Structural Properties of the Stability of Jamitons

Ramadan, Rabie A.; Rosales, Rodolfo R.; Seibold, Benjamin

doi:10.1007/978-3-030-66560-9_3

Cited by 5 publications

(16 citation statements)

References 50 publications

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“…It is straightforward to check that our condition ( 26) is consistent with the stability condition derived in [29] for a generic inhomogeneous ARZ model, in which however the traffic pressure is not precisely related to the sensitivity and the headway of the drivers.…”

Section: A Bando-type Stability Condition Of the Uniform Flowsupporting

confidence: 74%

“…were ρ, u are the traffic density and mean speed, respectively, and p(ρ) is a (pseudo) traffic pressure sometimes also called hesitation function [29]. The principle used in [1] consists in interpreting a discrete-in-time version of (3) as a space-time discretisation of (4) for a suitably chosen traffic pressure p and in showing that the discrete solutions thereby produced converge to weak solutions of (4) as the discretisation parameters become infinitesimal.…”

Section: Introductionmentioning

confidence: 99%

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A statistical mechanics approach to macroscopic limits of car-following traffic dynamics

Chiarello,

Piccoli,

Tosin

2021

Preprint

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We study the derivation of macroscopic traffic models from car-following vehicle dynamics by means of hydrodynamic limits of an Enskog-type kinetic description. We consider the superposition of Follow-the-Leader (FTL) interactions and relaxation towards a trafficdependent Optimal Velocity (OV) and we show that the resulting macroscopic models depend on the relative frequency between these two microscopic processes. If FTL interactions dominate then one gets an inhomogeneous Aw-Rascle-Zhang model, whose (pseudo) pressure and stability of the uniform flow are precisely defined by some features of the microscopic FTL and OV dynamics. Conversely, if the rate of OV relaxation is comparable to that of FTL interactions then one gets a Lighthill-Whitham-Richards model ruled only by the OV function. We further confirm these findings by means of numerical simulations of the particle system and the macroscopic models. Unlike other formally analogous results, our approach builds the macroscopic models as physical limits of particle dynamics rather than assessing the convergence of microscopic to macroscopic solutions under suitable numerical discretisations.

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Section: A Bando-type Stability Condition Of the Uniform Flowsupporting

confidence: 74%

Section: Introductionmentioning

confidence: 99%

A statistical mechanics approach to macroscopic limits of car-following traffic dynamics

Chiarello,

Piccoli,

Tosin

2021

Preprint

View full text Add to dashboard Cite

show abstract

“…If the diffusion coefficient µ(ρ) is negative then the advection-diffusion equation is ill-posed and therefore has solutions with unbounded growth even starting from small perturbations of equilibrium states [13,21,24,37,39]. The following result summarized sufficient conditions on the Maxwellian leading to this macroscopic effect.…”

Section: Macroscopic Behavior Via a Chapman-enskog Expansionmentioning

confidence: 95%

“…Sign of the diffusion coefficient in the Chapman-Enskog expansion: Stop and go waves are experimentally observed [16,19,40] unstable phenomena occurring in some regimes of traffic as the result of vehicles' inability of reaching an equilibrium state. The key observation in [21,37,39] is that relaxation models possess a phase transition behavior determined by properties of the diffusion function arising in the relaxation or diffusive limit. For certain densities, where the diffusion function is positive, the solutions are very close to equilibrium; while for other densities, where the diffusion is negative, the solutions develop backward propagating traffic waves that can be regarded as models for stop and go waves.…”

Section: Indicators Of Instability In Traffic Flowmentioning

confidence: 99%

“…Stop and go waves are periodic disturbances in the density of vehicles [37,39], which propagate backwards with respect to the direction of the flow, and with an amplitude increasing in time. Mathematically, the classical description of traffic flow is given by the Lighthill-Whitham-Richards (LWR) model [29,38], which is based on a single conservation law.…”

Section: Introductionmentioning

confidence: 99%

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Model of vehicle interactions with autonomous cars and its properties

Herty¹,

Puppo²,

Visconti³

2021

Preprint

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We study a hierarchy of models based on kinetic equations for the descriptions of traffic flow in presence of automated and human-driven vehicles. For a sufficiently large penetration rate of autonomous vehicles stabilization of traffic phenomena, like stop-and-go waves, is observed. The influence of the penetration rate is investigated analytically and numerically. Even though the microscopic interaction rules of autonomous cars are simplistic, features observed in real world traffic can be recovered on a macroscopic modeling level.

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Multiscale Properties of Traffic Flow: The Macroscopic Impact of Traffic Waves

Khoudari

Seibold²

2022

Mathematics Online First Collections

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Traffic waves can rise even from single lane car-following behaviour. To better understand and mitigate traffic waves, it is necessary to use analytical tools like mathematical models, data analysis, and micro-simulations that can capture the dynamics of real traffic flow. In this study, we isolate car-following dynamics and present a systematic hierarchy of tests that connect the microscopic scale with the meaningful macroscopic effective state in the presence of waves. This allows insights with precise attributable cause-to-effect relationships of specific observed traffic patterns. We establish a principled way of generating macroscopic flow quantities from microscopic models in the unstable regime. Those quantities are then used to study how the corresponding non-equilibrium wave structures manifest in the fundamental diagram, based on three basic scenarios that can serve as building blocks for understanding more complex micro-simulation studies. Finally, this study gives insight on the shapes of the reduced fundamental diagrams for different commonly used microscopic models.

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Structural Properties of the Stability of Jamitons

Cited by 5 publications

References 50 publications

A statistical mechanics approach to macroscopic limits of car-following traffic dynamics

A statistical mechanics approach to macroscopic limits of car-following traffic dynamics

Model of vehicle interactions with autonomous cars and its properties

Multiscale Properties of Traffic Flow: The Macroscopic Impact of Traffic Waves

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