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.

References

[1]	On the stability at all times of linearly extrapolated BDF2 timestepping for multiphysics incompressible flow problems. Numer. Methods Partial Differential Equations (2017) 33: 995-1017.
[2]	M. Akbas, L. G. Rebholz and C. Zerfas, Optimal vorticity accuracy in an efficient velocity-vorticity method for the 2D Navier-Stokes equations, Calcolo, 55 (2018), Paper No. 3, 29 pp. 1–29. doi: 10.1007/s10092-018-0246-7
[3]	Continuous data assimilation for the three-dimensional Navier–Stokes-$\alpha$ model. Asymptotic Anal. (2016) 97: 139-164.
[4]	Data assimilation and initialization of hurricane prediction models. J. Atmos. Sci. (1974) 31: 702-719.
[5]	Continuous data assimilation using general interpolant observables. Journal of Nonlinear Science (2014) 24: 277-304.
[6]	Continuous data assimilation with stochastically noisy data. Nonlinearity (2015) 28: 729-753.
[7]	Downscaling data assimilation algorithm with applications to statistical solutions of the Navier–Stokes equations. Ann. Inst. H. Poincaré Anal. Non Linéaire (2019) 36: 295-326.
[8]	Continuous data assimilation for the 2D magnetohydrodynamic equations using one component of the velocity and magnetic fields. Asymptot. Anal. (2018) 108: 1-43.
[9]	S. C. Brenner and L. R. Scott, The Mathematical Theory of Finite Element Methods, Third edition, Texts in Applied Mathematics, 15. Springer, New York, 2008. doi: 10.1007/978-0-387-75934-0
[10]	E. Carlson, J. Hudson and A. Larios, Parameter recovery for the 2 dimensional Navier-Stokes equations via continuous data assimilation, SIAM J. Sci. Comput., 42 (2020), A250–A270. doi: 10.1137/19M1248583
[11]	Spectral filtering of interpolant observables for a discrete-in-time downscaling data assimilation algorithm. SIAM J. Appl. Dyn. Syst. (2019) 18: 1118-1142.
[12]	On conservation laws of Navier-Stokes Galerkin discretizations. Journal of Computational Physics (2017) 337: 289-308.
[13]	Synchronization to big-data: Nudging the Navier-Stokes equations for data assimilation of turbulent flows. Physical Review X (2020) 10: 1-15.
[14]	R. Daley, Atmospheric Data Analysis, Cambridge Atmospheric and Space Science Series, Cambridge University Press, 1993. doi: 10.4267/2042/51948
[15]	Efficient dynamical downscaling of general circulation models using continuous data assimilation. Quarterly Journal of the Royal Meteorological Society (2019) 145: 3175-3194.
[16]	A. Ern and J.-L. Guermond, Theory and Practice of Finite Elements, Applied Mathematical Sciences, 159. Springer-Verlag, New York, 2004. doi: 10.1007/978-1-4757-4355-5
[17]	Data assimilation in large Prandtl Rayleigh–Bénard convection from thermal measurements. SIAM J. Appl. Dyn. Syst. (2020) 19: 510-540.
[18]	Continuous data assimilation for the 2D Bénard convection through velocity measurements alone. Phys. D (2015) 303: 59-66.
[19]	A. Farhat, E. Lunasin and E. S. Titi, A data assimilation algorithm: The paradigm of the 3D Leray-α model of turbulence, Partial Differential Equations Arising from Physics and Geometry, London Math. Soc. Lecture Note Ser., Cambridge Univ. Press, Cambridge, 450 (2019), 253-273.
[20]	A discrete data assimilation scheme for the solutions of the two-dimensional Navier-Stokes equations and their statistics. SIAM J. Appl. Dyn. Syst. (2016) 15: 2109-2142.
[21]	Uniform in time error estimates for a finite element method applied to a downscaling data assimilation algorithm for the Navier-Stokes equations. SIAM Journal on Numerical Analysis (2020) 58: 410-429.
