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12
A GENERAL PHASE TRANSITION MODEL FOR VEHICULAR TRAFFIC
"... Abstract. An extension of the Colombo phase transition model is proposed. The congestion phase is described by a twodimensional zone defined around an equilibrium flux known as the classical fundamental diagram. General criteria to build such a setvalued fundamental diagram are enumerated, and ins ..."
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Abstract. An extension of the Colombo phase transition model is proposed. The congestion phase is described by a twodimensional zone defined around an equilibrium flux known as the classical fundamental diagram. General criteria to build such a setvalued fundamental diagram are enumerated, and instantiated on several equilibrium fluxes with different concavity properties. The solution of the Riemann problem in the presence of phase transitions is obtained through the construction of a Riemann solver, which enables the definition of the solution of the Cauchy problem using wavefront tracking. The freeflow phase is described using a NewellDaganzo fundamental diagram, which allows for a more tractable definition of phase transition compared to the original Colombo phase transition model. The accuracy of the numerical solution obtained by a modified Godunov scheme is assessed on benchmark scenarios for the different flux functions constructed. Key words. partial differential equations, hyperbolic systems of conservation laws, macroscopic highway traffic flow model, phase transition, numerical scheme, riemann solver AMS subject classifications. 35L65, 35F25, 65M12, 90B20, 76T99
TransportEquilibrium Schemes for Computing Nonclassical Shocks
 I. Scalar Conservation Laws, preprint, Laboratoire JacquesLouis Lions
, 2005
"... This paper presents a very efficient numerical strategy for computing the weak solutions of scalar conservation laws which fail to be genuinely nonlinear. We concentrate on the typical situation of either concaveconvex or convexconcave flux functions. In such a situation, nonclassical shocks viola ..."
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Cited by 18 (7 self)
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This paper presents a very efficient numerical strategy for computing the weak solutions of scalar conservation laws which fail to be genuinely nonlinear. We concentrate on the typical situation of either concaveconvex or convexconcave flux functions. In such a situation, nonclassical shocks violating the classical Oleinik entropy criterion must be taken into account since they naturally arise as limits of certain diffusivedispersive regularizations to hyperbolic conservation laws. Such discontinuities play an important part in the resolution of the Riemann problem and their dynamics turns out to be driven by a prescribed kinetic function which acts as a selection principle. It aims at imposing the entropy dissipation rate across nonclassical discontinuities, or equivalently their speed of propagation. From a numerical point of view, the serious difficulty consists in enforcing the kinetic criterion, that is in controling the numerical entropy dissipation of the nonclassical shocks for any given discretization. This is known to be a very challenging issue. By means of an algorithm made of two steps, namely an Equilibrium step and a Transport step, we show how to force the validity of the kinetic criterion at the discrete level. The resulting scheme provides in addition sharp profiles. Numerical evidences illustrate the validity of our approach. 1
General constrained conservation laws. Application to pedestrian flow modeling
, 2012
"... (Communicated by Benedetto Piccoli) Abstract. We extend the results on conservation laws with local flux constraint obtained in [2, 12] to general (nonconcave) flux functions and nonclassical solutions arising in pedestrian flow modeling [15]. We first provide a wellposedness result based on wave ..."
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Cited by 9 (2 self)
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(Communicated by Benedetto Piccoli) Abstract. We extend the results on conservation laws with local flux constraint obtained in [2, 12] to general (nonconcave) flux functions and nonclassical solutions arising in pedestrian flow modeling [15]. We first provide a wellposedness result based on wavefront tracking approximations and the Kruˇzhkov doubling of variable technique for a general conservation law with constrained flux. This provides a sound basis for dealing with nonclassical 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 “fronttracking ” finite volume scheme allowing to sharply capture classical and nonclassical discontinuities. Numerical simulations illustrating the Braess paradox are presented as validation of the method. 1. Introduction. Several
Conservation laws with unilateral constraints in traffic modeling
 Transport Management and LandUse Effects in Presence of Unusual Demand”, Atti del convegno SIDT 2009
, 2009
"... Macroscopic models for both vehicular and pedestrian traffic are based on conservation laws. The mathematical description of toll gates along roads or of the escape dynamics for crowds needs the introduction of unilateral constraints on the observable flow. This note presents a rigorous approach to ..."
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Cited by 6 (3 self)
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Macroscopic models for both vehicular and pedestrian traffic are based on conservation laws. The mathematical description of toll gates along roads or of the escape dynamics for crowds needs the introduction of unilateral constraints on the observable flow. This note presents a rigorous approach to these constraints, and numerical integrations of the resulting models are included to show their practical usability. 1.
A general phase transition model for vehicular traffic
"... Istituto per le Applicazioni del Calcolo ‘Mauro Picone’ We propose an extension of the Colombo 2 × 2 phase transition model. We define the notion of equilibrium fundamental diagram and establish conditions to derive a perturbed fundamental diagram to model more accurately transport phenomena observe ..."
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Cited by 5 (0 self)
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Istituto per le Applicazioni del Calcolo ‘Mauro Picone’ We propose an extension of the Colombo 2 × 2 phase transition model. We define the notion of equilibrium fundamental diagram and establish conditions to derive a perturbed fundamental diagram to model more accurately transport phenomena observed on highways. The solution of the 2 × 2 system of partial differential equations is built through the definition of a Riemann solver, and a modified Godunov scheme is used to construct the numerical solution.
Modeling, estimation and control of distributed parameter systems: application to transportation networks
, 2012
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INVESTIGATION OF TRAFFIC FLOW DYNAMIC PROCESSES USING DISCRETE MODEL
"... Modelling the process of traffic flow was previously studied from different points of view and different mathematical methods where used to describe the same process. All authors have an agreement on basic traffic flow parameters like, traffic flow density, traffic flow rate or the average speed of ..."
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Modelling the process of traffic flow was previously studied from different points of view and different mathematical methods where used to describe the same process. All authors have an agreement on basic traffic flow parameters like, traffic flow density, traffic flow rate or the average speed of traffic flow. Besides, a lot of different investigations into the use of traffic flow models to deal with various problems of engineering are carried out. A comparison of different continuum models has drawn that a number of scientific works were based on fluid dynamic theory and gas kinetic traffic flow theory. The kinetic traffic flow theory is used in ‘microscopic ’ or “macroscopic”, traffic flow models.
Modified Suliciu relaxation system and exact resolution of isolated shock waves
, 2012
"... We present a new Approximate Riemann Solver (ARS) for the gas dynamics equations in Lagrangian coordinates and with general non linear pressure laws. The design of this new ARS relies on a generalized Suliciu pressure relaxation approach. It gives by construction the exact solutions for isolated ent ..."
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We present a new Approximate Riemann Solver (ARS) for the gas dynamics equations in Lagrangian coordinates and with general non linear pressure laws. The design of this new ARS relies on a generalized Suliciu pressure relaxation approach. It gives by construction the exact solutions for isolated entropic shocks and we prove that it is Lipschitzcontinuous and satisfies an entropy inequality. Finally, the ARS is used to develop either a classical entropy conservative Godunovtype method, or a Glimmtype (random sampling based Godunovtype) method able to generate infinitely sharp discrete shock profiles. Numerical experiments are proposed to prove the validity of these approaches. 1