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

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.