Convergence rate for the approximation of the limit law of weakly interacting particles: application to the Burgers equation

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.