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 (Ann. Probab. 46 (2018) 1042-1069) 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 (Math. Comp. 73 (2004) 777-812) to check the convergence in L1(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.