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

doi:10.3934/era.2020113

Electronic Research Archive

2021, Volume 29, Issue 3: 2223-2247. doi: 10.3934/era.2020113

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Continuous data assimilation applied to a velocity-vorticity formulation of the 2D Navier-Stokes equations

1.
School of Mathematical and Statistical Sciences, Clemson University, Clemson, SC, 29634, USA
2.
Department of Mathematics, University of Nebraska-Lincoln, Lincoln, NE, 68588, USA

Received: 01 June 2020 Revised: 01 August 2020 Published: 19 October 2020
Primary: 65M60; Secondary: 76D05

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.
- Data assimilation,
- Navier-Stokes equations,
- velocity-vorticity scheme
Citation: Matthew Gardner, Adam Larios, Leo G. Rebholz, Duygu Vargun, Camille Zerfas. Continuous data assimilation applied to a velocity-vorticity formulation of the 2D Navier-Stokes equations[J]. Electronic Research Archive, 2021, 29(3): 2223-2247. doi: 10.3934/era.2020113

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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 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.

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