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

TL;DR

This paper analyzes the convergence rates of mean-field rank-based particle systems approximating one-dimensional viscous scalar conservation laws, providing theoretical error bounds and numerical validation.

Contribution

It extends existing analysis by establishing optimal convergence rates and bias estimates for particle approximations of viscous scalar conservation laws.

Findings

Trajectorial propagation of chaos with rate N^{-1/2}

Convergence of empirical CDF with rate O(1/√N + h)

Bias of particle method behaves as O(1/N + h)

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 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.