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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
MathSciNet review: 492039
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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.


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  • [1] A. K. BATCHELOR, Introduction to Fluid Dynamics, Cambridge Univ. Press, London, 1967.
  • [2] Alexandre Joel Chorin, Numerical study of slightly viscous flow, J. Fluid Mech. 57 (1973), no. 4, 785–796. MR 0395483
  • [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] Theodore E. Dushane, Convergence for a vortex method for solving Euler’s equation, Math. Comp. 27 (1973), 719–728. MR 0339675, 10.1090/S0025-5718-1973-0339675-6
  • [5] Philip Hartman, Ordinary differential equations, John Wiley & Sons, Inc., New York-London-Sydney, 1964. MR 0171038
  • [6] Tosio Kato, On classical solutions of the two-dimensional nonstationary Euler equation, Arch. Rational Mech. Anal. 25 (1967), 188–200. MR 0211057
  • [7] F. J. McGrath, Nonstationary plane flow of viscous and ideal fluids, Arch. Rational Mech. Anal. 27 (1967), 329–348. MR 0221818
  • [8] F. Milinazzo and P. G. Saffman, The calculation of large Reynolds number two-dimensional flow using discrete vortices with random walk, J. Computational Phys. 23 (1977), no. 4, 380–392. MR 0452145
  • [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 theorème sur l’existence du mouvement plan d’un fluide parfait, homogène, incompressible, pendant un temps infiniment long, Math. Z. 37 (1933), no. 1, 698–726 (French). MR 1545430, 10.1007/BF01474610

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