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.
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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The paper presents a review of the main analytical results available on the traffic flow model with phase transitions
described in . We also introduce a forthcoming existence result on road networks .
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.
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.
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.
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 ,
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
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.
In this paper, we illustrate how second order traffic flow models, in our case the Aw-Rascle equations, can be used to reproduce empirical observations such as the capacity drop at merges and solve related optimal control problems. To this aim, we propose a model for on-ramp junctions and derive suitable coupling conditions. These are associated to the first order Godunov scheme to numerically study the well-known capacity drop effect, where the outflow of the system is significantly below the expected maximum. Control issues such as speed and ramp meter control are also addressed in a first-discretize-then-optimize framework.
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 .
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 .
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.