A Macroscopic Model to Reproduce Self-organization at Bottlenecks

Abstract

We propose a model for self-organized trac ow at bottlenecks that consists of a scalar conservation law with a nonlocal constraint on the ux. The constraint is a function of an organization marker which evolves through an ODE depending on the upstream trac density and its variations. We prove well-posedness for the problem, construct and analyze a nite volume scheme, perform numerical simulations and discuss the model and related perspectives.

Entropy solutions for a two-phase transition model for vehicular traffic with metastable phase and time depending point constraint on the density flow

“…In particular our setting allows to deal with constraint functions whose values, updated at fixed times, depend on the past evolution of the solution: we refer to Section 5 for precise assumptions of F as function of t and ρ. While models with continuously varying constraints allow to reproduce capacity drop leading to non-monotone empirical features of real traffic flows (see [1,3]), here we highlight the situations where the constraint and its variations do not result from the intrinsic disorganization (or, on the contrary, from a self-organization, see [2,7]) of the flow at high densities. Indeed, having in mind traffic management, we underline that the constraint and its variations under the form considered here naturally arise from operation of bottlenecks (such as toll gates or traffic lights) at discrete times as a function of data collected non-locally in time upstream the flow.…”

“…The model we deal with (see (7) below for for the short-cut PDE formulation and Definition 2.1 for the precise meaning given to it) is situated at the crossroads of two lines of research in macroscopic traffic modeling and model analysis. The first line consists in combining the classical LWR scalar equation (Lighthill, Whitham and Richards [30,33]) and the well-known ARZ system (Aw, Rascle and Zhang [8,36]), within a unique model featuring transitions between the "free flow" phase Ω f described by LWR and the "congested flow" phase Ω c described by ARZ.…”

“…[4] for the ARZ case) allows to model the traffic in presence, for instance, of traffic lights or construction sites operating only during a part of the day. In the case of non-local dependence of F on ρ, these models allow for possible adaptive management of such situations; moreover, they are able to reproduce capacity drop and its avatars like "Faster is Slower", the Braess paradoxes, see [1], and self-organization phenomena, see [2,7]. We refer to [4,5,16,22] for LWR and ARZ with point constraints, and to [2][3][4] for LWR with non-local point constraints and applications.…”

“…An informal way to describe our model (7) is to say that it combines the fundamental flow conservation relation (1), the point constraint (3) with specific assumptions on F, and (roughly speaking) transport equation (2) for the Lagrangian marker w along the flow. The closure relation is provided by linking the Lagrangian marker w to the state variables (ρ, v) in two distinct ways in the free phase Ω f and in the congested phase Ω c -compatible choices being made in the metastable phase Ω f ∩ Ω c .…”

“…As successful examples of such characterization, we refer to [5,16] and [6] for LWR with point constraints and for discontinuous-flux conservation laws, respectively; see also [4] for the "Kruzhkov-like" entropy characterization of solutions of the ARZ model, originally defined in [8] by means of the Riemann solver. Providing a characterization for model (7) in PDE terms is the main goal of our work.…”