[22]	A computational study of a data assimilation algorithm for the two-dimensional Navier–Stokes equations. Commun. Comput. Phys. (2016) 19: 1094-1110.
[23]	P. Gresho and R. Sani, Incompressible Flow and the Finite Element Method, Vol. 2, Wiley, 1998.
[24]	The Scott-Vogelius finite elements revisited. Math. Comp. (2019) 88: 515-529.
[25]	Unconditional long-time stability of a velocity-vorticity method for the 2D Navier-Stokes equations. Numer. Math. (2017) 135: 143-167.
[26]	The initialization of numerical models by a dynamic-initialization technique. Monthly Weather Review (1976) 104: 1551-1556.
[27]	H. A. Ibdah, C. F. Mondaini and E. S. Titi, Fully discrete numerical schemes of a data assimilation algorithm: Uniform-in-time error estimates, IMA Journal of Numerical Analysis, Drz043, (2019). doi: 10.1093/imanum/drz043
[28]	A second order ensemble method based on a blended BDF time-stepping scheme for time dependent Navier-Stokes equations. Numerical Methods for Partial Differential Equations (2017) 33: 34-61.
[29]	A new approach to linear filtering and prediction problems. Trans. ASME Ser. D. J. Basic Engrg. (1960) 82: 35-45.
[30]	Global in time stability and accuracy of IMEX-FEM data assimilation schemes for Navier-Stokes equations. Computer Methods in Applied Mechanics and Engineering (2019) 345: 1077-1093.
[31]	A. Larios and C. Victor, Continuous data assimilation with a moving cluster of data points for a reaction diffusion equation: A computational study, Commun. Comp. Phys., (accepted for publication).
[32]	K. Law, A. Stuart and K. Zygalakis, A Mathematical Introduction to Data Assimilation, Texts in Applied Mathematics, 62. Springer, Cham, 2015. doi: 10.1007/978-3-319-20325-6
[33]	On the accuracy of the rotation form in simulations of the Navier-Stokes equations. Journal of Computational Physics (2009) 228: 3433-3447.
[34]	On error analysis for the 3D Navier-Stokes equations in velocity-vorticity-helicity form. SIAM Journal on Numerical Analysis (2011) 49: 711-732.
[35]	Uniform-in-time error estimates for the postprocessing Galerkin method applied to a data assimilation algorithm. SIAM J. Numer. Anal. (2018) 56: 78-110.
[36]	Natural vorticity boundary conditions on solid walls. Computer Methods in Applied Mechanics and Engineering (2015) 297: 18-37.
[37]	Velocity-vorticity-helicity formulation and a solver for the Navier-Stokes equations. Journal of Computational Physics (2010) 229: 4291-4303.
[38]	On well-posedness of a velocity-vorticity formulation of the Navier-Stokes equations with no-slip boundary conditions. Discrete Contin. Dyn. Syst. (2018) 38: 3459-3477.
[39]	Grad-div stabilization for the Stokes equations. Math. Comp. (2004) 73: 1699-1718.
[40]	Continuous data assimilation for the 3D primitive equations of the ocean. Comm. Pure Appl. Math. (2019) 18: 643-661.
[41]	L. Rebholz and C. Zerfas, Simple and efficient continuous data assimilation of evolution equations via algebraic nudging, Submitted.
[42]	Towards computable flows and robust estimates for inf-sup stable fem applied to the time dependent incompressible Navier-Stokes equations. SeMA J. (2018) 75: 629-653.
[43]	C. Zerfas, Numerical Methods and Analysis for Continuous Data Assimilation in Fluid Models, PhD thesis, Clemson University, 2019,132 pp, https://tigerprints.clemson.edu/all_dissertations/2428.
[44]	C. Zerfas, L. G. Rebholz, M. Schneier and T. Iliescu, Continuous data assimilation reduced order models of fluid flow, Computer Methods in Applied Mechanics and Engineering, 357 (2019), 112596, 18 pp. doi: 10.1016/j.cma.2019.112596

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