Continuous data assimilation applied to a velocity-vorticity formulation of the 2D Navier-Stokes equations
Continuous data assimilation applied to a velocity-vorticity formulation of the 2D Navier-Stokes equations
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DOI：
10.3934/era.2020113
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发表时间：
2020-06
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通讯作者：
M. Gardner;Adam Larios;L. Rebholz;Duygu Vargun;C. Zerfas
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作者：
M. Gardner;Adam Larios;L. Rebholz;Duygu Vargun;C. Zerfas
本文研究了两种情况下二维Navier-Stokes方程速度-涡度方程的连续数据同化（CDA）算法：分别对速度和涡度进行微推，以及仅对速度进行微推。我们证明，在一个典型的有限元空间离散和向后欧拉时间离散，CDA的应用程序保持无条件的长时间稳定性的速度-涡量方法，并提供最佳的长时间精度。如果轻推仅应用于速度，并且如果轻推也应用于涡量，则在时间上更快地实现最佳长时间精度，则这些性质保持。数值试验验证了理论的正确性，并在平板槽道流动的应用问题上显示了其有效性。
We study a continuous data assimilation (CDA) algorithm for a velocity-vorticity formulation of the 2D Navier-Stokes equations in two cases: nudging applied to the velocity and vorticity, and nudging applied to the velocity only. We prove that under a typical finite element spatial discretization and backward Euler temporal discretization, application of CDA preserves the unconditional long-time stability property of the velocity-vorticity method and provides optimal long-time accuracy. These properties hold if nudging is applied only to the velocity, and if nudging is also applied to the vorticity then the optimal long-time accuracy is achieved more rapidly in time. Numerical tests illustrate the theory, and show its effectiveness on an application problem of channel flow past a flat plate.