Homogenization of second order discrete model with local perturbation and application to traffic flow

Forcadel, Nicolas; Salazar, Wilfredo; Zaydan, Mamdouh

doi:10.3934/dcds.2017060

Cited by 16 publications

(28 citation statements)

References 22 publications

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“…and using Lemma 4.1, and Definition 3.1, we can see that ρ * is a discontinuous viscosity super-solution of (19). We obtain a similar result for ρ * , therefore, ρ is a discontinuous viscosity solution of (19).…”

Section: Lemma 41 (Link Between the Velocities) Assume (A) Let ((Usupporting

confidence: 75%

“…To our knowledge, this result is the first in the presence of a local perturbation. Note that this result has also been extended by the authors to second order microscopic models ( [19]). This local perturbation can be constant in time and represent a slowdown near a school or due to a car crash near the road.…”

supporting

confidence: 61%

“…Remark 5 (Extension to second order microscopic model). The results presented in this paper have been extended by the authors to the case of second order microscopic models in [19].…”

Section: Remarkmentioning

confidence: 84%

“…We denote by U j (t) the position of the j th vehicle and we assume that the velocity of each vehicle is given by the function V . In order to obtain our homogenization result, we proceed as in [15,18,17,16] and rescale the microscopic model which describes the dynamics of each vehicle, to obtain a macroscopic model that describes the density of vehicles. If the local perturbation is located around zero, at the macroscopic scale it is natural to get an Hamilton-Jacobi equation with a junction condition at the origin (see Figure 2, u 0…”

mentioning

confidence: 99%

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Specified homogenization of a discrete traffic model leading to an effective junction condition

Forcadel

Salazar

Zaydan

2018

Communications on Pure &Amp; Applied Analysis

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In this paper, we focus on deriving traffic flow macroscopic models from microscopic models containing a local perturbation such as a traffic light. At the microscopic scale, we consider a first order model of the form "follow the leader" i.e. the velocity of each vehicle depends on the distance to the vehicle in front of it. We consider a local perturbation located at the origin that slows down the vehicles. At the macroscopic scale, we obtain an explicit Hamilton-Jacobi equation left and right of the origin and a junction condition at the origin (in the sense of [25]) which keeps the memory of the local perturbation. As it turns out, the macroscopic model is equivalent to a LWR model, with a flux limiting condition at the junction. Finally, we also present qualitative properties concerning the flux limiter at the junction.

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Section: Lemma 41 (Link Between the Velocities) Assume (A) Let ((Usupporting

confidence: 75%

supporting

confidence: 61%

“…Remark 5 (Extension to second order microscopic model). The results presented in this paper have been extended by the authors to the case of second order microscopic models in [19].…”

Section: Remarkmentioning

confidence: 84%

mentioning

confidence: 99%

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Specified homogenization of a discrete traffic model leading to an effective junction condition

Forcadel

Salazar

Zaydan

2018

Communications on Pure &Amp; Applied Analysis

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“…Instead Colombo and Rossi [9], Rossi [47], Di Francesco and Rosini [18], and Di Francesco et al [16] investigated the many-particle limit in the framework of first-order traffic models, deriving the macroscopic Lighthill-Whitham-Richards (LWR) model [39,45] as the limit of a first-order FtL model. Let us also mention the papers by Forcadel et al [24], Forcadel and Salazar [23] which investigate the many-particle limit exploiting the link between conservation laws and Hamilton-Jacobi equations.…”

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confidence: 99%

An interface-free multi-scale multi-order model for traffic flow

Cristiani¹,

Iacomini²

2019

Discrete &Amp; Continuous Dynamical Systems - B

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In this paper we present a new multi-scale method for reproducing traffic flow which couples a first-order macroscopic model with a second-order microscopic model, avoiding any interface or boundary conditions between them. The multi-scale model is characterized by the fact that microscopic and macroscopic descriptions are not spatially separated. On the contrary, the macro-scale is always active while the micro-scale is activated only if needed by the traffic conditions. The Euler-Godunov scheme associated to the model is conservative and it is able to reproduce typical traffic phenomena like stop & go waves.

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Comparing comparisons between vehicular traffic states in microscopic and macroscopic first‐order models

Cristiani

Saladino

2018

Math Methods in App Sciences

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In this paper, we deal with the analysis of the solutions of traffic flow models at multiple scales in both cases of a single road and road networks. We are especially interested in measuring the distance between traffic states (as they result from the mathematical modeling) and investigating whether these distances are somehow preserved passing from the microscopic to the macroscopic scale. By means of both theoretical and numerical investigations, we show that, on a single road, the notion of Wasserstein distance fully catches the human perception of distance independently of the scale, while in the case of networks it partially loses its nice properties. KEYWORDS earth mover's distance, follow-the-leader model, LWR model, many-particle limit, multi-path model, networks, traffic flow, Wasserstein distance INTRODUCTIONIn this paper, we deal with the analysis of the solutions of traffic flow models at multiple scales. More precisely, we are interested in measuring the distance between traffic states (as they result from the mathematical modeling) and investigating whether these distances are somehow preserved passing from the microscopic to the macroscopic scale. We will consider both cases of a single road and road networks.Connections between microscopic (agent-based) and macroscopic (fluid dynamics) traffic flow models are already well established. Aw et al, 1 Greenberg, 2 and Di Francesco et al 3 investigated the many-particle limit in the framework of second-order traffic models, deriving the macroscopic Aw-Rascle-Zhang (ARZ) 4,5 model from a particular second-order microscopic Follow-the-Leader (FtL) 6,7 model. On the other hand, Colombo and Rossi, 8 Rossi, 9 Di Francesco and Rosini, 10 and Di Francesco et al 11 investigated the many-particle limit in the framework of first-order traffic models, deriving the Ligthill-Whitham-Richards (LWR) 12,13 model as the limit of a specific first-order FtL model. Let us also mention the papers by Forcadel et al 14 and Forcadel and Salazar,15 which investigate the many-particle limit exploiting the link between conservation laws and Hamilton-Jacobi equations.Moving to road networks, analogous connections are rarer. This is probably because of the fact that macroscopic traffic models on networks are in general ill-posed, since the conservation of mass is not sufficient alone to characterize a unique solution at junctions. This ambiguity makes more difficult to find the right limit of the microscopic model, which, in turn, can be defined in different ways near the junctions. In this context let us mention the paper by Cristiani and Sahu, 16 which investigates the many-particle limit of a first-order FtL model suitably extended to a road network. The corresponding macroscopic model appears to be the extension of the LWR model on networks introduced by Hilliges and Weidlich 17 and then extensively studied by Briani and Cristiani 18 and Bretti et al. 19 This paper focuses in particular on the comparison of solutions to the equations associated to first-order traffic models....

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Homogenization of second order discrete model with local perturbation and application to traffic flow

Cited by 16 publications

References 22 publications

Specified homogenization of a discrete traffic model leading to an effective junction condition

Specified homogenization of a discrete traffic model leading to an effective junction condition

An interface-free multi-scale multi-order model for traffic flow

Comparing comparisons between vehicular traffic states in microscopic and macroscopic first‐order models

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