surface","authors":"G. Barrenechea, E. Audusse, A. Decoene, Pierrick Quemar","doi":"10.1051/m2an/2023096","DOIUrl":null,"url":null,"abstract":"In this work we study the numerical approximation of incompressible Navier-Stokes equations with free surface. The evolution of the free surface is driven by the kinematic boundary condition, and an Arbitrary Lagrangian Eulerian (ALE) approach is used to derive a (formal) weak formulation which involves three fields, namely, velocity, pressure, and the function describing the free surface. This formulation is discretised using finite elements in space and a time-advancing explicit finite difference scheme in time. In fact, the domain tracking algorithm is explicit: first, we solve the equation for the free surface, then move the mesh according to the sigma transform, and finally we compute the velocity and pressure in the updated domain. This explicit strategy is built in such a way that global conservation can be proven, which plays a pivotal role in the proof of stability of the discrete problem. The well-posedness and stability results are independent of the viscosity of the fluid, but while the proof of stability for the velocity is valid for all time steps, and all geometries, the stability for the free surface requires a CFL condition. The performance of the current approach is presented via numerical results and comparisons with the characteristics finite element method.","PeriodicalId":505020,"journal":{"name":"ESAIM: Mathematical Modelling and Numerical Analysis","volume":null,"pages":null},"PeriodicalIF":0.0000,"publicationDate":"2023-11-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"ESAIM: Mathematical Modelling and Numerical Analysis","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.1051/m2an/2023096","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 0

Abstract

In this work we study the numerical approximation of incompressible Navier-Stokes equations with free surface. The evolution of the free surface is driven by the kinematic boundary condition, and an Arbitrary Lagrangian Eulerian (ALE) approach is used to derive a (formal) weak formulation which involves three fields, namely, velocity, pressure, and the function describing the free surface. This formulation is discretised using finite elements in space and a time-advancing explicit finite difference scheme in time. In fact, the domain tracking algorithm is explicit: first, we solve the equation for the free surface, then move the mesh according to the sigma transform, and finally we compute the velocity and pressure in the updated domain. This explicit strategy is built in such a way that global conservation can be proven, which plays a pivotal role in the proof of stability of the discrete problem. The well-posedness and stability results are independent of the viscosity of the fluid, but while the proof of stability for the velocity is valid for all time steps, and all geometries, the stability for the free surface requires a CFL condition. The performance of the current approach is presented via numerical results and comparisons with the characteristics finite element method.
带自由表面的纳维-斯托克斯方程有限元近似的稳定性分析
在这项工作中,我们研究了具有自由表面的不可压缩纳维-斯托克斯方程的数值近似。自由表面的演化由运动边界条件驱动,并采用任意拉格朗日欧拉(ALE)方法推导出一种(形式)弱公式,其中涉及三个场,即速度、压力和描述自由表面的函数。该公式在空间上使用有限元进行离散,在时间上使用时间递增显式有限差分方案。事实上,域跟踪算法是显式的:首先,我们求解自由表面方程,然后根据西格玛变换移动网格,最后计算更新域中的速度和压力。这种显式策略可以证明全局守恒,这对证明离散问题的稳定性起着关键作用。拟合度和稳定性结果与流体的粘度无关,但速度的稳定性证明适用于所有时间步长和所有几何形状,而自由表面的稳定性则需要 CFL 条件。通过数值结果以及与特征有限元法的比较,介绍了当前方法的性能。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
求助全文
约1分钟内获得全文 求助全文
来源期刊
CiteScore
3.00
自引率
0.00%
发文量
0
×
引用
GB/T 7714-2015
复制
MLA
复制
APA
复制
导出至
BibTeX EndNote RefMan NoteFirst NoteExpress
×
提示
您的信息不完整,为了账户安全,请先补充。
现在去补充
×
提示
您因"违规操作"
具体请查看互助需知
我知道了
×
提示
确定
请完成安全验证×
copy
已复制链接
快去分享给好友吧!
我知道了
右上角分享
点击右上角分享
0
联系我们:info@booksci.cn Book学术提供免费学术资源搜索服务,方便国内外学者检索中英文文献。致力于提供最便捷和优质的服务体验。 Copyright © 2023 布克学术 All rights reserved.
京ICP备2023020795号-1
ghs 京公网安备 11010802042870号
Book学术文献互助
Book学术文献互助群
群 号:481959085
Book学术官方微信