Fundamental diagrams for kinetic equations of traffic flow

Fermo, Luisa; Tosin, Andrea

doi:10.3934/dcdss.2014.7.449

Cited by 13 publications

(46 citation statements)

References 17 publications

(38 reference statements)

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“…[39]. From the theoretical point of view, a few mathematical models have been able to explain the emergence of such large scale characteristics of traffic from a microscopic description of vehicle interactions [24,31]. In some cases, models have also successfully investigated the origin of the data scattering typically seen in measured traffic diagrams [51,61].…”

Section: Homogeneous Kinetic Modelling Of Traffic Flowmentioning

confidence: 99%

“…In fact purely controlled dynamics are not aware of the stochastic fluctuations, because u * has been deduced deterministically by averaging with respect to η. Notice that when Θ = 1 and ν → 0 + the first microscopic rule in (24) can be seen as a model for a fully automated, or autonomous, vehicle.…”

Section: Microscopic Binary Control For Road Risk Mitigationmentioning

confidence: 99%

“…where ( v, w) are the pre-interaction speeds which generate the post-interaction speeds (v, w) according to the interaction rules (24) and J is the Jacobian of the transformation from ( v, w) to (v, w). We now briefly account for the numerical scheme by which we solve (48).…”

Section: Inhomogeneous Kinetic Modelmentioning

confidence: 99%

See 2 more Smart Citations

Kinetic-Controlled Hydrodynamics for Traffic Models with Driver-Assist Vehicles

Tosin¹,

Zanella²

2019

Multiscale Model. Simul.

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We develop a hierarchical description of traffic flow control by means of driver-assist vehicles aimed at the mitigation of speed-dependent road risk factors. Microscopic feedback control strategies are designed at the level of vehicle-to-vehicle interactions and then upscaled to the global flow via a kinetic approach based on a Boltzmann-type equation. Then first and second order hydrodynamic traffic models, which naturally embed the microscopic control strategies, are consistently derived from the kinetic-controlled framework via suitable closure methods. Several numerical examples illustrate the effectiveness of such a hierarchical approach at the various scales.

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Section: Homogeneous Kinetic Modelling Of Traffic Flowmentioning

confidence: 99%

Section: Microscopic Binary Control For Road Risk Mitigationmentioning

confidence: 99%

See 1 more Smart Citation

Kinetic-Controlled Hydrodynamics for Traffic Models with Driver-Assist Vehicles

Tosin¹,

Zanella²

2019

Multiscale Model. Simul.

Self Cite

View full text Add to dashboard Cite

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“…This means that the model cannot reproduce and explain the scattering behavior of the experimental measurements, at least not at equilibrium. Mathematical models being able to describe multivalued fundamental diagrams were also studied and they link the observed scattered data to several reasons: for instance the presence of heterogeneous components, such as different drivers [31,33] or populations of vehicles [42], or the deviation of microscopic speeds at equilibrium [13], or the presence of non-equilibrium phenomena such as stop and go waves [25,46,49]. In this work we will focus on the last motivation and consider non-equilibrium phenomena that might lead to complex traffic phenomena as e.g.…”

Section: Motivation: Non-equilibrium Regimesmentioning

confidence: 99%

The BGK approximation of kinetic models for traffic

Herty¹,

Puppo²,

Roncoroni³

et al. 2020

Kinetic &Amp; Related Models

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We study spatially non-homogeneous kinetic models for vehicular traffic flow. Classical formulations, as for instance the BGK equation, lead to unconditionally unstable solutions in the congested regime of traffic. We address this issue by deriving a modified formulation of the BGK-type equation. The new kinetic model allows to reproduce conditionally stable nonequilibrium phenomena in traffic flow. In particular, stop and go waves appear as bounded backward propagating signals occurring in bounded regimes of the density where the model is unstable. The BGK-type model introduced here also offers the mesoscopic description between the microscopic follow-the-leader model and the macroscopic Aw-Rascle and Zhang model.MSC 90B20, 35Q20, 35Q70

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“…and therefore the continuity equation in Lagrangian coordinates (13) Recalling that τ = 1/ρ, we get the following equation:…”

Section: From Lagrangian To Eulerian Coordinatesmentioning

confidence: 99%

Macroscopic Modeling of Multilane Motorways Using a Two-Dimensional Second-Order Model of Traffic Flow

Herty¹,

Moutari²,

Visconti³

2018

SIAM J. Appl. Math.

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Lane changing is one of the most common maneuvers on motorways. Although, macroscopic traffic models are well known for their suitability to describe fast moving crowded traffic, most of these models are generally developed in a one dimensional framework, henceforth lane changing behavior is either not explicitly modeled or explicitly forbidden. In this paper, we propose a macroscopic model, which accounts for lane-changing behavior on motorway, based on a two-dimensional extension of the Aw and Rascle [4] and Zhang [39] macroscopic model for traffic flow. Under conditions, when lane changing maneuvers are no longer possible, the model "relaxes" to the one-dimensional Aw-Rascle-Zhang model. Following the same approach as in [3], we derive the two-dimensional macroscopic model through scaling of time discretization of a microscopic follow-the-leader model with driving direction. We provide a detailed analysis of the space-time discretization of the proposed macroscopic model as well as an approximation of the solution to the associated Riemann problem. Furthermore, we illustrate some promising features of the proposed model through some numerical experiments.

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Fundamental diagrams for kinetic equations of traffic flow

Cited by 13 publications

References 17 publications

Kinetic-Controlled Hydrodynamics for Traffic Models with Driver-Assist Vehicles

Kinetic-Controlled Hydrodynamics for Traffic Models with Driver-Assist Vehicles

The BGK approximation of kinetic models for traffic

Macroscopic Modeling of Multilane Motorways Using a Two-Dimensional Second-Order Model of Traffic Flow

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