A non-local traffic flow model for 1-to-1 junctions

Abstract

We present a model for a class of non-local conservation laws arising in traffic flow modelling at road junctions. Instead of a single velocity function for the whole road, we consider two different road segments, which may differ for their speed law and number of lanes (hence their maximal vehicle density). We use an upwind type numerical scheme to construct a sequence of approximate solutions, and we provide uniform L∞ and total variation estimates. In particular, the solutions of the proposed model stay positive and below the maximum density of each road segment. Using a Lax–Wendroff type argument and the doubling of variables technique, we prove the well-posedness of the proposed model. Finally, some numerical simulations are provided and compared with the corresponding (discontinuous) local model.

Influence of a slow moving vehicle on traffic: Well-posedness and approximation for a mildly nonlocal model

“…Theorem 4.5. Fix ρ 0 ∈ L 1 (R; [0, R])∩BV(R) and y 0 ∈ R. Suppose that f satisfies (2)-( 7)- (16) and that Q satisfies (17). Suppose also that in (25), we use the Godunov flux when j = 0 and any other monotone consistent and Lipschitz numerical flux when j = 0.…”

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

“…Other extensions of the LWR model involve non-local speed dependencies. In recent years, nonlocal conservation laws have become of interest in modelling traffic flow [3,5,6,14]. These models take into account the look-ahead distance of drivers, such that vehicles adapt their speed with respect to the downstream traffic.…”

“…These models take into account the look-ahead distance of drivers, such that vehicles adapt their speed with respect to the downstream traffic. In literature, two different modelling approaches are proposed: either drivers react to the mean downstream traffic density [3,6], or to the mean downstream velocity [5,14]. The well-posedness of these models is proved in [6,14].…”

An Overview of Non-local Traffic Flow Models

“…We observe that when the look-ahead distance η → ∞, the non-local problem (1)-( 5) becomes a classical transport equation (9). Besides the mathematical implications, Corollary 1 may give information on connected autonomous vehicle flow characteristics.…”

“…This is the main difference with the models in [7,10], where drivers adapt their velocity with respect to a mean downstream traffic density. In particular, the model in [19] allows to capture changes in the velocity function and for this reason, it could be extended to junctions, as in [9].…”