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October 2002 Rate of convergeance of a particle method for the solution of a 1D viscous scalar conservation law in a bounded interval
Mireille Bossy, Benjamin Jourdain
Ann. Probab. 30(4): 1797-1832 (October 2002). DOI: 10.1214/aop/1039548372

Abstract

In this paper, we give a probabilistic interpretation of a viscous scalar conservation law in a bounded interval thanks to a nonlinear martingale problem. The underlying nonlinear stochastic process is reflected at the boundary to take into account the Dirichlet conditions. After proving uniqueness for the martingale problem, we show existence thanks to a propagation of chaos result. Indeed we exhibit a system of N interacting particles, the empirical measure of which converges to the unique solution of the martingale problem as $N\to+\infty$. As a consequence, the solution of the viscous conservation law can be approximated thanks to a numerical algorithm based on the simulation of the particle system. When this system is discretized in time thanks to the Euler-Lépingle scheme, we show that the rate of convergence of the error is in $\OO(\Delta t +1/\sqrt{N})$, where $\Delta t$ denotes the time step. Finally, we give numerical results which confirm this theoretical rate.

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Mireille Bossy. Benjamin Jourdain. "Rate of convergeance of a particle method for the solution of a 1D viscous scalar conservation law in a bounded interval." Ann. Probab. 30 (4) 1797 - 1832, October 2002. https://doi.org/10.1214/aop/1039548372

Information

Published: October 2002
First available in Project Euclid: 10 December 2002

zbMATH: 1013.60080
MathSciNet: MR1944006
Digital Object Identifier: 10.1214/aop/1039548372

Subjects:
Primary: 60F99 , 60K35 , 65C35 , 65N12

Keywords: Euler discretization scheme , nonlinear martingale problem , Reflected stochastic processes , weak convergence rate

Rights: Copyright © 2002 Institute of Mathematical Statistics

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Vol.30 • No. 4 • October 2002
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