Convergence of a higherorder vortex method for twodimensional Euler equations
Authors:
C. Chiu and R. A. Nicolaides
Journal:
Math. Comp. 51 (1988), 507534
MSC:
Primary 65N30; Secondary 7608, 76C05
MathSciNet review:
935078
Fulltext PDF Free Access
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Abstract: There has been considerable interest recently in the convergence properties of point vortex methods. In this paper, we define a vortex method using vortex multipoles and obtain error estimates for it. In the case of a nonuniform mesh, the rate of convergence of the dipolar algorithm is shown to be of higher order of accuracy than obtained with the simple vortex methods.
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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]
 J. T. Beale & A. Majda, "Vortex methods I: Convergence in three dimensions," Math. Comp., v. 39, 1982, pp. 127. 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. 2952. MR 658213 (83i:65069b)
 [4]
 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)
 [5]
 G. H. Cottet, Méthodes Particulaires pour l'équation d'Euler dans le Plan, Thèse de 3ème cycle, Université Paris VI, 1982.
 [6]
 A. J. Chorin, "Numerical study of slightly viscous flow," J. Fluid Mech., v. 57, 1973, pp. 785796. MR 0395483 (52:16280)
 [7]
 A. J. Chorin & J. Marsden, A Mathematical Introduction to Fluid Mechanics, SpringerVerlag, New York, 1979. MR 551053 (81m:76001)
 [8]
 S. Choudhury & R. A. Nicolaides, "Vortex multipole methods for viscous incompressible flows," 10th International Conf. on Numerical Methods in Fluid Dynamics (F. G. Zhuang and Y. L. Zhu, eds.), Lecture Notes in Phys., vol. 264, SpringerVerlag, Berlin, 1986.
 [9]
 P. G. Ciarlet, The Finite Element Method for Elliptic Problems, NorthHolland, Amsterdam, 1978. MR 0520174 (58:25001)
 [10]
 O. H. Hald, "Convergence of vortex methods for Euler's equations. II," SIAM J. Numer. Anal., v. 16, 1979, pp. 726755. MR 543965 (81b:76015b)
 [11]
 J. K. Hale, Ordinary Differential Equations, Wiley Interscience, New York, 1969. MR 0419901 (54:7918)
 [12]
 F. H. Harlow, "The particle in cell computing method for fluid dynamics," Methods in Computational Physics (B. Alder, S. Fernbach & M. Rotenberg, eds.), Vol. 3, Academic Press, New York, 1964.
 [13]
 R. W. Hockney & J. W. Eastwood, Computer Simulation Using Particles, McGrawHill, New York, 1981.
 [14]
 T. Kato, "Nonstationary flows of viscous and ideal fluids in ," J. Funct. Anal., v. 9, 1972, pp. 296305. MR 0481652 (58:1753)
 [15]
 A. Leonard, "Vortex methods for flow simulations," J. Comput. Phys., v. 37, 1980, pp. 289335. MR 588256 (81i:76016)
 [16]
 F. J. McGrath, "Nonstationary plane flow of viscous and ideal fluids," Arch. Rational Mech. Anal., v. 27, 1968, pp. 328348. MR 0221818 (36:4870)
 [17]
 R. A. Nicolaides, "Construction of higher order accurate vortex and particle methods," Appl. Numer. Math., v. 2, 1986, pp. 313320. MR 863990 (87k:65119)
 [18]
 P. A. Raviart, "An analysis of particle methods," Numerical Methods in Fluid Dynamics, Como, July, 1983.
 [19]
 R.Temam, "Local existence of solutions of the Euler equations of incompressible perfect fluids," Turbulence and NavierStokes Equations (R. Temam, ed.), Lecture Notes in Math., vol. 565, SpringerVerlag, Berlin, 1976. MR 0467033 (57:6902)
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00255718198809350782
PII:
S 00255718(1988)09350782
Article copyright:
© Copyright 1988
American Mathematical Society
