Computational Synergetics and Innovation in Wave and Vortex Dynamics

  • N. J. Zabusky
Conference paper
Part of the Springer Series in Synergetics book series (SSSYN, volume 24)


It is demonstrated how the computer, used in a heuristic mode, has greatly augmented our understanding of the mathematics of nonlinear dynamical processes. Examples are given of recent work in soli ton mathematics (waves) and contour dynamics - a boundary integral evolutionary method that is applicable to a wide class of 2D flows. The role of good graphics in enhancing the discovery, retention and communication of new mathematical properties of equations is illustrated.


Euler Equation Vortex Street Vortex Method Progressive Wave Contour Dynamic 
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  1. 1.
    H. H. Goldstine and J. von Neumann, On the principles of large scale computing machines, in “Collected Works of John von Neumann” (A. Taub, Ed.), Vol. 5, pp. 1–32. Macmillan, New York, 1963. The material in this paper was first given as a talk on May 15, 1946. In the opening sections the authors discuss the difficulty of nonlinear problems and describe how they propose to use the digital computer to break the deadlock. Also see Recent theories in turbulence, in “Collected Works of John von Neumann” (A. Taub, Ed.), Vol. 6, pp. 437–472. This paper was issued as a report in 1949. On p. 469, we find a lucid formulation of the synergetic approach.Google Scholar
  2. 2.
    S. M. Ulam, “A Collection of Mathematical Problems,” Wiley-Interscience, New York, 1960. See Chap. VII, Sect. 8: “Physical Systems, Nonlinear Problems”; and Chap. VIII, Sect. 10: “Computing Machines as a Heuristic Aid-Synergesis.”zbMATHGoogle Scholar
  3. 3.
    S. M. Ulam, Introduction to “Studies of Nonlinear Problems” by E. Fermi, J. Pasta and S. M. Ulam in “Collected Papers of Enrico Fermi,” Vol. II, Univ. of Chicago Press, Chicago, 1965.Google Scholar
  4. 4.
    N. J. Zabusky, “Computational synergetics and mathematical innovation,” J. Comput. Phys. 43, (1981), 195–249.MathSciNetADSzbMATHCrossRefGoogle Scholar
  5. 5.
    N. J. Zabusky, Solitons and energy transport in nonlinear lattices, Comp. Phys. Comm. 5, 1 (1973).ADSCrossRefGoogle Scholar
  6. 6.
    N. J. Zabusky, Coherent structures in fluid mechanics. In The Significance of Nonlinearity in the Natural Sciences. Plenum Press, New York, 1977.Google Scholar
  7. 7.
    P. G. Saffman and G. R. Baker, “Vortex interactions,” Ann. Rev. Fluid Mech. 11, 95–122 (1979).ADSCrossRefGoogle Scholar
  8. 8.
    A. Leonard, “Vortex methods for flow simulation,” J. Comput. Phys. 37, 289–335 (1980).MathSciNetADSzbMATHCrossRefGoogle Scholar
  9. 9.
    H. Aref, “Integrable, chaotic, and turbulent vortex motion in two- dimensional flows,” Ann. Rev. Fluid Mech. 15, 345–89 (1983).MathSciNetADSCrossRefGoogle Scholar
  10. 10.
    N. J. Zabusky, M. H. Hughes, and K. V. Roberts, “Contour dynamics for the Euler equations in two dimensions,” J. Comput. Phys. 30, 96–106 (1979).MathSciNetADSzbMATHCrossRefGoogle Scholar
  11. 11.
    G. S. Deem and N. J. Zabusky, “Vortex waves: Stationary ‘V-states’, interactions, recurrence, and breaking,” Phys. Rev. Lett. 40, 859–62 (1978).ADSCrossRefGoogle Scholar
  12. 12.
    H. M. Wu, E. A. Overman, II, and N. J. Zabusky, “Steady-state solutions of the Euler equations in two dimensions: Rotating and translating V-states with limiting cases. I. Numerical algorithms and results,” J. Comput. Phys. In press (1983).Google Scholar
  13. 13.
    P. G. Saffman and J. C. Schatzman, “Stability of a vortex street of finite vortices,” J. Fluid Mech. 117, 171–85 (1982).MathSciNetADSzbMATHCrossRefGoogle Scholar
  14. 14.
    S. Kida, “Stabilizing effects of finite core on Karman vortex street,” J. Fluid Mech. 122, 487–504 (1982).MathSciNetADSzbMATHCrossRefGoogle Scholar
  15. 15.
    E. A. Overman, II and N. J. Zabusky, “Evolution and merger of isolated vortex structures,” Phys. Fluids 25, 1297–1305 (1982).MathSciNetADSzbMATHCrossRefGoogle Scholar
  16. 16.
    E. A. Overman, II and N. J. Zabusky, “Coaxial scattering of Euler- equation translating V-states via contour dynamics,” J. Fluid Mech. 125, 187–202 (1982).MathSciNetADSzbMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1984

Authors and Affiliations

  • N. J. Zabusky
    • 1
  1. 1.Department of Mathematics and StatisticsUniversity of PittsburghPittsburghUSA

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