Convergence for a vortex method for solving Euler's equation

Author:
Theodore E. Dushane

Journal:
Math. Comp. **27** (1973), 719-728

MSC:
Primary 76.65

MathSciNet review:
0339675

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Abstract: We consider a new vortex approximation for solving the initial-value 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.

**[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. 241-258.**[3]**Alexandre Joel Chorin,*On the convergence of discrete approximations to the Navier-Stokes equations*, Math. Comp.**23**(1969), 341–353. MR**0242393**, 10.1090/S0025-5718-1969-0242393-5**[4]**A. J. Chorin, "Computational aspects of the turbulence problem,"*Proc. Second International Conference on Numerical Methods in Fluid Dynamics*, Springer-Verlag, Berlin, 1970.**[5]**Alexandre Joel Chorin,*A vortex method for the study of rapid flow*, Proceedings of the Third International Conference on Numerical Methods in Fluid Mechanics (Univ. Paris VI and XI, Paris, 1972) Springer, Berlin, 1973, pp. 100–104. Lecture Notes in Phys., Vol. 19. MR**0479005****[6]**Alexandre Joel Chorin,*Numerical study of slightly viscous flow*, J. Fluid Mech.**57**(1973), no. 4, 785–796. MR**0395483****[7]**O. P. Christiansen, "Numerical simulation of hydrodynamics by the method of point vortices." (To appear.)**[8]**James Glimm,*Solutions in the large for nonlinear hyperbolic systems of equations*, Comm. Pure Appl. Math.**18**(1965), 697–715. MR**0194770****[9]**A. Krzhivitski and O. A. Ladyzhenskaya,*A grid method for the Navier-Stokes equations*, Soviet Physics Dokl.**11**(1966), 212–213. MR**0211619****[10]**R. H. Levy & R. W. Hockney, "Computer experiments on low density crossed field electron beams,"*Phys. Fluids*, v. 11, 1968, pp. 766-771.**[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. 170-192.**[13]**H. Takami,*A Numerical Experiment With Discrete-Vortex 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:
https://doi.org/10.1090/S0025-5718-1973-0339675-6

Keywords:
Vortex approximation,
Euler's equation,
two dimensions,
incompressible fluid flow,
convergence proof

Article copyright:
© Copyright 1973
American Mathematical Society