Abstract
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.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
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.
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.”
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.
N. J. Zabusky, “Computational synergetics and mathematical innovation,” J. Comput. Phys. 43, (1981), 195–249.
N. J. Zabusky, Solitons and energy transport in nonlinear lattices, Comp. Phys. Comm. 5, 1 (1973).
N. J. Zabusky, Coherent structures in fluid mechanics. In The Significance of Nonlinearity in the Natural Sciences. Plenum Press, New York, 1977.
P. G. Saffman and G. R. Baker, “Vortex interactions,” Ann. Rev. Fluid Mech. 11, 95–122 (1979).
A. Leonard, “Vortex methods for flow simulation,” J. Comput. Phys. 37, 289–335 (1980).
H. Aref, “Integrable, chaotic, and turbulent vortex motion in two- dimensional flows,” Ann. Rev. Fluid Mech. 15, 345–89 (1983).
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).
G. S. Deem and N. J. Zabusky, “Vortex waves: Stationary ‘V-states’, interactions, recurrence, and breaking,” Phys. Rev. Lett. 40, 859–62 (1978).
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).
P. G. Saffman and J. C. Schatzman, “Stability of a vortex street of finite vortices,” J. Fluid Mech. 117, 171–85 (1982).
S. Kida, “Stabilizing effects of finite core on Karman vortex street,” J. Fluid Mech. 122, 487–504 (1982).
E. A. Overman, II and N. J. Zabusky, “Evolution and merger of isolated vortex structures,” Phys. Fluids 25, 1297–1305 (1982).
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).
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1984 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Zabusky, N.J. (1984). Computational Synergetics and Innovation in Wave and Vortex Dynamics. In: Kuramoto, Y. (eds) Chaos and Statistical Methods. Springer Series in Synergetics, vol 24. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-69559-9_27
Download citation
DOI: https://doi.org/10.1007/978-3-642-69559-9_27
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-69561-2
Online ISBN: 978-3-642-69559-9
eBook Packages: Springer Book Archive