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# Rigorous derivation of the Lighthill–Whitham–Richards model from the follow the-leader model as many particle limit

Abstract: We prove that the unique entropy solution to the macroscopic Lighthill–Witham–Richards model for traffic flow can be rigorously obtained as the large particle limit of the microscopic follow-the-leader model, which is interpreted as the discrete Lagrangian approximation of the former. More precisely, we prove that the empirical measure (respectively the discretised density) obtained from the follow-the-leader system converges in the 1-Wasserstein topology (respectively in $L^1_{\mathrm{loc}}$) to the unique entropy solution of the Lighthill–Witham–Richards equation in the Kruzkov sense. The initial data are taken in $L^\infty$ with compact support, hence we are able to handle densities with vacuum. Our result holds for a reasonably general class of velocity maps (including all the relevant examples in the applications) with possible degenerate slope near the vacuum state. The proof of the result is based on discrete BV estimates and on a discrete version of the one-sided Oleinik-type condition. In particular, we prove that the $L^\infty\to\mathrm{BV}$ regularizing effect is intrinsic of the discrete model.

Paper:
Available as PDF (409 Kbytes).
Author(s):
Marco Di Francesco
Massimiliano D. Rosini
Submitted by:
; 2014-04-28.