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Mathematics of Computation

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A random space-time birth particle method for 2d vortex equations with external field

Authors: J. Fontbona and S. Méléard
Journal: Math. Comp. 77 (2008), 1525-1558
MSC (2000): Primary 65C35, 76D17, 76M23; Secondary 82C22
Published electronically: February 22, 2008
MathSciNet review: 2398779
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Abstract: We consider incompressible 2d Navier-Stokes equations in the whole plane with external nonconservative forces fields. The initial data and external field are functions assumed to satisfy only slight integrability properties. We develop a probabilistic interpretation of these equations based on the associated vortex equation, in order to construct a numerical particle method to approximate the solutions. More precisely, we relate the vortex equation with additional term to a nonlinear process with random space-time birth, which provides a probabilistic description of the creation of vorticity. We then introduce interacting particle systems defined for a regularized interaction kernel, whose births are chosen randomly in time and space. By a coupling method, we show that these systems are approximations of the nonlinear process and obtain precise convergence estimates. From this result, we deduce a stochastic numerical particle method to obtain the vorticity and also to recover the velocity field. The results are either pathwise or of weak convergence, depending on the integrability of the data. We illustrate our results with simulations.

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Additional Information

J. Fontbona
Affiliation: CMM-DIM, UMI(2807) UCHILE-CNRS, Universidad de Chile. Casilla 170-3, Correo 3, Santiago-Chile

S. Méléard
Affiliation: CMAP, ECOLE POLYTECHNIQUE, CNRS, Route de Saclay 91128 Palaiseau Cedex France

Received by editor(s): November 6, 2006
Received by editor(s) in revised form: June 22, 2007
Published electronically: February 22, 2008
Additional Notes: The first author was supported by Fondecyt Projects 1040689 and 1070743, Millennium Nucleus ICM P04-069-F, ECOS C05E02 and FONDAP Applied Mathematics
The second author was supported by Fondecyt International Cooperation 7050142 and ECOS C05E02.
Article copyright: © Copyright 2008 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.