Convergence Rate for the Approximation of the Limit Law of Weakly Interacting Particles 2: Application to the Burgers Equation

Mireille Bossy 1 Denis Talay 1
1 OMEGA - Probabilistic numerical methods
CRISAM - Inria Sophia Antipolis - Méditerranée , UHP - Université Henri Poincaré - Nancy 1, Université Nancy 2, CNRS - Centre National de la Recherche Scientifique : UMR7502
Abstract : In this paper, we construct a stochastic particles method for the Burgers equation with a monotonic initial condition; we prove that the convergence rate is $\displaystyleO\left(\frac1\sqrtN +\sqrt\D\right)$ for the $L^1(I\!\!R \times \Omega)$-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, equal to the Heaviside function. In~\citebossy_talay-93, we show how the algorithm and the result extend to the case of non monotonic initial conditions for the Burgers equation; we also treat the case of nonlinear PDE's related to particles systems with Lipschitz interaction kernels. Our next objective is to adapt our methodology to the (more difficult) case of the 2-D inviscid Navier-Stokes equation.
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Rapport
[Research Report] RR-2410, INRIA. 1994, pp.49
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https://hal.inria.fr/inria-00074265
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Soumis le : mercredi 24 mai 2006 - 14:53:53
Dernière modification le : samedi 27 janvier 2018 - 01:31:03
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Mireille Bossy, Denis Talay. Convergence Rate for the Approximation of the Limit Law of Weakly Interacting Particles 2: Application to the Burgers Equation. [Research Report] RR-2410, INRIA. 1994, pp.49. 〈inria-00074265〉

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