# Convergence of the Random Vortex Method

## Abstract

This will be rather informal and imprecise. For precise statements and proofs, see the references. I will first explain my motives and goals in studying the particle system that is the random vortex method. Then I will outline the proof of the convergence theorem. For more basic discussion of vortex methods, see the book [2]. Information on the practical success of vortex methods for computing viscous and inviscid flows can be found in [6]. Despite many published demonstrations of the effectiveness of vortex methods, the methods still have many vocal opponents. I hope this confusion will be clarified over the next few years, but I will not comment on it here. Recently, Long [7] has announced a different proof that leads to a sharper convergence theorem than the one discussed here.

## Keywords

Vortex Core Pseudodifferential Operator Particle Simulation Inviscid Flow Monte Carlo Approximation## Preview

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## References

- [1]J.T. Beale and A. Majda, “Vortex methods II: Higher order accuracy in two and three dimensions”, Math. Comp.39 (1982)pp.29–52MathSciNetMATHGoogle Scholar
- [2]A.J. Chorin and J. Marsden, A Mathematical Introduction to Fluid Mechanics, Springer Verlag, New York, 1979Google Scholar
- [3]J.D. Dawson, “Particle simulations of plasmas”, Rev. Mod. Phys., v. 55 (1983), pp. 403–447CrossRefGoogle Scholar
- [4]J. Goodman, “Convergence of the random vortex method”, preprintGoogle Scholar
- [5]A. Leonard, “Computing three-dimensional incompressible flows with vortex elements”, in Annual Review of Fluid Mechanics, ed. by M. van Dyke, J.V. Wehausen, and J.L. Lumley, vol. 17 (1985), pp. 523–559Google Scholar
- [6]O. Hald, “The convergence of vortex methods, II” SIAM J. Num. Anal., 16 (1979) pp. 762–755MathSciNetGoogle Scholar
- [7]Long, PhD thesis, Princeton University, in preparationGoogle Scholar
- [8]H. Osada, “Propagation of chaos for the two dimensional Navier-Stokes equation”, preprint. See also these proceedings.Google Scholar