Vortex methods. I. Convergence in three dimensions
Authors:
J. Thomas Beale and Andrew Majda
Journal:
Math. Comp. 39 (1982), 127
MSC:
Primary 65M15; Secondary 76C05
MathSciNet review:
658212
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Abstract: Recently several different approaches have been developed for the simulation of threedimensional incompressible fluid flows using vortex methods. Some versions use detailed tracking of vortex filament structures and often local curvatures of these filaments, while other methods require only crude information, such as the vortex blobs of the twodimensional case. Can such "crude" algorithms accurately account for vortex stretching and converge? We answer this question affirmatively by constructing a new class of "crude" threedimensional vortex methods and then proving that these methods are stable and convergent, and can even have arbitrarily high order accuracy without being more expensive than other "crude" versions of the vortex algorithm.
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 [1]
 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)
 [2]
 J. T. Beale & A. Majda, The Design and Analysis of Vortex Methods, Proc. Conf. on Transonic, Shock, and MultiDimensional Flows, Madison, Wise., May 1981. (To appear.)
 [3]
 A. J. Chorin, "Numerical study of slightly viscous flow," J. Fluid Mech., v. 57, 1973, pp. 785796. MR 0395483 (52:16280)
 [4]
 A. J. Chorin, "Vortex models and boundary layer instability," SIAM J. Sci. Statist. Comput., v. 1, 1980, pp. 121. MR 572539 (81e:76043)
 [5]
 A. J. Chorin, "Estimates of intermittency, spectra, and blowup in developed turbulence," Comm. Pure Appl. Math. (To appear.) MR 634288 (82m:76036)
 [6]
 A. J. Chorin & P. Bernard, "Discretization of vortex sheet with an example of rollup," J. Comput. Phys., v. 13, 1973, pp. 423428.
 [7]
 A. J. Chorin & J. Marsden, A Mathematical Introduction to Fluid Mechanics, SpringerVerlag, New York, 1979. MR 551053 (81m:76001)
 [8]
 V. Del Prete, Numerical Simulation of Vortex Breakdown, L.B.L. Math, and Comp. Report, 1978.
 [9]
 G. Folland, Introduction to Partial Differential Equations, Princeton Univ. Press, Princeton, N. J., 1978. MR 1357411 (96h:35001)
 [10]
 O. Hald, "The convergence of vortex methods. II," SIAM J. Numer. Anal., v. 16, 1979, pp. 726755. MR 543965 (81b:76015b)
 [11]
 O. Hald, "Convergence of Fourier methods for the NavierStokes equations," J. Comput. Phys. (To appear.) MR 617100 (82k:76006)
 [12]
 J. K. Hale, Ordinary Differential Equations, WileyInterscience, New York, 1969. MR 0419901 (54:7918)
 [13]
 T. Kato, "Nonstationary flows of viscous and ideal fluids in ," J. Funct. Anal., v. 9, 1972, pp. 296305. MR 0481652 (58:1753)
 [14]
 H. O. Kreiss & J. Oliger, "Comparison of accurate methods for the integration of hyperbolic equations," Tellus, v. 24, 1972, pp. 199215. MR 0319382 (47:7926)
 [15]
 H. O. Kreiss & J. Oliger, "Stability of the Fourier method," SIAM J. Numer. Anal., v. 16, 1979, pp. 421433. MR 530479 (80i:65130)
 [16]
 A. Leonard, "Vortex methods for flow simulations," J. Comput. Phys., v. 37, 1980, pp. 289335. MR 588256 (81i:76016)
 [17]
 A. Leonard, Numerical Simulation of Interacting ThreeDimensional Vortex Filaments, Proc. 4th Internat. Conf. Numer. Methods Fluid Dynamics, SpringerVerlag, New York, 1975, pp. 245249.
 [18]
 A. Leonard, Simulation of ThreeDimensional Separated Flows with Vortex Filaments, Proc. 5th Internat. Conf. Numer. Methods Fluid Dynamics, SpringerVerlag, New York, 1977, pp. 280284.
 [19]
 A. Majda, J. McDonough & S. Osher, "The Fourier method for nonsmooth initial data," Math. Comp., v. 32, 1978, pp. 10411081. MR 501995 (80a:65197)
 [20]
 L. Rosenhead, "The point vortex approximation of a vortex sheet," Proc. Roy. Soc. London Ser. A, v. 134, 1932, pp. 170192.
 [21]
 E. Stein & G. Weiss, Fourier Analysis on Euclidean Space, Princeton Univ. Press, Princeton, N. J., 1971. MR 0304972 (46:4102)
 [22]
 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)
 [23]
 R. Temam, The NavierStokes Equations, NorthHolland, Amsterdam, 1977. MR 0609732 (58:29439)
 [24]
 K. Kuwahara & H. Takami, "Numerical studies of twodimensional vortex motion by a system of point vortices," J. Phys. Soc. Japan, v. 34, 1973, pp. 247253.
 [25]
 L. M. MilneThompson, Theoretical Hydrodynamics, 4th ed., Macmillan, New York, 1960. MR 0112435 (22:3286)
 [26]
 J. Serrín, Mathematical Principles of Classical Fluid Mechanics, Handbuch der Physik VIII/I, SpringerVerlag, Berlin, 1959. MR 0108116 (21:6836b)
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00255718198206582125
PII:
S 00255718(1982)06582125
Keywords:
Vortex method, incompressible flow
Article copyright:
© Copyright 1982
American Mathematical Society
