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

Abstract

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 ${\mathit{N}}^{-1/2}$ 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 ${\mathit{L}}^1(\mathbb{R})$ with rate $\mathcal{O}(\frac{1}{\sqrt{\mathit{N}}}+\mathit{h})$ 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 $\mathcal{O}(\frac{1}{\mathit{N}}+\mathit{h})$. We provide numerical results which confirm our theoretical estimates.