Hostname: page-component-76fb5796d-vvkck Total loading time: 0 Render date: 2024-04-25T11:50:11.252Z Has data issue: false hasContentIssue false
  • English
  • Français

A Computational Study of a Data Assimilation Algorithm for the Two-dimensional Navier-Stokes Equations

Part of: Smooth dynamical systems: general theory Stability theory Equations of mathematical physics and other areas of application Incompressible inviscid fluids Control systems

Published online by Cambridge University Press:  12 April 2016

Masakazu Gesho ,
Eric Olson  and
Edriss S. Titi
Masakazu Gesho
Affiliation:
Department of Chemical and Petroleum Engineering, University of Wyoming, 1000 E. University Ave, Dept. 3295, Laramie, WY 82071, USA
Eric Olson*
Affiliation:
Department of Mathematics and Statistics, University of Nevada, Reno, NV 89557, USA
Edriss S. Titi
Affiliation:
Department of Mathematics, Texas A&M University, 3368–TAMU, College Station, TX 77843, USA The Department of Computer Science and Applied Mathematics, Weizmann Institute of Science, Rehovot 76100, Israel
*
*Corresponding author. Email addresses:mgesho@uwyo.edu (M. Gesho), ejolson@unr.edu (E. Olson), titi@math.tamu.edu, edriss.titi@weizmann.ac.il (E. S. Titi)
Get access

Abstract

We study the numerical performance of a continuous data assimilation (downscaling) algorithm, based on ideas from feedback control theory, in the context of the two-dimensional incompressible Navier-Stokes equations. Our model problem is to recover an unknown reference solution, asymptotically in time, by using continuous-in-time coarse-mesh nodal-point observational measurements of the velocity field of this reference solution (subsampling), as might be measured by an array of weather vane anemometers. Our calculations show that the required nodal observation density is remarkably less than what is suggested by the analytical study; and is in fact comparable to the number of numerically determining Fourier modes, which was reported in an earlier computational study by the authors. Thus, this method is computationally efficient and performs far better than the analytical estimates suggest.

Keywords

Continuous data assimilationdetermining nodessignal synchronizationtwo-dimensional Navier-Stokes equationsdownscaling

MSC classification

Secondary: 35Q30: Navier-Stokes equations 93C20: Systems governed by partial differential equations 37C50: Approximate trajectories (pseudotrajectories, shadowing, etc.) 76B75: Flow control and optimization 34D06: Synchronization
Type
Research Article
Information
Communications in Computational Physics , Volume 19 , Issue 4 , April 2016 , pp. 1094 - 1110
Copyright
Copyright © Global-Science Press 2016 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1]Azouani, A., Olson, E., Titi, E.S., Continuous data assimilation using general interpolant observables, Journal of Nonlinear Science, Vol. 24, 2014, 277–304.CrossRefGoogle Scholar
[2]Basdevant, C., Technical Improvements for Direct Numerical Simulation of Homogeneous Three-Dimensional Turbulence, Journal of Computational Physics, Vol. 50, 1983, pp. 209–214.CrossRefGoogle Scholar
[3]Bessaih, H., Olson, E., Titi, E.S., Continuous Data Assimilation with Stochastically Noisy Data, Nonlinearity, Vol. 28, 2015, pp. 729–753.CrossRefGoogle Scholar
[4]Blömker, D., Law, K., Stuart, A.M., Zygalakis, K.C., Accuracy and stability of the continuous-time 3DVAR filter for the Navier-Stokes equation.Google Scholar
[5]Charney, J. and Halem, M. and Jastrow, R., Use of incomplete historical data to infer the present state of the atmosphere, J. Atmos. Sci., Vol. 26, 1160–1163.2.0.CO;2>CrossRefGoogle Scholar
[6]Constantin, P., Foias, C., Navier-Stokes Equations, University of Chicago Press, 1988.CrossRefGoogle Scholar
[7]Farhat, A., Jolly, M.S., Titi, E.S., Continuous data assimilation for 2D Bénard convetion through velocity measurements alone, Physica D, to appear.Google Scholar
[8]Farhat, A., Lunasin, E., Titi, E.S., Abridged dynamic continuous data assimilation for the 2D Navier-Stokes equations. arXiv:1504.05978 [Math.AP]Google Scholar
[9]Foias, C., Temam, R., Determination of the solutions of the Navier-Stokes equations by a set of nodal values, Math. Comp., Vol. 43, No. 167, 1984, pp. 117–133.CrossRefGoogle Scholar
[10]Gesho, M., A Numerical Study of Continuous Data Assimilation Using Nodal Points in Space for the Two-dimensional Navier-Stokes Equations, Masters Thesis, University of Nevada, Department of Mathematics and Statistics, 2013.Google Scholar
[11]Jolly, M., Martinez, V., Titi, E.S., A data assimilation algorithm for the subcritical surface quasi- geostrophic equation. Preprint.Google Scholar
[12]Jones, D.A., Titi, E.S., Upper bounds on the number of determining modes, nodes and volume elements for the Navier-Stokes equations, Indiana Univ. Math. J., Vol. 42, No. 3, 1993, pp. 875–887.CrossRefGoogle Scholar
[13]Kalnay, E., Atmospheric modeling, data assimilation and predictability, Cambridge University Press, 2003.Google Scholar
[14]Law, K., Shukla, A., Stuart, A., Analysis of the 3DVAR filter for the partially observed Lorenz'63 model. Discrete Contin. Dyn. Syst. vol. 34, no.3, 2014, pp. 1061–1078.CrossRefGoogle Scholar
[15]Majda, A., Harlim, J., Filtering Complex Turbulent Systems, Cambridge University Press, 2012.CrossRefGoogle Scholar
[16]CUDA C Programming Guide, http://www.nvidia.com, 2012, pp. 1–175.Google Scholar
[17]Olson, E., Titi, E.S., Determining modes for continuous data assimilation in 2D turbulence, Journal of Statistical Physics, Vol. 113, No. 5–6, 2003, pp. 799–840.CrossRefGoogle Scholar
[18]Olson, E., Titi, E.S., Determining modes and Grashof number in 2D turbulence, Theoretical and Computational Fluid Dynamics, Vol. 22, No. 5, 2008, pp. 327–339.CrossRefGoogle Scholar
[19]Robinson, J., Infinite-Dimensional Dynamical Systems, Cambridge Texts in Applied Mathematics, 2001.CrossRefGoogle Scholar
[20]Temam, R., Navier-Stokes Equations and Nonlinear Functional Analysis, CBMS Regional Conference Series, No. 41, SIAM, Philadelphia, 1983.Google Scholar
[21]Xiao, Z., Wan, M., Chen, S., Eyink, G., Physical mechanism of the inverse energy cascade in two-dimensional turbulence: a numerical investigation, J. Fluid Mech., Vol. 619, 2009, pp. 1–44.CrossRefGoogle Scholar