Convergence for a vortex method for solving Euler's equation
Author:
Theodore E. Dushane
Journal:
Math. Comp. 27 (1973), 719728
MSC:
Primary 76.65
MathSciNet review:
0339675
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Abstract: We consider a new vortex approximation for solving the initialvalue problem for the Euler equations in two dimensions. We assume there exists a smooth solution to these equations and that the vorticity has compact support. Then we show that our approximation to the velocity field converges uniformly in space and time for a short time interval.
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H. Takami, A Numerical Experiment With DiscreteVortex Approximation, With Reference to the Rolling Up of a Vortex Sheet, Dept. of Aero. and Astro., Stanford Univ. Report SUDAER 202, Sept. 1964.
 [1]
 G. K. Batchellor, An Introduction to Fluid Dynamics, Cambridge Univ. Press, Cambridge, 1967.
 [2]
 C. K. Birdsall, A. B. Langdon & H. Okuda, "Finite size particle physics applied to plasma simulation," Methods in Computational Physics, v. 9, 1970, pp. 241258.
 [3]
 A. J. Chorin, "On the convergence of discrete approximations to the NavierStokes equations," Math. Comp., v. 23, 1969, pp. 341353. MR 39 #3724. MR 0242393 (39:3724)
 [4]
 A. J. Chorin, "Computational aspects of the turbulence problem," Proc. Second International Conference on Numerical Methods in Fluid Dynamics, SpringerVerlag, Berlin, 1970.
 [5]
 A. J. Chorin, "A vortex method for the study of rapid flows," Proc. Third International Conference on Numerical Methods in Fluid Dynamics, SpringerVerlag, Berlin, 1973. MR 0479005 (57:18461)
 [6]
 A. J. Chorin, "Numerical study of slightly viscous flows." (To appear.) MR 0395483 (52:16280)
 [7]
 O. P. Christiansen, "Numerical simulation of hydrodynamics by the method of point vortices." (To appear.)
 [8]
 J. Glimm, "Solutions in the large for nonlinear hyperbolic systems of equations," Comm. Pure Appl. Math., v. 18, 1965, pp. 697715. MR 33 #2976. MR 0194770 (33:2976)
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 A. Kržywicki & O. Ladyženskaja, "A grid method for the NavierStokes equations," Dokl. Akad. Nauk SSSR, v. 167, 1966, pp. 309311 = Soviet Physics Dokl. v. 11, 1966, pp. 212213. MR 35 #2497. MR 0211619 (35:2497)
 [10]
 R. H. Levy & R. W. Hockney, "Computer experiments on low density crossed field electron beams," Phys. Fluids, v. 11, 1968, pp. 766771.
 [11]
 D. W. Moore, "The discrete approximation of a finite vortex sheet." (Preprint.)
 [12]
 L. Rosenhead, "The formation of vortices from a surface of discontinuity," Proc. Roy. Soc. Ser. A, v. 134, 1932, pp. 170192.
 [13]
 H. Takami, A Numerical Experiment With DiscreteVortex Approximation, With Reference to the Rolling Up of a Vortex Sheet, Dept. of Aero. and Astro., Stanford Univ. Report SUDAER 202, Sept. 1964.
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00255718197303396756
PII:
S 00255718(1973)03396756
Keywords:
Vortex approximation,
Euler's equation,
two dimensions,
incompressible fluid flow,
convergence proof
Article copyright:
© Copyright 1973
American Mathematical Society
