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13
Congestion redux
 SIAM J. Appl. Math
"... Abstract. In this paper we analyze a class of secondorder traffic models and show that these models support stable oscillatory traveling waves typical of the waves observed on a congested roadway. The basic model has trivial or constant solutions where cars are uniformly spaced and travel at a cons ..."
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Cited by 7 (1 self)
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Abstract. In this paper we analyze a class of secondorder traffic models and show that these models support stable oscillatory traveling waves typical of the waves observed on a congested roadway. The basic model has trivial or constant solutions where cars are uniformly spaced and travel at a constant equilibrium velocity that is determined by the car spacing. The stable traveling waves arise because there is an interval of car spacing for which the constant solutions are unstable. These waves consist of a smooth part where both the velocity and spacing between successive cars are increasing functions of a Lagrange mass index. These smooth portions are separated by shock waves that travel at computable negative velocity.
A MultiClass Homogenized Hyperbolic Model of Traffic Flow
 SIAM J. Math. Anal
, 2003
"... We introduce a new homogenized hyperbolic (multiclass) traffic flow model which allows to take into account the behaviors of different type of vehicles (cars, trucks, buses, etc.) and drivers. We discretize the starting Lagrangian system introduced below with a Godunov scheme, and we let the mesh s ..."
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Cited by 7 (2 self)
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We introduce a new homogenized hyperbolic (multiclass) traffic flow model which allows to take into account the behaviors of different type of vehicles (cars, trucks, buses, etc.) and drivers. We discretize the starting Lagrangian system introduced below with a Godunov scheme, and we let the mesh size h in (x; t) go to 0: the typical length (of a vehicle) and time vanish. Therefore, the variables  here (w; a)  which describe the heterogeneity of the reactions of the different cardriver pairs in the traffic, develop large oscillations when h ! 0. These (know) oscillations in (w; a) persist in time, and we describe the homogenized relations between velocity and density. We show that the velocity is the unique solution "à la Krukov" of a scalar conservation law, with variable coe cients, discontinuous in x. Finally, we prove that the same macroscopic homogenized model is also the hydrodynamic limit of the corresponding multiclass FollowtheLeader model.
EXISTENCE OF SOLUTIONS FOR THE AW–RASCLE TRAFFIC FLOW MODEL WITH VACUUM
"... Abstract. We consider the macroscopic model for traffic flow proposed by Aw and Rascle in 2000. The model is a 2×2 system of hyperbolic conservation laws, or, when the model includes a relaxation term, a 2 × 2 system of hyperbolic balance laws. The main difficulty is the presence of vacuum, which ma ..."
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Abstract. We consider the macroscopic model for traffic flow proposed by Aw and Rascle in 2000. The model is a 2×2 system of hyperbolic conservation laws, or, when the model includes a relaxation term, a 2 × 2 system of hyperbolic balance laws. The main difficulty is the presence of vacuum, which makes us unable to control the total variation of the conservative variables. We allow vacuum to appear and prove existence of a weak entropy solution to the Cauchy problem. 1.
CARFOLLOWING AND THE MACROSCOPIC AW–RASCLE TRAFFIC FLOW MODEL
"... Abstract. We consider a semidiscrete carfollowing model and the macroscopic Aw–Rascle model for traffic flow given in Lagrangian form. The solution of the carfollowing model converges to a weak entropy solution of the system of hyperbolic balance laws with Cauchy initial data. For the homogeneous ..."
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Abstract. We consider a semidiscrete carfollowing model and the macroscopic Aw–Rascle model for traffic flow given in Lagrangian form. The solution of the carfollowing model converges to a weak entropy solution of the system of hyperbolic balance laws with Cauchy initial data. For the homogeneous system, we allow vacuum in the initial data. By using properties of the semidiscrete model, we show that this solution of the hyperbolic system is stable in the L 1norm. 1.
TRAFFIC CONGESTION AN INSTABILITY IN A HYPERBOLIC SYSTEM BY
, 2004
"... In this paper we analyze a class of second order traffic models and show these models support stable oscillatory traveling waves typical of the waves observed on a congested roadway. The basic model has trivial or constant solutions where cars are uniformly spaced and travel at a constant equilibri ..."
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Cited by 1 (0 self)
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In this paper we analyze a class of second order traffic models and show these models support stable oscillatory traveling waves typical of the waves observed on a congested roadway. The basic model has trivial or constant solutions where cars are uniformly spaced and travel at a constant equilibrium velocity that is determined by the car spacing. The stable traveling waves arise because there is an interval of car spacing for which the constant solutions are unstable. These waves consist of a smooth part where both the velocity and spacing between successive cars are increasing functions of a Lagrange mass index. These smooth portions are separated by shock waves that travel at computable negative velocity. 1.
A rigorous treatment of a followtheleader traffic model with traffic lights present
 SIAM J. Appl. Math
"... Abstract. Traffic flow on a unidirectional roadway in the presence of traffic lights is modeled. Individual car responses to green, yellow, and red lights are postulated and these result in rules governing the acceleration and deceleration of individual cars. The essence of the model is that only sp ..."
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Abstract. Traffic flow on a unidirectional roadway in the presence of traffic lights is modeled. Individual car responses to green, yellow, and red lights are postulated and these result in rules governing the acceleration and deceleration of individual cars. The essence of the model is that only specific cars are directly affected by the lights. The other cars behave according to simple followtheleader rules which limit their speed by the spacing between them and the car directly ahead. The model has a number of desirable properties; namely, cars do not run red lights, cars do not smash into one another, and cars exhibit no velocity reversals. In a situation with multiple lights operating inphase, we get, after an initial startup period, a constant number of cars through each light during any greenyellow period. Moreover, this flux is less by one or two cars per period than the flux obtained in discretized versions of the idealized Lighthill–Whitham–Richards model which allows for infinite accelerations.
The AwRascle traffic model with locally constrained flow
, 2010
"... We consider solutions of the AwRascle model for traffic flow fulfilling a constraint on the flux at x = 0. Two different kinds of solutions are proposed: at x = 0 the first one conserves both the number of vehicles and the generalized momentum, while the second one conserves only the number of cars ..."
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We consider solutions of the AwRascle model for traffic flow fulfilling a constraint on the flux at x = 0. Two different kinds of solutions are proposed: at x = 0 the first one conserves both the number of vehicles and the generalized momentum, while the second one conserves only the number of cars. We study the invariant domains for these solutions and we compare the two Riemann solvers in terms of total variation of relevant quantities. Finally we construct ad hoc finite volume numerical schemes to compute these solutions.
Critical thresholds in a . . .
"... In this paper, we consider a hyperbolic relaxation system arising from a dynamic continuum traffic flow model. The equilibrium characteristic speed resonates with one characteristic speed of the full relaxation system in this model. Thus the usual subcharacteristic condition only holds marginally. ..."
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In this paper, we consider a hyperbolic relaxation system arising from a dynamic continuum traffic flow model. The equilibrium characteristic speed resonates with one characteristic speed of the full relaxation system in this model. Thus the usual subcharacteristic condition only holds marginally. In spite of this obstacle, we prove global in time regularity and finite time singularity formation of solutions simultaneously by showing the critical threshold phenomena associated with the underlying relaxation system. We identify five upper thresholds for finite time singularity in solutions and three lower thresholds for global existence of smooth solutions. The set of initial data leading to global smooth solutions is large, in particular allowing initial velocity of negative slope. Our results show that the shorter the drivers ’ responding time to the traffic, the larger the set of initial conditions leading to global smooth solutions which correctly predicts the empirical findings for traffic flows.