A Model for the Formation and Evolution of Traffic Jams

Berthelin, Florent; Degond, Pierre; Delitala, Marcello Edoardo; Rascle, Michel

doi:10.1007/s00205-007-0061-9

Cited by 156 publications

(186 citation statements)

References 24 publications

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“…Therefore, in contrast with the model introduced in [5], here the maximal density n * is a functional of the velocity u. However, this natural consideration imparts to the SOMC model a particular property: a double behaviour.…”

Section: The Second Order Model With Constraint (Somc)mentioning

confidence: 99%

“…The other one is to choose a velocity offset p which is singular at n = n * . One of the good candidate proposed in [5] is…”

Section: The Case N * =Constantmentioning

confidence: 99%

“…(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%

“…

AbstractRecently, 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.

…”

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confidence: 99%

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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: The Second Order Model With Constraint (Somc)mentioning

confidence: 99%

“…The other one is to choose a velocity offset p which is singular at n = n * . One of the good candidate proposed in [5] is…”

Section: The Case N * =Constantmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

“…

…”

mentioning

confidence: 99%

See 2 more Smart Citations

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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show abstract

“…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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On the incompressible limit for a tumour growth model incorporating convective effects

David,

Schmidtchen

2023

Comm Pure Appl Math

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In this work we study a tissue growth model with applications to tumour growth. The model is based on that of Perthame, Quirós, and Vázquez proposed in 2014 but incorporates the advective effects caused, for instance, by the presence of nutrients, oxygen, or, possibly, as a result of self‐propulsion. The main result of this work is the incompressible limit of this model which builds a bridge between the density‐based model and a geometry free‐boundary problem by passing to a singular limit in the pressure law. The limiting objects are then proven to be unique.

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A Model for the Formation and Evolution of Traffic Jams

Cited by 156 publications

References 24 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

Coupling Traffic Flow Networks to Pedestrian Motion

On the incompressible limit for a tumour growth model incorporating convective effects

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