  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 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 $L^1(\mathbb{R})$ with rate ${\mathcal O}(\frac{1}{\sqrt N} +
h)$ 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 in ${\mathcal O}(\frac{1}{N} + h)$. We
provide numerical results which confirm our theoretical estimates.
