Traffic modeling and management: Trends and perspectives
Alexandre Bayen Rinaldo M. Colombo Paola Goatin Benedetto Piccoli
The present issue of Discrete and Continuous Dynamical Systems -- Series S is devoted to Traffic Modeling and Management. This subject dramatically developed in recent years. On one hand, the successes of the analytical theory of conservation laws have provided new tools to traffic researchers while, on the other hand, the requirements coming from the applications have grown dramatically. Remarkably, two of the papers that opened the way to this decades long development date the same year. In 1995 ``The Unique Limit of the Glimm Scheme'' by A. Bressan (Archive for Rational Mechanics and Analysis, 130, 3, 205--230) gave a basis for several well posedness results for 1D systems of conservation laws. In the same year, ``Requiem for High-Order Fluid Approximations of Traffic Flow'' by C. Daganzo (Transportation Research Part B: Methodological, 29B, 4, 277--287) posed serious criticisms to models studied at that time and started to fix minimal requirements for a traffic model to be seriously considered.

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Traffic flow models with phase transitions on road networks
Paola Goatin
The paper presents a review of the main analytical results available on the traffic flow model with phase transitions described in [10]. We also introduce a forthcoming existence result on road networks [14].
keywords: Hyperbolic Conservation Laws Riemann Problem Phase Transitions Continuum Traffic Models.
A mixed system modeling two-directional pedestrian flows
Paola Goatin Matthias Mimault
In this article, we present a simplified model to describe the dynamics of two groups of pedestrians moving in opposite directions in a corridor. The model consists of a $2\times 2$ system of conservation laws of mixed hyperbolic-elliptic type. We study the basic properties of the system to understand why and how bounded oscillations in numerical simulations arise. We show that Lax-Friedrichs scheme ensures the invariance of the domain and we investigate the existence of measure-valued solutions as limit of a subsequence of approximate solutions.
keywords: macroscopic models for pedestrian flows Systems of conservation laws finite volume schemes mixed hyperbolic-elliptic systems Young measures.
$L^1$ continuous dependence for the Euler equations of compressible fluids dynamics
Paola Goatin Philippe G. LeFloch
We prove the $L^1$ continuous dependence of entropy solutions for the $2 \times 2$ (isentropic) and the $3\times 3$ (non-isentropic) systems of inviscid fluid dynamics in one-space dimension. We follow the approach developed by the second author for solutions with small total variation to general systems of conservation laws in [11, 14]. For the systems of fluid dynamics under consideration here, our estimates are more precise and we cover entropy solutions with large total variation.
keywords: large amplitude conservation law compressible fluids continuous dependence entropy solution Euler equations large total variation. uniqueness
A front tracking method for a strongly coupled PDE-ODE system with moving density constraints in traffic flow
Maria Laura Delle Monache Paola Goatin
In this paper we introduce a numerical method for tracking a bus trajectory on a road network. The mathematical model taken into consideration is a strongly coupled PDE-ODE system: the PDE is a scalar hyperbolic conservation law describing the traffic flow while the ODE, that describes the bus trajectory, needs to be intended in a Carathéodory sense. The moving constraint is given by an inequality on the flux which accounts for the bottleneck created by the bus on the road. The finite volume algorithm uses a locally non-uniform moving mesh which tracks the bus position. Some numerical tests are shown to describe the behavior of the solution.
keywords: PDE-ODE model Conservation laws with constraints numerical simulations front-tracking methods. traffic flow modeling
The Cauchy problem at a node with buffer
Mauro Garavello Paola Goatin
We consider the Lighthill-Whitham-Richards traffic flow model on a network composed by an arbitrary number of incoming and outgoing arcs connected together by a node with a buffer. Similar to [15], we define the solution to the Riemann problem at the node and we prove existence and well posedness of solutions to the Cauchy problem, by using the wave-front tracking technique and the generalized tangent vectors.
keywords: traffic flow at junctions wave-front tracking. Scalar conservation laws macroscopic models
Well-posedness and finite volume approximations of the LWR traffic flow model with non-local velocity
Paola Goatin Sheila Scialanga
We consider an extension of the traffic flow model proposed by Lighthill, Whitham and Richards, in which the mean velocity depends on a weighted mean of the downstream traffic density. We prove well-posedness and a regularity result for entropy weak solutions of the corresponding Cauchy problem, and use a finite volume central scheme to compute approximate solutions. We perform numerical tests to illustrate the theoretical results and to investigate the limit as the convolution kernel tends to a Dirac delta function.
keywords: finite volume schemes. Scalar conservation laws macroscopic traffic flow models non-local flux
Stability estimates for scalar conservation laws with moving flux constraints
Maria Laura Delle Monache Paola Goatin

We study well-posedness of scalar conservation laws with moving flux constraints. In particular, we show the Lipschitz continuous dependence of BV solutions with respect to the initial data and the constraint trajectory. Applications to traffic flow theory are detailed.

keywords: Scalar conservation laws moving flux constraints PDE-ODE system
General constrained conservation laws. Application to pedestrian flow modeling
Christophe Chalons Paola Goatin Nicolas Seguin
We extend the results on conservation laws with local flux constraint obtained in [2, 12] to general (non-concave) flux functions and non-classical solutions arising in pedestrian flow modeling [15]. We first provide a well-posedness result based on wave-front tracking approximations and the Kružhkov doubling of variable technique for a general conservation law with constrained flux. This provides a sound basis for dealing with non-classical solutions accounting for panic states in the pedestrian flow model introduced by Colombo and Rosini [15]. In particular, flux constraints are used here to model the presence of doors and obstacles. We propose a "front-tracking" finite volume scheme allowing to sharply capture classical and non-classical discontinuities. Numerical simulations illustrating the Braess paradox are presented as validation of the method.
keywords: finite volume schemes macroscopic pedestrian flow models non-classical shocks Braess paradox. Constrained scalar conservation laws
Moving bottlenecks for the Aw-Rascle-Zhang traffic flow model
Stefano Villa Paola Goatin Christophe Chalons

We introduce a second order model for traffic flow with moving bottlenecks. The model consists of the × 2$ Aw-Rascle-Zhang system with a point-wise flow constraint whose trajectory is governed by an ordinary differential equation. We define two Riemann solvers, characterize the corresponding invariant domains and propose numerical strategies, which are effective in capturing the non-classical shocks due to the constraint activation.

keywords: Traffic flow models moving bottlenecks unilateral flux constraints Riemann solvers finite volume schemes

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