Convergence of the vortex filament method

Author:
Claude Greengard

Journal:
Math. Comp. **47** (1986), 387-398

MSC:
Primary 65M15; Secondary 76-08

DOI:
https://doi.org/10.1090/S0025-5718-1986-0856692-7

MathSciNet review:
856692

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Abstract: Fully discrete convergence estimates have previously been given for the three-dimensional vortex method proposed by Beale and Majda. It is shown in this paper that vortex filament methods of the kind used in practice converge, provided smooth vortex structures consisting of closed filaments are appropriately discretized, and the stretching of the discrete filaments is computed sufficiently accurately. The error estimates obtained are those of the previous theory.

**[1]**C. Anderson & C. Greengard, "On vortex methods,"*SIAM J. Numer. Anal.*, v. 22, 1985, pp. 413-440. MR**787568 (86j:76016)****[2]**J. T. Beale & A. Majda, "Vortex methods. I: Convergence in three dimensions,"*Math. Comp.*, v. 39, 1982, pp. 1-27. MR**658212 (83i:65069a)****[3]**J. T. Beale & A. Majda, "Vortex methods. II: Higher order accuracy in two and three dimensions,"*Math. Comp.*, v. 39, 1982, pp. 29-52. MR**658213 (83i:65069b)****[4]**J. T. Beale & A. Majda, "High order accurate vortex methods with explicit velocity kernels,"*J. Comput. Phys.*, v. 58, 1985, pp. 188-208.**[5]**A. J. Chorin, "Estimates of intermittency, spectra, and blow-up in developed turbulence,"*Comm. Pure Appl. Math.*, v. 34, 1981, pp. 641-672. MR**634288 (82m:76036)****[6]**G. H. Cottet,*Méthodes Particulaires pour l'Equation d'Euler dans le Plan*, Thèse de 3ème cycle, Université P. et M. Curie, Paris, 1982.**[7]**G. H. Cottet & P. A. Raviart, "Particle methods for one-dimensional Vlasov-Poisson equations,"*SIAM J. Numer. Anal.*, v. 21, 1984, pp. 52-76. MR**731212 (85c:82048)****[8]**C. Greengard,*Three Dimensional Vortex Methods*, Ph.D. Thesis, Univ. of California, Berkeley, 1984.**[9]**O. Hald, "The convergence of vortex methods, II,"*SIAM J. Numer. Anal.*, v. 16, 1979, pp. 726-755. MR**543965 (81b:76015b)****[10]**T. Kato, "Nonstationary flows of viscous and ideal fluids in ,"*J. Funct. Anal.*, v. 9, 1972, pp. 296-305. MR**0481652 (58:1753)****[11]**A. Leonard, "Computing three-dimensional incompressible flows with vortex elements,"*Annual Review of Fluid Mechanics*, v. 17, 1985, pp. 523-559.**[12]**P. A. Raviart,*An Analysis of Particle Methods*, CIME Course in Numerical Methods in Fluid Dynamics, Como, July 1983.

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Additional Information

DOI:
https://doi.org/10.1090/S0025-5718-1986-0856692-7

Article copyright:
© Copyright 1986
American Mathematical Society