Vehicular Traffic: A Review of Continuum Mathematical Models

Piccoli, Benedetto; Tosin, Andrea

doi:10.1007/978-1-4614-1806-1_112

Cited by 32 publications

(40 citation statements)

References 57 publications

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“…Motivated by the introduction of the traffic pressure defined by (31), which allows one to rewrite the Aw-Rascle model in the form (32), we introduce now a generalised traffic pressure P = P (ρ, u) defined by the relationships…”

Section: Generalisations Of the Aw-rascle Modelmentioning

confidence: 99%

“…We recall here, in particular, the well known follow-the-leader and optimal velocity models [2,16]. On the other hand, there are the macroscopic models, which, inspired by the hyperbolic conservation/balance laws of fluid dynamics, treat the vehicles as a continuum with density [31]. In this case, one distinguishes between first order models, which rely on the mass conservation only, cf.…”

Section: Introductionmentioning

confidence: 99%

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The Aw–Rascle Traffic Model: Enskog-Type Kinetic Derivation and Generalisations

Dimarco

Tosin

2019

J Stat Phys

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We study the derivation of second order macroscopic traffic models from kinetic descriptions. In particular, we recover the celebrated Aw-Rascle model as the hydrodynamic limit of an Enskog-type kinetic equation out of a precise characterisation of the microscopic binary interactions among the vehicles. Unlike other derivations available in the literature, our approach unveils the multiscale physics behind the Aw-Rascle model. This further allows us to generalise it to a new class of second order macroscopic models complying with the Aw-Rascle consistency condition, namely the fact that no wave should travel faster than the mean traffic flow.

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Section: Generalisations Of the Aw-rascle Modelmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

The Aw–Rascle Traffic Model: Enskog-Type Kinetic Derivation and Generalisations

Dimarco

Tosin

2019

J Stat Phys

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“…To close the above system of two equations and three unknowns, a further condition has to be imposed. Two main approaches are used in the literature (see [31,42,45] for a survey): first order and second order models. The former are based on a constitutive law, which expresses one of the three unknowns as function of the remaining two.…”

Section: 4mentioning

confidence: 99%

Rigorous Derivation of Nonlinear Scalar Conservation Laws from Follow-the-Leader Type Models via Many Particle Limit

Francesco

Rosini

2015

Arch Rational Mech Anal

100

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Abstract. We prove that the unique entropy solution to a scalar nonlinear conservation law with strictly monotone velocity and nonnegative initial condition can be rigorously obtained as the large particle limit of a microscopic follow-the-leader type model, which is interpreted as the discrete Lagrangian approximation of the nonlinear scalar conservation law. More precisely, we prove that the empirical measure (respectively the discretised density) obtained from the follow-the-leader system converges in the 1-Wasserstein topology (respectively in L 1 loc ) to the unique Kruzkov entropy solution of the conservation law. The initial data are taken in L ∞ , nonnegative, and with compact support, hence we are able to handle densities with vacuum. Our result holds for a reasonably general class of velocity maps (including all the relevant examples in the applications, e.g. in the Lighthill-Whitham-Richards model for traffic flow) with possible degenerate slope near the vacuum state. The proof of the result is based on discrete BV estimates and on a discrete version of the one-sided Oleinik-type condition. In particular, we prove that the regularizing effect L ∞ → BV for nonlinear scalar conservation laws is intrinsic of the discrete model. Keywords: Micro-macro limit and Scalar conservation laws and Follow-the-leader models and Oleinik condition and Entropy solutions and Particle method. 35L65 and 35L45 and 90B20 and 65N75 and 82C22 . AMS Subject classification:

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“…Macroscopic models of traffic flows are nowadays a consolidated and nonetheless continuously in ferment field of mathematical research from both theoretical and applied points of view, as the surveys [5,17,18,20] and the books [7,13,19] demonstrate.…”

Section: Introductionmentioning

confidence: 99%

Existence of BV solutions for a non-conservative constrained Aw–Rascle–Zhang model for vehicular traffic

Dymski

Goatin

Rosini

2018

Journal of Mathematical Analysis and Applications

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In this paper we study the Aw-Rascle-Zhang (ARZ) model with non-conservative local point constraint on the density flux introduce in [Garavello, M., and Goatin, P. The Aw-Rascle traffic model with locally constrained flow. Journal of Mathematical Analysis and Applications 378, 2 (2011), 634-648], its motivation being, for instance, the modeling of traffic across a toll gate. We prove the existence of weak solutions under assumptions that result to be more general than those required in [Garavello, M., and Villa, S. The Cauchy problem for the Aw-Rascle-Zhang traffic model with locally constrained flow. Journal of Hyperbolic Differential Equations 14, 03 (2017), 393-414]. More precisely, we do not require that the waves of the first characteristic family have strictly negative speeds of propagation. The result is achieved by showing the convergence of a sequence of approximate solutions constructed via the wave-front tracking algorithm. The case of solutions attaining values at the vacuum is considered. We also present an explicit numerical example to describe some qualitative features of the solutions.

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Vehicular Traffic: A Review of Continuum Mathematical Models

Cited by 32 publications

References 57 publications

The Aw–Rascle Traffic Model: Enskog-Type Kinetic Derivation and Generalisations

The Aw–Rascle Traffic Model: Enskog-Type Kinetic Derivation and Generalisations

Rigorous Derivation of Nonlinear Scalar Conservation Laws from Follow-the-Leader Type Models via Many Particle Limit

Existence of BV solutions for a non-conservative constrained Aw–Rascle–Zhang model for vehicular traffic

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