Solutions of the Aw-Rascle-Zhang system with point constraints

Andreïanov, Boris; Donadello, Carlotta; Razafison, Ulrich; Rolland, Julien Yves; Rosini, Massimiliano D.

doi:10.3934/nhm.2016.11.29

Cited by 15 publications

(1 citation statement)

References 28 publications

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“…[16] provides the definition and analysis of two Riemann solvers, as well as the corresponding numerical strategies to compute approximate solutions by a modified Godunov scheme. Existence results for the ARZ model with fixed constraints are provided in [3,5,17].…”

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confidence: 99%

Moving bottlenecks for the Aw-Rascle-Zhang traffic flow model

Villa

Goatin²,

Chalons

2017

Discrete &Amp; Continuous Dynamical Systems - B

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We introduce a second order model for traffic flow with moving bottlenecks. The model consists of the 2 × 2 Aw-Rascle-Zhang system with a point-wise flow constraint whose trajectory is governed by an ordinary differential equation. We define two Riemann solvers, characterize the corresponding invariant domains and propose numerical strategies, which are effective in capturing the non-classical shocks due to the constraint activation.

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mentioning

confidence: 99%

Moving bottlenecks for the Aw-Rascle-Zhang traffic flow model

Villa

Goatin²,

Chalons

2017

Discrete &Amp; Continuous Dynamical Systems - B

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General phase transition models for vehicular traffic with point constraints on the flow

Santo

Rosini

Dymski

et al. 2017

Math Methods in App Sciences

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In this paper, we present a general phase transition model that describes the evolution of vehicular traffic along a one-lane road. Two different phases are taken into account, according to whether the traffic is low or heavy. The model is given by a scalar conservation law in the free-flow phase and by a system of 2 conservation laws in the congested phase. The free-flow phase is described by a one-dimensional fundamental diagram corresponding to a Newell-Daganzo type flux. The congestion phase is described by a two-dimensional fundamental diagram obtained by perturbing a general fundamental flux. In particular, we study the resulting Riemann problems in the case a local point constraint on the flow of the solutions is enforced. KEYWORDS conservation laws, phase transitions, Lighthill-Whitham-Richards model, Aw-Rascle-Zhang model, vehicular traffic, unilateral point constraint, Riemann problem INTRODUCTIONThis paper deals with phase transition models (PT models for short) of hyperbolic conservation laws for vehicular traffic. More precisely, we focus on macroscopic models that describe vehicular traffic along a unidirectional one-lane road, which has neither entrances nor exits and where overtaking is not allowed.In the specialized literature, vehicular traffic is shown to behave differently depending on whether it is free or congested. This leads to consider 2 different regimes corresponding to a free-flow phase (denoted by Ω f ) and a congested phase (denoted by Ω c ). The PT models analysed here are given by a scalar conservation law in the free-flow phase, coupled with a 2 × 2 system of conservation laws in the congested phase. The coupling is achieved via phase transitions, namely, discontinuities that separate 2 states belonging to different phases and that satisfy the Rankine-Hugoniot condition.This two-phase approach was introduced in Colombo 1 and is motivated by experimental observations; according to which for low densities, the flow of vehicles is free and approximable by a one-dimensional flux function, while at high densities, the flow is congested and covers a two-dimensional domain in the fundamental diagram, see Colombo 1, Figure 1.1 or Blandin et al. 2, Figure 3.1 Hence, it is reasonable to describe the dynamics in the free regime by a first-order model (a scalar PDE) and those in the congested regime by a second-order model (a 2 × 2 system of PDEs).In literature, 1 Colombo proposed to let the dynamics in free-flow phase Ω f be governed by the classical LWR model by Lighthill, Whitham, and Richards 3,4 ; on the other hand, the congested phase Ω c includes one more equation for the conservation of a linearized momentum. We recall that the LWR model expresses the conservation of the number of vehicles and assumes that the velocity is a function of the density alone. Furthermore, Colombo's model uses a Greenshields type (strictly parabolic) flux function in the free-flow regime and one consequence is that Ω f cannot intersect Ω c , see Colombo. 1, Remark 2Math Meth Appl Sci. 2017;40:6623-6641.wile...

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Lack of BV bounds for approximate solutions to a two‐phase transition model arising from vehicular traffic

Benyahia

Rosini

2020

Math Methods in App Sciences

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We consider wave-front tracking approximate solutions to a two-phase transition model for vehicular traffic. We construct an explicit example showing that the total variation in space of the solution blows up in finite time even for an initial datum with bounded total variation. 2010 Mathematical Subject Classification. Primary: 35L65, 90B20, 35L45 Key words. Conservation laws, phase transitions, Lighthill-Whitham-Richards model, Aw-Rascle-Zhang model, point constraint on the density flux, wave-front tracking, blow up in finite time.

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Solutions of the Aw-Rascle-Zhang system with point constraints

Cited by 15 publications

References 28 publications

Moving bottlenecks for the Aw-Rascle-Zhang traffic flow model

Moving bottlenecks for the Aw-Rascle-Zhang traffic flow model

General phase transition models for vehicular traffic with point constraints on the flow

Lack of BV bounds for approximate solutions to a two‐phase transition model arising from vehicular traffic

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