Continuous Data Assimilation Applied To A Velocity-Vorticity Formulation Of The 2d Navier-Stokes Equations

Matthew Gardner; Adam Larios; Leo G. Rebholz; Duygu Vargun; Camille Zerfas

doi:10.3934/era.2020113

Continuous Data Assimilation Applied To A Velocity-Vorticity Formulation Of The 2d Navier-Stokes Equations

Matthew Gardner, Adam Larios, Leo G. Rebholz, Duygu Vargun, Camille Zerfas

Research output: Contribution to journal › Article › peer-review

18 Scopus citations

Abstract

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 ve-locity 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 ap-plied 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 electiveness on an application problem of channel ﬂow past a ﬂat plate.

Original language	English
Pages (from-to)	2223-2247
Number of pages	25
Journal	Electronic Research Archive
Volume	29
Issue number	3
DOIs	https://doi.org/10.3934/era.2020113
State	Published - Aug 2021
Externally published	Yes

Funding

2020 Mathematics Subject Classification. Primary: 65M60; Secondary: 76D05. Key words and phrases. Data assimilation, Navier-Stokes equations, velocity-vorticity scheme. The second author is supported by NSF Grants DMS 1716801 and CMMI 1953346. The third and fourth authors are supported by NSF Grant DMS 2011490.

Funders	Funder number
National Science Foundation	DMS 2011490, CMMI 1953346, DMS 1716801
Directorate for Mathematical and Physical Sciences	1716801

Keywords

Data assimilation
Navier-Stokes equations
velocity-vorticity scheme.

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10.3934/era.2020113

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title = "Continuous Data Assimilation Applied To A Velocity-Vorticity Formulation Of The 2d Navier-Stokes Equations",

abstract = "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 ve-locity 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 ap-plied 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 electiveness on an application problem of channel ﬂow past a ﬂat plate.",

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AU - Larios, Adam

AU - Rebholz, Leo G.

AU - Vargun, Duygu

AU - Zerfas, Camille

PY - 2021/8

Y1 - 2021/8

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AB - 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 ve-locity 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 ap-plied 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 electiveness on an application problem of channel ﬂow past a ﬂat plate.

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Continuous Data Assimilation Applied To A Velocity-Vorticity Formulation Of The 2d Navier-Stokes Equations

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