Vortex methods. II. Higher order accuracy in two and three dimensions
Authors:
J. Thomas Beale and Andrew Majda
Journal:
Math. Comp. 39 (1982), 2952
MSC:
Primary 65M15; Secondary 76C05
MathSciNet review:
658213
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Abstract: In an earlier paper the authors introduced a new version of the vortex method for threedimensional, incompressible flows and proved that it converges to arbitrarily high order accuracy, provided we assume the consistency of a discrete approximation to the BiotSavart Law. We prove this consistency statement here, and also derive substantially sharper results for twodimensional flows. A complete, simplified proof of convergence in two dimensions is included.
 [1]
J.
Thomas Beale and Andrew
Majda, Vortex methods. I. Convergence in
three dimensions, Math. Comp.
39 (1982), no. 159, 1–27. MR 658212
(83i:65069a), http://dx.doi.org/10.1090/S00255718198206582125
 [2]
J. T. Beale & A. Majda, The Design and Analysis of Vortex Methods, Proc. Conf. on Transonic, Shock, and MultiDimensional Flows, Madison, Wisc., May 1981.
 [3]
J.
Thomas Beale and Andrew
Majda, Rates of convergence for viscous
splitting of the NavierStokes equations, Math.
Comp. 37 (1981), no. 156, 243–259. MR 628693
(82i:65056), http://dx.doi.org/10.1090/S00255718198106286930
 [4]
Lipman
Bers, Fritz
John, and Martin
Schechter, Partial differential equations, Lectures in Applied
Mathematics, Vol. III, Interscience Publishers John Wiley & Sons, Inc.
New YorkLondonSydney, 1964. MR 0163043
(29 #346)
 [5]
Alexandre
Joel Chorin, Numerical study of slightly viscous flow, J.
Fluid Mech. 57 (1973), no. 4, 785–796. MR 0395483
(52 #16280)
 [6]
Gerald
B. Folland, Introduction to partial differential equations,
2nd ed., Princeton University Press, Princeton, NJ, 1995. MR 1357411
(96h:35001)
 [7]
Ole
H. Hald, Convergence of vortex methods for Euler’s equations.
II, SIAM J. Numer. Anal. 16 (1979), no. 5,
726–755. MR
543965 (81b:76015b), http://dx.doi.org/10.1137/0716055
 [8]
Ole
Hald and Vincenza
Mauceri del Prete, Convergence of vortex methods for
Euler’s equations, Math. Comp.
32 (1978), no. 143, 791–809. MR 492039
(81b:76015a), http://dx.doi.org/10.1090/S00255718197804920391
 [9]
Jack
K. Hale, Ordinary differential equations, WileyInterscience
[John Wiley & Sons], New YorkLondonSydney, 1969. Pure and Applied
Mathematics, Vol. XXI. MR 0419901
(54 #7918)
 [10]
L. Hörmander, "Fourier integral operators. I," Acta Math., v. 127, 1971, pp. 79183.
 [11]
K. Kuwahara & H. Takami, "Numerical studies of a twodimensional vortex motion by a system of point vortices," J. Phys. Soc. Japan, v. 34, 1973, pp. 247253.
 [12]
A.
Leonard, Vortex methods for flow simulation, J. Comput. Phys.
37 (1980), no. 3, 289–335. MR 588256
(81i:76016), http://dx.doi.org/10.1016/00219991(80)900406
 [13]
F.
J. McGrath, Nonstationary plane flow of viscous and ideal
fluids, Arch. Rational Mech. Anal. 27 (1967),
329–348. MR 0221818
(36 #4870)
 [14]
Elias
M. Stein and Guido
Weiss, Introduction to Fourier analysis on Euclidean spaces,
Princeton University Press, Princeton, N.J., 1971. Princeton Mathematical
Series, No. 32. MR 0304972
(46 #4102)
 [15]
Michael
Taylor, Pseudo differential operators, Lecture Notes in
Mathematics, Vol. 416, SpringerVerlag, BerlinNew York, 1974. MR 0442523
(56 #905)
 [1]
 J. T. Beale & A. Majda, "Vortex methods. I: Convergence in three dimensions," Math. Comp., v. 39, 1982, pp. 127. MR 658212 (83i:65069a)
 [2]
 J. T. Beale & A. Majda, The Design and Analysis of Vortex Methods, Proc. Conf. on Transonic, Shock, and MultiDimensional Flows, Madison, Wisc., May 1981.
 [3]
 J. T. Beale & A. Majda, "Rates of convergence for viscous splitting of the NavierStokes equations," Math. Comp., v. 37, 1981, pp. 243260. MR 628693 (82i:65056)
 [4]
 L. Bers, F. John & M. Schechter, Partial Differential Equations, Lectures in Appl. Math., Vol. 3A, Amer. Math. Soc., Providence, R.I., 1964. MR 0163043 (29:346)
 [5]
 A. J. Chorin, "Numerical study of slightly viscous flow," J. Fluid Mech., v. 57, 1973, pp. 785796. MR 0395483 (52:16280)
 [6]
 G. Folland, Introduction to Partial Differential Equations. Princeton Univ. Press, Princeton, N.J., 1978. MR 1357411 (96h:35001)
 [7]
 O. Hald, "The convergence of vortex methods. II," SIAM J. Numer. Anal., v. 16, 1979, pp. 726755. MR 543965 (81b:76015b)
 [8]
 O. Hald & V. M. Del Prete, "Convergence of vortex methods for Euler's equations," Math. Comp., v. 32, 1978, pp. 791809. MR 492039 (81b:76015a)
 [9]
 J. K. Hale, Ordinary Differential Equations, WileyInterscience, New York, 1969. MR 0419901 (54:7918)
 [10]
 L. Hörmander, "Fourier integral operators. I," Acta Math., v. 127, 1971, pp. 79183.
 [11]
 K. Kuwahara & H. Takami, "Numerical studies of a twodimensional vortex motion by a system of point vortices," J. Phys. Soc. Japan, v. 34, 1973, pp. 247253.
 [12]
 A. Leonard, "Vortex methods for flow simulations," J. Comput. Phys., v. 37, 1980, pp. 289335. MR 588256 (81i:76016)
 [13]
 F. J. McGrath, "Nonstationary plane flow of viscous and ideal fluids," Arch. Rational Mech. Anal., v. 27, 1968, pp. 328348. MR 0221818 (36:4870)
 [14]
 E. M. Stein & G. Weiss, Introduction to Fourier Analysis on Euclidean Spaces, Princeton Univ. Press, Princeton, N.J., 1971. MR 0304972 (46:4102)
 [15]
 M. Taylor, Pseudo Differential Operators, Lecture Notes in Math., Vol. 417, SpringerVerlag, Berlin, 1974. MR 0442523 (56:905)
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00255718198206582137
PII:
S 00255718(1982)06582137
Keywords:
Vortex method,
incompressible flow
Article copyright:
© Copyright 1982
American Mathematical Society
