Convergence of vortex methods for Euler's equations
Authors:
Ole Hald and Vincenza Mauceri del Prete
Journal:
Math. Comp. 32 (1978), 791809
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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 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. 785796. MR 0395483 (52:16280)
 [3]
 A. J. CHORIN & P. S. BERNARD, "Discretization of a vortex sheet, with an example of rollup," J. Computational Phys., v. 13, 1973, pp. 423429.
 [4]
 T. E. DUSHANE, "Convergence for a vortex method for solving Euler's equation," Math. Comp., v. 27, 1973, pp. 719728. 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 twodimensional nonstationary Euler equation," Arch. Rational Mech. Anal., v. 25, 1967, pp. 188200. MR 0211057 (35:1939)
 [7]
 F. J. McGRATH, "Nonstationary plane flow of viscous and ideal fluids," Arch. Rational Mech. Anal., v. 27, 1968, pp. 329348. 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. 380392. MR 0452145 (56:10426)
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 D. W. MOORE, The Discrete Vortex Approximation of a Finite Vortex Sheet, California Inst. of Tech. Report AFOSR180469, 1971.
 [10]
 L. ROSENHEAD, "The formation of vortices from a surface of discontinuity," Proc. Roy. Soc. London Ser. A, v. 134, 1932, pp. 170192.
 [11]
 A. I. SHESTAKOV, Numerical Solution of the NavierStokes 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 SUDAER202, 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. 698726. MR 1545430
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00255718197804920391
PII:
S 00255718(1978)04920391
Article copyright:
© Copyright 1978
American Mathematical Society
