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

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

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)

Abstract

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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

Information

Type
Research Article
Information
Communications in Computational Physics , Volume 19 , Issue 4 , April 2016 , pp. 1094 - 1110
Copyright
Copyright © Global-Science Press 2016 

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