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PROBABILISTIC INTERPRETATION AND PARTICLE METHOD FOR VORTEX EQUATIONS WITH NEUMANN’S BOUNDARY CONDITION

Published online by Cambridge University Press:  09 November 2004

Benjamin Jourdain  and
Sylvie Méléard
Benjamin Jourdain
Affiliation:
ENPC-CERMICS, 6–8 av. Blaise Pascal, Cité Descartes, Champs sur Marne, 77455 Marne-la-Vallée Cedex 2, France (jourdain@cermics.enpc.fr)
Sylvie Méléard
Affiliation:
Université Paris 10, MODALX, 200 av. de la République, 92000 Nanterre, France (sylvie.meleard@u-paris10.fr)
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Abstract

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We are interested in proving the convergence of Monte Carlo approximations for vortex equations in bounded domains of $\mathbb{R}^2$ with Neumann’s condition on the boundary. This work is the first step towards justifying theoretically some numerical algorithms for Navier–Stokes equations in bounded domains with no-slip conditions.

We prove that the vortex equation has a unique solution in an appropriate energy space and can be interpreted from a probabilistic point of view through a nonlinear reflected process with space-time random births on the boundary of the domain.

Next, we approximate the solution $w$ of this vortex equation by the weighted empirical measure of interacting diffusive particles with normal reflecting boundary conditions and space-time random births on the boundary. The weights are related to the initial data and to the Neumann condition. We prove a trajectorial propagation-of-chaos result for these systems of interacting particles. We can deduce a simple stochastic particle algorithm to simulate $w$.

AMS 2000 Mathematics subject classification: Primary 60K35; 76D05

Keywords

vortex equation on a bounded domainMonte Carlo approximationinteracting particle systems with reflectionspace-time random births
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 47 , Issue 3 , October 2004 , pp. 597 - 624
Copyright
Copyright © Edinburgh Mathematical Society 2004