Extensions and Amplifications of a Traffic Model of Aw and Rascle

Greenberg, J. M.

doi:10.1137/s0036139900378657

Cited by 139 publications

(126 citation statements)

References 5 publications

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“…(i) Microscopic models or Carfollowing models e.g. [11,2]: they are based on supposed mechanisms describing the process of one vehicle following another; (ii) Kinetic models [22,20,19,14,18,13]: they describe the dynamics of the velocity distribution of vehicles, in the traffic flow; (iii) Fluid-dynamical models [17,23,21,9,22,1,25,6,8,5,4]:…”

Section: Introductionmentioning

confidence: 99%

A Traffic-Flow Model With Constraints for the Modeling of Traffic Jams

Berthelin

Degond

Blanc

et al. 2008

Math. Models Methods Appl. Sci.

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Recently, Berthelin et al [5] introduced a traffic flow model describing the formation and the dynamics of traffic jams. This model which consists of a Constrained Pressureless Gas Dynamics system assumes that the maximal density constraint is independent of the velocity. However, in practice, the distribution of vehicles on a highway depends on their velocity. In this paper we propose a more realistic model namely the Second Order Model with Constraint (SOMC model), derived from the Aw & Rascle model [1] abd which takes into this feature. Moreover, when the maximal density constraint is saturated, the SOMC model "relaxes" to the Lighthill & Whitham model [17]. We prove an existence result of weak solutions for this model by means of cluster dynamics in order to construct a sequence of approximations and we solve completely the associated Riemann problem.

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Section: Introductionmentioning

confidence: 99%

A Traffic-Flow Model With Constraints for the Modeling of Traffic Jams

Berthelin

Degond

Blanc

et al. 2008

Math. Models Methods Appl. Sci.

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“…Steady state solutions can be linked by a shock wave, if the quantities ρ − , v − left to the interface and the corre- sponding quantities ρ + , v + right of the interface satisfy the following conditions [9,11]:…”

Section: Quasi Steady State Solutionsmentioning

confidence: 99%

“…Instead, the main focus has been laid on the principal part of the equations [9], i.e. the collection of terms in the partial differential equation containing derivatives of order equal to the order of the partial differential equation [10], and systems with constant relaxation time [11,12,13,14,15]. In [16], we presented the balanced vehicular traffic model (BVT model), which generalizes the model of Aw, Rascle and Greenberg [9,11] by prescribing a more general source term subsumed under an effective relaxation coefficient.…”

Section: Introductionmentioning

confidence: 99%

Synchronized flow and wide moving jams from balanced vehicular traffic

Siebel¹,

Mauser²

2006

Phys. Rev. E

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Recently we proposed an extension to the traffic model of Aw, Rascle and Greenberg. The extended traffic model can be written as a hyperbolic system of balance laws and numerically reproduces the reverse λ shape of the fundamental diagram of traffic flow. In the current work we analyze the steady state solutions of the new model and their stability properties. In addition to the equilibrium flow curve the trivial steady state solutions form two additional branches in the flow-density diagram. We show that the characteristic structure excludes parts of these branches resulting in the reverse λ shape of the flow-density relation. The upper branch is metastable against the formation of synchronized flow for intermediate densities and unstable for high densities, whereas the lower branch is unstable for intermediate densities and metastable for high densities. Moreover, the model can reproduce the typical speed of the downstream front of wide moving jams. It further reproduces a constant outflow from wide moving jams, which is far below the maximum free flow. Applying the model to simulate traffic flow at a bottleneck we observe a general pattern with wide moving jams traveling through the bottleneck.

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“…Traffic and pedestrian flow equations on the mesoscopic or kinetic level can be found for example in [35,33,29,20,14]. Macroscopic traffic and pedestrian flow equations involving equations for density and mean velocity of the flow are derived in [40,3,13,2,17,16,19] and [20,6]. The classical macroscopic traffic flow model based on scalar continuity equations is described in [37].…”

Section: Introductionmentioning

confidence: 99%

Coupling Traffic Flow Networks to Pedestrian Motion

Borsche

Klar

Kühn

et al. 2013

Math. Models Methods Appl. Sci.

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In the present paper scalar macroscopic models for traffic and pedestrian flows are coupled and the resulting system is investigated numerically. For the traffic flow the classical Lighthill-Whitham model on a network of roads and for the pedestrian flow the Hughes model are used. These models are coupled via terms in the fundamental diagrams modeling an influence of the traffic and pedestrian flow on the maximal velocities of the corresponding models. Several physical situations, where pedestrians and cars interact, are investigated.

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Extensions and Amplifications of a Traffic Model of Aw and Rascle

Cited by 139 publications

References 5 publications

A Traffic-Flow Model With Constraints for the Modeling of Traffic Jams

A Traffic-Flow Model With Constraints for the Modeling of Traffic Jams

Synchronized flow and wide moving jams from balanced vehicular traffic

Coupling Traffic Flow Networks to Pedestrian Motion

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