Convergence of the Random Vortex Method

• Jonathan Goodman
Chapter
Part of the The IMA Volumes in Mathematics and Its Applications book series (IMA, volume 9)

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
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

References

1. [1]
J.T. Beale and A. Majda, “Vortex methods II: Higher order accuracy in two and three dimensions”, Math. Comp.39 (1982)pp.29–52
2. [2]
A.J. Chorin and J. Marsden, A Mathematical Introduction to Fluid Mechanics, Springer Verlag, New York, 1979Google Scholar
3. [3]
J.D. Dawson, “Particle simulations of plasmas”, Rev. Mod. Phys., v. 55 (1983), pp. 403–447
4. [4]
J. Goodman, “Convergence of the random vortex method”, preprintGoogle Scholar
5. [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. [6]
O. Hald, “The convergence of vortex methods, II” SIAM J. Num. Anal., 16 (1979) pp. 762–755
7. [7]
Long, PhD thesis, Princeton University, in preparationGoogle Scholar
8. [8]

© Springer-Verlag New York Inc. 1987

Authors and Affiliations

• Jonathan Goodman
• 1
1. 1.Courant Institute of Mathematical SciencesNew York UniversityNew YorkUSA