Micro-Macro Limit of a Nonlocal Generalized Aw-Rascle Type Model

Abstract

We introduce a Follow-the-Leader approximation of a non-local generalized Aw-Rascle-Zhang (GARZ) model for traffic flow. We prove the convergence to weak solutions of the corresponding macroscopic equations deriving L ∞ and BV estimates. We also provide numerical simulations illustrating the micro-macro convergence and we investigate numerically the non-local to local limit for both the microscopic and macroscopic models.

Microscopic derivation of a traffic flow model with a bifurcation

“…The inequality θ m e ptq ´C1 ď θ e ptq can be checked in the same way, by using points 2) and 3) of Lemma 3.5. This proves (22). As (22) holds in the event tJ n pm ´1q ě 0u, we get P r|θ e ptq ´θm e ptq| ą Cs ď P rJ n pm ´1q ă 0s ď P rJ n pmq ă 0s .…”

“…This proves (22). As (22) holds in the event tJ n pm ´1q ě 0u, we get P r|θ e ptq ´θm e ptq| ą Cs ď P rJ n pm ´1q ă 0s ď P rJ n pmq ă 0s .…”

“…Indeed there exists many different continuous models of traffic flow on a junction or with a local perturbation in the literature [5,13,23,32,33,48] and the relation between these models is not completely clear. If the basic continuous model on a single straight road (the socalled LWR model, from Lighthill and Whitham [38] and Richards [44]) is well understood and justified by micro-macro limits in several contexts [8,22,26,34,35], there is no consensus for problems with a junction or a bifurcation: the models are only obtained so far by heuristic arguments, with the exception of [28] discussed below. In this paper we show that the continuous model suggested in [31,36] pops up as the natural limit of follow-the-leader models.…”

Network models for nonlocal traffic flow

“…Future work will include the investigation of a general M -to-N junction. We also intend to derive coupling conditions for other nonlocal modeling equations, such as the second order traffic flow model proposed in [10].…”

“…Nonlocal models for traffic flow are widely studied in current research concerning existence of solutions [4,11,24,29,36], numerical schemes [4,8,23,24,29] or convergence to local conservation laws [5,6,15,18,38] -even, in general, this question is still an open problem. Modeling approaches include microscopic models [10,13,28,45], second order models [10], multiclass models [12], multilane models [3,22] and also time delay models [37]. But to the best of our knowledge only a few works deal with network models.…”