A convergent D vortex method with gridfree stretching
Author:
J. Thomas Beale
Journal:
Math. Comp. 46 (1986), 401424, S15
MSC:
Primary 76C05; Secondary 65M15, 7608
MathSciNet review:
829616
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Abstract: We prove the convergence of a vortex method for threedimensional, incompressible, inviscid flow without boundaries. This version differs from an earlier one whose convergence was shown in [4] in that the calculation does not depend explicitly on the arrangement of the vorticity elements in a Lagrangian frame. Thus, it could be used naturally in a more general context in which boundaries and viscosity are present. It is also shown that previous estimates for the velocity approximation can be improved by taking into account the fact that the integral kernel has average value zero. Implications for the design of the method are discussed.
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M. Stein, Singular integrals and differentiability properties of
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 [1]
 C. Anderson & C. Greengard, "On vortex methods," SIAM J. Numer. Anal., v. 22, 1985, pp. 413440. MR 787568 (86j:76016)
 [2]
 C. Anderson, Vortex methods for Flows of Variable Density, Ph. D. Thesis, Univ. of California, Berkeley, 1983.
 [3]
 C. R. Anderson, "A vortex method for flows with slight density variation," J. Comput. Phys., v. 61, 1985, pp. 417444. MR 816663 (87h:76034)
 [4]
 J. T. Beale & A. Majda, "Vortex methods, I: Convergence in three dimensions," Math. Comp., v. 39, 1982, pp. 127. MR 658212 (83i:65069a)
 [5]
 J. T. Beale & A. Majda, "Vortex methods, II: Higher order accuracy in two and three dimensions," Math. Comp., v. 39, 1982, pp. 2952. MR 658213 (83i:65069b)
 [6]
 J. T. Beale & A. Majda, "High order accurate vortex methods with explicit velocity kernels," J. Comput. Phys., v. 58, 1985, pp. 188208.
 [7]
 J. T. Beale & A. Majda, "Vortex methods for fluid flow in two or three dimensions," Contemp. Math., v. 28, 1984, pp. 221229. MR 751986 (86e:76023)
 [8]
 J. T. Beale & A. Majda, "Rates of convergence for viscous splitting of the NavierStokes equations," Math. Comp., v. 37, 1981, pp. 243259. MR 628693 (82i:65056)
 [9]
 A. Cheer, "Numerical analysis of time dependent flow structure generated by an impulsively started circular cylinder in a slightly viscous incompressible fluid," SIAM J. Sci. Statist. Comput. (To appear.)
 [10]
 A. Chorin, "Vortex methods and boundary layer instability," SIAM J. Sci. Statist. Comput., v. 1, 1980, pp. 121. MR 572539 (81e:76043)
 [11]
 A. Chorin, "The evolution of a turbulent vortex", Comm. Math. Phys., v. 83, 1982, pp. 517535. MR 649815 (83g:76042)
 [12]
 A. Chorin & J. Marsden, A Mathematical Introduction to Fluid Mechanics, SpringerVerlag, New York, 1979. MR 551053 (81m:76001)
 [13]
 A. Chorin, T. Hughes, M. McCracken & J. Marsden, "Product formulas and numerical algorithms," Comm. Pure Appl. Math., v. 31, 1978, pp. 205256. MR 0488713 (58:8230)
 [14]
 G.H. Cottet & P.A. Raviart, "Particle methods for the onedimensional VlasovPoisson equations," SIAM J. Numer. Anal., v. 21, 1984, pp. 5276. MR 731212 (85c:82048)
 [15]
 G.H. Cottet, Méthodes Particulaires pour l'Equation d'Euler dans le Plan, Thèse de 3e cycle, Université P. et M. Curie, Paris, 1982.
 [16]
 G. Folland, Introduction to Partial Differential Equations, Princeton Univ. Press, Princeton, N. J., 1978. MR 1357411 (96h:35001)
 [17]
 C. Greengard, ThreeDimensional Vortex Methods, Ph. D. thesis, Univ. of California, Berkeley, 1984.
 [18]
 O. Hald & V. Del Prete, "Convergence of vortex methods for Euler's equations," Math. Comp., v. 32, 1978, pp. 791809. MR 492039 (81b:76015a)
 [19]
 O. Hald, "The convergence of vortex methods, II," SIAM J. Numer. Anal., v. 16, 1979, pp. 726755. MR 543965 (81b:76015b)
 [20]
 T. Kato, "Nonstationary flows of viscous and ideal fluids in ," J. Funct. Anal., v. 9, 1972, pp. 296305. MR 0481652 (58:1753)
 [21]
 A. Leonard, "Numerical simulation of interacting threedimensional vortex filaments," Proc. 4th Internat. Conf. Numer. Methods Fluid Dynamics, Lecture Notes in Physics, vol. 35, SpringerVerlag, New York, 1975, pp. 245249.
 [22]
 A. Leonard, "Vortex methods for flow simulations," J. Comput. Phys., v. 37, 1980, pp. 289335. MR 588256 (81i:76016)
 [23]
 A. Leonard, "Computing threedimensional incompressible flows with vortex elements," Ann. Rev. Fluid Mech., v. 17, 1985, pp. 523529.
 [24]
 C. Marchioro & M. Pulvirenti, "Hydrodynamics in two dimensions and vortex theory," Comm. Math. Phys., v. 84, 1982, pp. 483503. MR 667756 (84e:35126)
 [25]
 Y. Nakamura, A. Leonard & P. Spalart, Numerical Simulation of Vortex Breakdown by the VortexFilament Method, AGARD Proceedings #342, Rotterdam, 1983.
 [26]
 P. A. Raviart, An Analysis of Particle Methods, CIME Course, Como, Italy, 1983.
 [27]
 P. G. Saffman, "Vortex interactions and coherent structures in turbulence," Transition and Turbulence (R. Meyer, ed.), Academic Press, New York, 1981, pp. 149166.
 [28]
 E. D. Siggia, "Collapse and amplification of a vortex filament," Phys. Fluids, v. 28, 1985, pp. 794805.
 [29]
 P. Spalart, Numerical Simulation of Separated Flows, Ph. D. thesis, Stanford University, 1982.
 [30]
 E. M. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton Univ. Press, Princeton, N. J., 1970. MR 0290095 (44:7280)
 [31]
 R. Temam, "Local existence of solutions of the Euler equations of incompressible perfect fluids," in Turbulence and the NavierStokes Equations, SpringerVerlag, New York, 1976, pp. 184194. MR 0467033 (57:6902)
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00255718198608296166
PII:
S 00255718(1986)08296166
Article copyright:
© Copyright 1986
American Mathematical Society
