Weak and strong error analysis for mean-field rank based particle approximations of one dimensional viscous scalar conservation laws

In this talk, we analyse the rate of convergence of a system of N interacting particles with mean-field rank based interaction in the drift coefficient and constant diffusion coefficient. We first adapt arguments by Kolli and Shkolnikhov to check trajectorial propagation of chaos with optimal rate N^{-1/2} to the associated stochastic differential equations nonlinear in the sense of McKean. We next relax the assumptions needed by Bossy to check convergence in L^1(R) with rate O(1/N^{1/2} + h) of the empirical cumulative distribution function of the Euler discretization with step h of the particle system to the solution of a one dimensional viscous scalar conservation law. Last, we prove that the bias of this stochastic particle method behaves in O(1/N + h). We provide numerical results which confirm our theoretical estimates.