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August 1996 Convergence rate for the approximation of the limit law of weakly interacting particles: application to the Burgers equation
Mireille Bossy, Denis Talay
Ann. Appl. Probab. 6(3): 818-861 (August 1996). DOI: 10.1214/aoap/1034968229

Abstract

In this paper we construct a stochastic particle method for the Burgers equation with a monotone initial condition; we prove that the convergence rate is $O(1/ \sqrt{N} + \sqrt{\Delta t})$ for the $L^1 (\mathbb{R} \times \Omega)$ norm of the error. To obtain that result, we link the PDE and the algorithm to a system of weakly interacting stochastic particles; the difficulty of the analysis comes from the discontinuity of the interaction kernel, which is equal to the Heaviside function.

In a previous paper we showed how the algorithm and the result extend to the case of nonmonotone initial conditions for the Burgers equation. We also treated the case of nonlinear PDE's related to particle systems with Lipschitz interaction kernels. Our next objective is to adapt our methodology to the (more difficult) case of the two-dimensional inviscid Navier-Stokes equation.

Citation

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Mireille Bossy. Denis Talay. "Convergence rate for the approximation of the limit law of weakly interacting particles: application to the Burgers equation." Ann. Appl. Probab. 6 (3) 818 - 861, August 1996. https://doi.org/10.1214/aoap/1034968229

Information

Published: August 1996
First available in Project Euclid: 18 October 2002

zbMATH: 0860.60038
MathSciNet: MR1410117
Digital Object Identifier: 10.1214/aoap/1034968229

Subjects:
Primary: 60H10 , 60K35 , 65C20 , 65M15 , 65U05

Keywords: Burgers equation , interacting particle systems , Stochastic particle methods

Rights: Copyright © 1996 Institute of Mathematical Statistics

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Vol.6 • No. 3 • August 1996
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