The Annals of Probability

Rate of convergeance of a particle method for the solution of a 1D viscous scalar conservation law in a bounded interval

Mireille Bossy and Benjamin Jourdain

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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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Ann. Probab., Volume 30, Number 4 (2002), 1797-1832.

First available in Project Euclid: 10 December 2002

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Primary: 65N12: Stability and convergence of numerical methods 65C35: Stochastic particle methods [See also 82C80] 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43] 60F99: None of the above, but in this section

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


Bossy, Mireille; Jourdain, Benjamin. Rate of convergeance of a particle method for the solution of a 1D viscous scalar conservation law in a bounded interval. Ann. Probab. 30 (2002), no. 4, 1797--1832. doi:10.1214/aop/1039548372.

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