A review of conservation laws on networks
Mauro Garavello
This paper deals with various applications of conservation laws on networks. In particular we consider the car traffic, described by the Lighthill-Whitham-Richards model and by the Aw-Rascle-Zhang model, the telecommunication case, by using the model introduced by D'Apice-Manzo-Piccoli and, finally, the case of a gas pipeline, modeled by the classical $p$-system. For each of these models we present a review of some results about Riemann and Cauchy problems in the case of a network, formed by a single vertex with $n$ incoming and $m$ outgoing arcs.
keywords: car traffic Conservation laws gas pipeline telecommunication. networks
Comparison among different notions of solution for the $p$-system at a junction
Rinaldo M. Colombo Mauro Garavello
We consider $n$ tubes exiting a junction and filled with a non viscous isentropic or isothermal fluid. In each tube a copy of the $p$-system in Euler coordinates is considered. The aim of the presentation is to compare three different notions of solutions at the junctions: p-solutions, Q-solutions and P-solutions.
keywords: $p$-System Gas Dynamics Hyperbolic Systems of Conservation Laws Compressible Fluids Riemann Problem
On fluido-dynamic models for urban traffic
Mauro Garavello Benedetto Piccoli
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.
keywords: fluido-dynamic models car traffic urban traffic networks.
Stability and optimization in structured population models on graphs
Rinaldo M. Colombo Mauro Garavello
We prove existence and uniqueness of solutions, continuous dependence from the initial datum and stability with respect to the boundary condition in a class of initial--boundary value problems for systems of balance laws. The particular choice of the boundary condition allows to comprehend models with very different structures. In particular, we consider a juvenile-adult model, the problem of the optimal mating ratio and a model for the optimal management of biological resources. The stability result obtained allows to tackle various optimal management/control problems, providing sufficient conditions for the existence of optimal choices/controls.
keywords: optimal mating ratio. management of biological resources juvenile-adult model Renewal equation balance laws
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
Conservation laws with discontinuous flux
Mauro Garavello Roberto Natalini Benedetto Piccoli Andrea Terracina
We consider a hyperbolic conservation law with discontinuous flux. Such a partial differential equation arises in different applications, in particular we are motivated by a model of traffic flow. We provide a new formulation in terms of Riemann Solvers. Moreover, we determine the class of Riemann Solvers which provide existence and uniqueness of the corresponding weak entropic solutions.
keywords: discontinuous flux traffic flow Conservation laws Riemann Solvers front tracking
The LWR traffic model at a junction with multibuffers
Mauro Garavello
We consider the Lighthill-Whitham-Richards traffic flow model on a network composed by a single junction $J$ with $n$ incoming roads, $m$ outgoing roads and $m$ buffers, one for each outgoing road. We introduce a concept solution at $J$, which is compared with that proposed in [14]. Finally we study the Cauchy problem and, in the special case of $n \le 2$ and $m \le 2$, we prove existence of solutions to the Cauchy problem, via the wave-front tracking method.
keywords: junction Lighthill-Whitham-Richards model Conservation laws multibuffer traffic problems.
Coupling of microscopic and phase transition models at boundary
Mauro Garavello Benedetto Piccoli
This paper deals with coupling conditions between the classical microscopic Follow The Leader model and a phase transition (PT) model. We propose a solution at the interface between the two models. We describe the solution to the Riemann problem.
keywords: coupling conditions Follow the Leader conservation laws Cauchy Problem. phase transition model
A Well Posed Riemann Problem for the $p$--System at a Junction
Rinaldo M. Colombo Mauro Garavello
This work is devoted to the solution to Riemann Problems for the $p$-system at a junction, the main goal being the extension to the case of an ideal junction of the classical results that hold in the standard case.
keywords: compressible fluids at junctions. Riemann problem gas $p$-system
The Riemann Problem at a Junction for a Phase Transition Traffic Model
Mauro Garavello Francesca Marcellini

We extend the Phase Transition model for traffic proposed in [8], by Colombo, Marcellini, and Rascle to the network case. More precisely, we consider the Riemann problem for such a system at a general junction with $n$ incoming and $m$ outgoing roads. We propose a Riemann solver at the junction which conserves both the number of cars and the maximal speed of each vehicle, which is a key feature of the Phase Transition model. For special junctions, we prove that the Riemann solver is well defined.

keywords: Phase transition model hyperbolic systems of conservation laws continuum traffic models Riemann problem Riemann solver

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