Remote Access Mathematics of Computation
Green Open Access

Mathematics of Computation

ISSN 1088-6842(online) ISSN 0025-5718(print)

 
 

 

Convergence of a higher-order vortex method for two-dimensional Euler equations


Authors: C. Chiu and R. A. Nicolaides
Journal: Math. Comp. 51 (1988), 507-534
MSC: Primary 65N30; Secondary 76-08, 76C05
DOI: https://doi.org/10.1090/S0025-5718-1988-0935078-2
MathSciNet review: 935078
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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.


References [Enhancements On Off] (What's this?)

  • [1] C. Anderson & C. Greengard, "On vortex methods," SIAM J. Numer. Anal., v. 22, 1985, pp. 413-440. MR 787568 (86j:76016)
  • [2] J. T. Beale & A. Majda, "Vortex methods I: Convergence in three dimensions," Math. Comp., v. 39, 1982, pp. 1-27. 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. 29-52. MR 658213 (83i:65069b)
  • [4] J. T. Beale & A. Majda, "Rates of convergence for viscous splitting of the Navier-Stokes equations," Math. Comp., v. 37, 1981, pp. 243-259. 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. 785-796. MR 0395483 (52:16280)
  • [7] A. J. Chorin & J. Marsden, A Mathematical Introduction to Fluid Mechanics, Springer-Verlag, 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, Springer-Verlag, Berlin, 1986.
  • [9] P. G. Ciarlet, The Finite Element Method for Elliptic Problems, North-Holland, 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. 726-755. 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, McGraw-Hill, New York, 1981.
  • [14] T. Kato, "Nonstationary flows of viscous and ideal fluids in $ {R^3}$," J. Funct. Anal., v. 9, 1972, pp. 296-305. MR 0481652 (58:1753)
  • [15] A. Leonard, "Vortex methods for flow simulations," J. Comput. Phys., v. 37, 1980, pp. 289-335. MR 588256 (81i:76016)
  • [16] F. J. McGrath, "Nonstationary plane flow of viscous and ideal fluids," Arch. Rational Mech. Anal., v. 27, 1968, pp. 328-348. MR 0221818 (36:4870)
  • [17] R. A. Nicolaides, "Construction of higher order accurate vortex and particle methods," Appl. Numer. Math., v. 2, 1986, pp. 313-320. 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 $ {C^\infty }$ solutions of the Euler equations of incompressible perfect fluids," Turbulence and Navier-Stokes Equations (R. Temam, ed.), Lecture Notes in Math., vol. 565, Springer-Verlag, Berlin, 1976. MR 0467033 (57:6902)

Similar Articles

Retrieve articles in Mathematics of Computation with MSC: 65N30, 76-08, 76C05

Retrieve articles in all journals with MSC: 65N30, 76-08, 76C05


Additional Information

DOI: https://doi.org/10.1090/S0025-5718-1988-0935078-2
Article copyright: © Copyright 1988 American Mathematical Society

American Mathematical Society