Vortex methods. I. Convergence in three dimensions

Authors:
J. Thomas Beale and Andrew Majda

Journal:
Math. Comp. **39** (1982), 1-27

MSC:
Primary 65M15; Secondary 76C05

DOI:
https://doi.org/10.1090/S0025-5718-1982-0658212-5

MathSciNet review:
658212

Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Recently several different approaches have been developed for the simulation of three-dimensional 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 two-dimensional case. Can such "crude" algorithms accurately account for vortex stretching and converge? We answer this question affirmatively by constructing a new class of "crude" three-dimensional 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.

**[1]**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)****[2]**J. T. Beale & A. Majda,*The Design and Analysis of Vortex Methods*, Proc. Conf. on Transonic, Shock, and Multi-Dimensional Flows, Madison, Wise., May 1981. (To appear.)**[3]**A. J. Chorin, "Numerical study of slightly viscous flow,"*J. Fluid Mech.*, v. 57, 1973, pp. 785-796. MR**0395483 (52:16280)****[4]**A. J. Chorin, "Vortex models and boundary layer instability,"*SIAM J. Sci. Statist. Comput.*, v. 1, 1980, pp. 1-21. MR**572539 (81e:76043)****[5]**A. J. Chorin, "Estimates of intermittency, spectra, and blow-up 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 roll-up,"*J. Comput. Phys.*, v. 13, 1973, pp. 423-428.**[7]**A. J. Chorin & J. Marsden,*A Mathematical Introduction to Fluid Mechanics*, Springer-Verlag, 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. 726-755. MR**543965 (81b:76015b)****[11]**O. Hald, "Convergence of Fourier methods for the Navier-Stokes equations,"*J. Comput. Phys.*(To appear.) MR**617100 (82k:76006)****[12]**J. K. Hale,*Ordinary Differential Equations*, Wiley-Interscience, 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. 296-305. 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. 199-215. MR**0319382 (47:7926)****[15]**H. O. Kreiss & J. Oliger, "Stability of the Fourier method,"*SIAM J. Numer. Anal.*, v. 16, 1979, pp. 421-433. MR**530479 (80i:65130)****[16]**A. Leonard, "Vortex methods for flow simulations,"*J. Comput. Phys.*, v. 37, 1980, pp. 289-335. MR**588256 (81i:76016)****[17]**A. Leonard,*Numerical Simulation of Interacting Three-Dimensional Vortex Filaments*, Proc. 4th Internat. Conf. Numer. Methods Fluid Dynamics, Springer-Verlag, New York, 1975, pp. 245-249.**[18]**A. Leonard,*Simulation of Three-Dimensional Separated Flows with Vortex Filaments*, Proc. 5th Internat. Conf. Numer. Methods Fluid Dynamics, Springer-Verlag, New York, 1977, pp. 280-284.**[19]**A. Majda, J. McDonough & S. Osher, "The Fourier method for nonsmooth initial data,"*Math. Comp.*, v. 32, 1978, pp. 1041-1081. 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. 170-192.**[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 Navier-Stokes Equations*, Springer-Verlag, New York, 1976, pp. 184-194. MR**0467033 (57:6902)****[23]**R. Temam,*The Navier-Stokes Equations*, North-Holland, Amsterdam, 1977. MR**0609732 (58:29439)****[24]**K. Kuwahara & H. Takami, "Numerical studies of two-dimensional vortex motion by a system of point vortices,"*J. Phys. Soc. Japan*, v. 34, 1973, pp. 247-253.**[25]**L. M. Milne-Thompson,*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, Springer-Verlag, Berlin, 1959. MR**0108116 (21:6836b)**

Retrieve articles in *Mathematics of Computation*
with MSC:
65M15,
76C05

Retrieve articles in all journals with MSC: 65M15, 76C05

Additional Information

DOI:
https://doi.org/10.1090/S0025-5718-1982-0658212-5

Keywords:
Vortex method, incompressible flow

Article copyright:
© Copyright 1982
American Mathematical Society