Remote Access Mathematics of Computation
Green Open Access

Mathematics of Computation

ISSN 1088-6842(online) ISSN 0025-5718(print)

 
 

 

Convergence of vortex methods for Euler's equations


Authors: Ole Hald and Vincenza Mauceri del Prete
Journal: Math. Comp. 32 (1978), 791-809
MSC: Primary 76C05; Secondary 65N99
DOI: https://doi.org/10.1090/S0025-5718-1978-0492039-1
MathSciNet review: 492039
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: A numerical method for approximating the flow of a two dimensional incompressible, inviscid fluid is examined. It is proved that for a short time interval Chorin's vortex method converges superlinearly toward the solution of Euler's equations, which govern the flow. The length of the time interval depends upon the smoothness of the flow and of the particular cutoff. The theory is supported by numerical experiments. These suggest that the vortex method may even be a second order method.


References [Enhancements On Off] (What's this?)

  • [1] A. K. BATCHELOR, Introduction to Fluid Dynamics, Cambridge Univ. Press, London, 1967.
  • [2] A. J. CHORIN, "Numerical study of slightly viscous flow," J. Fluid Mech., v. 57, 1973, pp. 785-796. MR 0395483 (52:16280)
  • [3] A. J. CHORIN & P. S. BERNARD, "Discretization of a vortex sheet, with an example of roll-up," J. Computational Phys., v. 13, 1973, pp. 423-429.
  • [4] T. E. DUSHANE, "Convergence for a vortex method for solving Euler's equation," Math. Comp., v. 27, 1973, pp. 719-728. MR 0339675 (49:4433)
  • [5] P. HARTMAN, Ordinary Differential Equations, Wiley, New York, 1964. MR 0171038 (30:1270)
  • [6] T. KATO, "On classical solutions of the two-dimensional non-stationary Euler equation," Arch. Rational Mech. Anal., v. 25, 1967, pp. 188-200. MR 0211057 (35:1939)
  • [7] F. J. McGRATH, "Nonstationary plane flow of viscous and ideal fluids," Arch. Rational Mech. Anal., v. 27, 1968, pp. 329-348. MR 0221818 (36:4870)
  • [8] F. MILINAZZO & P. G. SAFFMAN, "The calculation of large Reynolds number twodimensional flow using discrete vortices with random walk," J. Computational Phys., v. 23, 1977, pp. 380-392. MR 0452145 (56:10426)
  • [9] D. W. MOORE, The Discrete Vortex Approximation of a Finite Vortex Sheet, California Inst. of Tech. Report AFOSR-1804-69, 1971.
  • [10] L. ROSENHEAD, "The formation of vortices from a surface of discontinuity," Proc. Roy. Soc. London Ser. A, v. 134, 1932, pp. 170-192.
  • [11] A. I. SHESTAKOV, Numerical Solution of the Navier-Stokes Equations at High Reynolds Numbers, Ph. D. Thesis, Univ. of California, Berkeley, Calif.,1975.
  • [12] H. TAKAMI, Numerical Experiment with Discrete Vortex Approximation, with Reference to the Rolling Up of a Vortex Sheet, Dept. of Aero. and Astr., Stanford University Report SUDAER-202, 1964.
  • [13] F. L. WESTWATER, Aero.Res. Coun., Rep. and Mem. #1692, 1936. See also Batchelor [1, p. 590].
  • [14] W. WOLIBNER, "Un théorème sur l'existence du mouvement plan d'un fluide parfait, homogène, incompressible, pendent un temps infiniment long," Math. Z., v. 37, 1933, pp. 698-726. MR 1545430

Similar Articles

Retrieve articles in Mathematics of Computation with MSC: 76C05, 65N99

Retrieve articles in all journals with MSC: 76C05, 65N99


Additional Information

DOI: https://doi.org/10.1090/S0025-5718-1978-0492039-1
Article copyright: © Copyright 1978 American Mathematical Society

American Mathematical Society