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

In this paper, 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 Shkolnikov [22] 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 [6] to check the convergence in L (R) with rate O ( 1 √ N + 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 as O ( 1 N + h ) . We provide numerical results which confirm our theoretical estimates.