- Physical Review Journals
All JournalsPhysics Magazine
Access Provided by Xinjiang University

Physical Review Fluids

Export citation

Export citation

Choose format for download:

Download Citation
  • Rapid Communication
  • Access by Xinjiang University

Koopman analysis of Burgers equation

Jacob Page* and Rich R. Kerswell†

  • School of Mathematics, University of Bristol, Bristol BS8 1TW, United Kingdom and Centre for Mathematical Sciences, University of Cambridge, Cambridge CB3 0WA, United Kingdom

  • *jacob.page@damtp.cam.ac.uk
  • †r.r.kerswell@damtp.cam.ac.uk
PDF

Phys. Rev. Fluids 3, 071901(R) – Published 24 July, 2018

DOI: https://doi.org/10.1103/PhysRevFluids.3.071901

Abstract

The emergence of dynamic mode decomposition (DMD) as a practical way to attempt a Koopman mode decomposition of a nonlinear partial differential equation (PDE) presents exciting prospects for identifying invariant sets and slowly decaying transient structures buried in the PDE dynamics. However, there are many subtleties in connecting DMD to Koopman analysis, and it remains unclear how realistic Koopman analysis is for complex systems such as the Navier-Stokes equations. With this as motivation, we present here a full Koopman decomposition for the velocity field in the Burgers equation by deriving explicit expressions for the Koopman modes and eigenfunctions. As far as we are aware the first time this has been done for a nonlinear PDE. The decomposition highlights the fact that different observables can require different subsets of Koopman eigenfunctions to express them, and it presents a nice example in which (i) the Koopman modes are linearly dependent and so they cannot be fit a posteriori to snapshots of the flow without knowledge of the Koopman eigenfunctions, and (ii) the Koopman eigenvalues are highly degenerate, which means that computed Koopman modes become initial-condition-dependent. As a way of illustration, we discuss the form of the Koopman expansion with various initial conditions, and we assess the capability of DMD to extract the decaying nonlinear coherent structures in run-down simulations.

View figure in article

Physics Subject Headings (PhySH)

Article Text

References (21)

  1. B. O. Koopman, Hamiltonian systems and transformations in Hilbert space, Proc. Natl. Acad. Sci. USA 17, 315 (1931).
  2. I. Mezić, Spectral properties of dynamical systems, model reduction and decompositions, Nonlin. Dyn. 41, 309 (2005).
  3. P. J. Schmid, Dynamic mode decomposition of numerical and experimental data, J. Fluid Mech. 656, 5 (2010).
  4. S. Bagheri, Koopman-mode decomposition of the cylinder wake, J. Fluid Mech. 726, 596 (2013).
  5. M. R. Jovanović, P. J. Schmid, and J. W. Nichols, Sparsity-promoting dynamic mode decomposition, Phys. Fluids 26, 024103 (2014).
  6. J. N. Kutz, S. L. Brunton, B. W. Brunton, and J. L. Proctor, Dynamic Mode Decomposition: Data-Driven Modeling of Complex Systems, 1st ed. (SIAM, New York, 2016).
  7. C. W. Rowley, I. Mezić, S. Bagheri, P. Schlatter, and D. S. Henningson, Spectral analysis of nonlinear flows, J. Fluid Mech. 641, 115 (2009).
  8. J. H. Tu, C. W. Rowley, D. M. Luchtenburg, S. L. Brunton, and J. N. Kutz, On dynamic mode decomposition: Theory and applications, J. Comput. Dyn. 1, 391 (2014).
  9. C. W. Rowley and S. T. M. Dawson, Model reduction for flow analysis and control, Ann. Rev. Fluid Mech. 49, 387 (2017).
  10. A. S. Sharma, I. Mezić, and B. J. McKeon, Correspondence between Koopman mode decompositions, resolvent mode decomposition and invariant solutions of the Navier-Stokes equations, Phys. Rev. Fluids 1, 032402(R) (2016).
  11. H. Arbabi and I. Mezić, Study of dynamics in post-transient flows using Koopman mode decomposition, Phys. Rev. Fluids 2, 124402 (2017).
  12. M. O. Williams, I. G. Kevrekidis, and C. W. Rowley, A data-driven approximation of the Koopman operator: Extending dynamic mode decomposition, J. Nonlin. Sci. 25, 1307 (2015).
  13. S. L. Brunton, B. W. Brunton, J. L. Proctor, and J. N. Kutz, Koopman invariant subspaces and finite linear representations of nonlinear dynamical systems for control, PLoS ONE 11, e0150171 (2016).
  14. E. Hopf, The partial differential equation ut+uux=μuxx, Commun. Pure Appl. Math. 3, 201 (1950).
  15. J. D. Cole, On a quasi-linear parabolic equation occurring in aerodynamics, Quart. Appl. Math. 9, 225 (1951).
  16. J. N. Kutz, J. L. Proctor, and S. L. Brunton, Koopman theory for partial differential equations, arXiv:1607.07076.
  17. E. R. Benton and G. W. Platzman, A table of solutions of the one-dimensional Burgers equation, Quart. Appl. Math. 30, 195 (1972).
  18. I. Mezić, Analysis of fluid flows via spectral properties of the Koopman operator, Annu. Rev. Fluid Mech. 45, 357 (2013).
  19. R. C. Vaughan, Squares: Additive questions and partitions, Int. J. Numb. Theor. 1, 11 (2015).
  20. P. G. Drazin and R. S. Johnson, Solitons: An Introduction, 1st ed. (Cambridge University Press, Cambridge, 1989), Vol. 11.
  21. https://doi.org/10.17863/CAM.25339.

Outline

Information

Phys. Rev. Fluids 3, 071901(R)– Published 24 July, 2018
  • Received 15 December 2017
Crossmark - Check for updates button

©2018 American Physical Society

Sign In to Your Journals Account

Filter

Filter

Article Lookup

Enter a citation