Conservation law models for traffic flow on a network of roads (original) (raw)

On the variational theory of traffic flow: well-posedness, duality and applications

Networks and Heterogeneous Media, 2006

This paper describes some simplifications allowed by the variational theory of traffic flow (VT). It presents general conditions guaranteeing that the solution of a VT problem with bottlenecks exists, is unique and makes physical sense ABSTRACT This paper describes some simplifications allowed by the variational theory of traffic flow (VT). It presents general conditions guaranteeing that the solution of a VT problem with bottlenecks exists, is unique and makes physical senseB i.e., that the problem is well-posed. The requirements for well-posedness are mild and met by practical applications. They are consistent with narrower results available for kinematic wave or Hamilton-Jacobi theories. The paper also describes some duality ideas relevant to all these theories. Duality and VT are used to establish the equivalence of eight traffic models. Most of these are not new but VTduality considerations offer a new insight into their relationship.

Conservation laws with unilateral constraints in traffic modeling

2009

Abstract 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.

Non classical solution of a conservation law arising in vehicular traffic modelling

ESAIM, 2016

We are interested in this paper in the modelling and numerical simulation of some phenomena that are observed in the context of car dynamics, in particular the appearance of persistent jams upstream critical points with no real cause of flux limitation. We shall consider the case of a stable jam on a freeway upstream an accident that took place on the opposite lane. This situation is not properly handled by most models, either micro-or macroscopic ones, since it corresponds to a phenomenon that does not have a counterpart in gas dynamics, for which only entropy solution are usually considered as physically feasible. The approach we propose consists in accounting for the very behaviour of agents in the neighbourhood of the discontinuity, and makes it possible to numerically recover in a robust way steady traffic jams. Résumé. Nous nous intéressons dans cet article à la modélisation et à la simulation numérique de phénomènes particuliers observés dans le contexte du trafic routier, plus particulièrement le phénomène d'apparition et de persistance de bouchons en amont de points critiques sans cause objective de ralentissement. Nous considérerons notamment le cas d'un bouchon persistant sur une autoroute en amont d'un accident qui s'est produit sur la voie d'en face. Cette situation n'est pas reproduite par la plupart des modèles, qu'ils soient microscopiques ou macroscopiques, car elle correspond à un phénomène qui n'a pas d'équivalent en dynamique des gaz, pour lesquels seules les solutions dites entropiques sont en général considérées comme correspondant à un comportement observable dans la réalité. L'approche que nous proposons, aux niveaux microscopique et macroscopique, consiste à prendre en compte le comportement particulier des agents humains au voisinage de la discontinuité, et permet de retrouver numériquement de façon robuste des bouchons stables.

On the role of source terms in continuum traffic flow models

2006

We introduce some models for vehicular traffic flow based on hyperbolic balance laws. We focus in particular on source terms for modeling highway entries and exits or local changes of the traffic flow due to inhomogeneities of the road. Rigorous well-posedness results and numerical investigations are presented. We show in particular how real phenomena (eg the formation of a queue) that are not captured by models based on systems of conservation laws are instead observable with our models.

Non classical solution of a conservation law arising in vehicular traffic or crowd modelling

2016

Traffic Flow on a Road Network

SIAM Journal on Mathematical Analysis, 2005

This paper is concerned with a fluidodynamic model for traffic flow. More precisely, we consider a single conservation law, deduced from conservation of the number of cars, defined on a road network that is a collection of roads with junctions. The evolution problem is underdetermined at junctions, hence we choose to have some fixed rules for the distribution of traffic plus an optimization criteria for the flux. We prove existence, uniqueness and stability of solutions to the Cauchy problem.

On the modeling and management of traffic

2011

Abstract Several realistic situations in vehicular traffic that give rise to queues can be modeled through conservation laws with boundary and unilateral constraints on the flux. This paper provides a rigorous analytical framework for these descriptions, comprising stability with respect to the initial data, to the boundary inflow and to the constraint. We present a framework to rigorously state optimal management problems and prove the existence of the corresponding optimal controls.

A Fluid-Dynamic Traffic Model on Road Networks

Archives of Computational Methods in Engineering, 2007

We consider a mathematical model for fluiddynamic flows on networks which is based on conservation laws. Road networks are studied as graphs composed by arcs that meet at some nodes, corresponding to junctions, which play a key-role. Indeed interactions occur at junctions and there the problem is underdetermined. The approximation of scalar conservation laws along arcs is carried out by using conservative methods, such as the classical Godunov scheme and the more recent discrete velocities kinetic schemes with the use of suitable boundary conditions at junctions. Riemann problems are solved by means of a simulation algorithm which processes each junction. We present the algorithm and its application to some simple test cases and to portions of urban network.

Optima and equilibria for a model of traffic flow

2011

The paper is concerned with the Lighthill–Whitham model of traffic flow, where the density of cars is described by a scalar conservation law. A cost functional is introduced, depending on the departure and arrival times of each driver. Under natural assumptions, we prove the existence of a unique globally optimal solution, minimizing the total cost to all drivers. This solution contains no shocks and can be explicitly described. We also prove the existence of a Nash equilibrium solution, where no driver can lower his individual cost by changing his own departure time. A characterization of the Nash solution is provided, establishing its uniqueness. Some explicit examples are worked out, comparing the costs of the optimal and the equilibrium solutions. The analysis also yields a strategy for optimal toll pricing.

On fluido-dynamic models for urban traffic

2009

The aim of this paper is to address the following questions: which models, among fluido-dynamic ones, are more appropriate to describe urban traffic? While a rich debate was developed for the complicate dynamics of highway traffic, some basic problems of urban traffic are not always appropriately discussed. We analyze many recent, and less recent, models focusing on three basic properties. The latter are necessary to reproduce correctly queue formation at lights and junctions, and their backward propagation on an urban network. 2000 Mathematics Subject Classification: 35L65

Conservation law models for traffic flow on a network of roads (original) (raw)

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