A Fluidodynamic Model for Traffic in a Road Network

Garavello, Mauro; Piccoli, Benedetto

doi:10.1007/978-3-540-47641-2_34

Cited by 33 publications

(97 citation statements)

References 15 publications

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“…We assume that α = β, otherwise we may have more than one solutions to the Linear Programming problem, see [10] for details.…”

Section: The Case Of N = 2 Incoming Roads and M = 2 Outgoing Roadsmentioning

confidence: 99%

“…In that case the map providing a unique solution to Riemann problems is called Riemann solver. A fairly general theory for such models on networks is now available, see [10,11].…”

Section: Introductionmentioning

confidence: 99%

“…More precisely, the phenomenon of lack of Lipschitz continuous dependence is illustrated in Section 5.4 of the book [10]. However, some Riemann solvers do provide Lipschitz continuous dependence for Cauchy problems (and thus also uniqueness).…”

Section: Introductionmentioning

confidence: 99%

“…However, some Riemann solvers do provide Lipschitz continuous dependence for Cauchy problems (and thus also uniqueness). Examples can be found in [9], [10] (Chapter 9) and [11]. The specific solvers considered in this paper fall in this category, except for the case of two incoming and two outgoing roads (for which Lipschitz continuous dependence is false but uniqueness and continuous dependence are still open problems.…”

Section: Introductionmentioning

confidence: 99%

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Runge–Kutta Discontinuous Galerkin Method for Traffic Flow Model on Networks

et al. 2014

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Abstract. We propose a bound-preserving Runge-Kutta (RK) discontinuous Galerkin (DG) method as an efficient, effective and compact numerical approach for numerical simulation of traffic flow problems on networks, with arbitrary high order accuracy. Road networks are modeled by graphs, composed of a finite number of roads that meet at junctions. On each road, a scalar conservation law describes the dynamics, while coupling conditions are specified at junctions to define flow separation or convergence at the points where roads meet. We incorporate such coupling conditions in the RK DG framework, and apply an arbitrary high order bound preserving limiter to the RK DG method to preserve the physical bounds on the network solutions (car density). We showcase the proposed algorithm on several benchmark test cases from the literature, as well as several new challenging examples with rich solution structures. Modeling and simulation of Cauchy problems for traffic flows on networks is notorious for lack of uniqueness or (Lipschitz) continuous dependence. The discontinuous Galerkin method proposed here deals elegantly with these problems, and is perhaps the only realistic and efficient high-order method for network problems.

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“…We assume that α = β, otherwise we may have more than one solutions to the Linear Programming problem, see [10] for details.…”

Section: The Case Of N = 2 Incoming Roads and M = 2 Outgoing Roadsmentioning

confidence: 99%

“…In that case the map providing a unique solution to Riemann problems is called Riemann solver. A fairly general theory for such models on networks is now available, see [10,11].…”

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Runge–Kutta Discontinuous Galerkin Method for Traffic Flow Model on Networks

et al. 2014

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“…In particular, there have been several publications on the treatment of the boundary conditions at nodes (connections) of several network links (i.e. road sections) [25,7,28,14,10,9,4,22,23,24]. In particular, the modelling of merging and intersecting flows is not unique, as there are many possible forms of organization, including the use of traffic lights.…”

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confidence: 99%

Self-Organized Network Flows

Helbing

Siegmeier²,

Lämmer³

2012

Lecture Notes in Mathematics

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Abstract.A model for traffic flow in street networks or material flows in supply networks is presented, that takes into account the conservation of cars or materials and other significant features of traffic flows such as jam formation, spillovers, and load-dependent transportation times. Furthermore, conflicts or coordination problems of intersecting or merging flows are considered as well. Making assumptions regarding the permeability of the intersection as a function of the conflicting flows and the queue lengths, we find self-organized oscillations in the flows similar to the operation of traffic lights.

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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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A Fluidodynamic Model for Traffic in a Road Network

Cited by 33 publications

References 15 publications

Runge–Kutta Discontinuous Galerkin Method for Traffic Flow Model on Networks

Runge–Kutta Discontinuous Galerkin Method for Traffic Flow Model on Networks

Self-Organized Network Flows

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

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