133 research outputs found
Solutions of the Aw-Rascle-Zhang system with point constraints
We revisit the entropy formulation and the wave-front tracking construction of physically admissible solutions of the Aw-Rascle and Zhang (ARZ) " second-order " model for vehicular traffic. A Kruzhkov-like family of entropies is introduced to select the admissible shocks. This tool allows to define rigorously the appropriate notion of admissible weak solution and to approximate the solutions of the ARZ model with point constraint. Stability of solutions w.r.t. strong convergence is justified. We propose a finite volumes numerical scheme for the constrained ARZ, and we show that it can correctly locate contact discontinuities and take the constraint into account
Entropy solutions for a traffic model with phase transitions
In this paper, we consider the two phases macroscopic traffic model
introduced in [P. Goatin, The Aw-Rascle vehicular traffic flow with phase
transitions, Mathematical and Computer Modeling 44 (2006) 287-303]. We first
apply the wave-front tracking method to prove existence and a priori bounds for
weak solutions. Then, in the case the characteristic field corresponding to the
free phase is linearly degenerate, we prove that the obtained weak solutions
are in fact entropy solutions \`a la Kruzhkov. The case of solutions attaining
values at the vacuum is considered. We also present an explicit numerical
example to describe some qualitative features of the solutions
A class of multi-phase traffic theories for microscopic, kinetic and continuum traffic models
In the present paper a review and numerical comparison of a special class of
multi-phase traffic theories based on microscopic, kinetic and macroscopic
traffic models is given. Macroscopic traffic equations with multi-valued
fundamental diagrams are derived from different microscopic and kinetic models.
Numerical experiments show similarities and differences of the models, in
particular, for the appearance and structure of stop and go waves for highway
traffic in dense situations. For all models, but one, phase transitions can
appear near bottlenecks depending on the local density and velocity of the
flow
Two-way multi-lane traffic model for pedestrians in corridors
We extend the Aw-Rascle macroscopic model of car traffic into a two-way
multi-lane model of pedestrian traffic. Within this model, we propose a
technique for the handling of the congestion constraint, i.e. the fact that the
pedestrian density cannot exceed a maximal density corresponding to contact
between pedestrians. In a first step, we propose a singularly perturbed
pressure relation which models the fact that the pedestrian velocity is
considerably reduced, if not blocked, at congestion. In a second step, we carry
over the singular limit into the model and show that abrupt transitions between
compressible flow (in the uncongested regions) to incompressible flow (in
congested regions) occur. We also investigate the hyperbolicity of the two-way
models and show that they can lose their hyperbolicity in some cases. We study
a diffusive correction of these models and discuss the characteristic time and
length scales of the instability
Lack of BV bounds for approximate solutions to a two-phase transition model arising from vehicular traffic
We consider wave-front tracking approximate solutions to a two-phase transition model for vehicular traffic. We construct an explicit example showing that the total variation in space of the solution blows up in finite time even for an initial datum with bounded total variation
The Variational Formulation of a Non-equilibrium Traffic Flow Model: Theory and Implications
AbstractThe analysis and numerical solution of non-equilibrium traffic flow models in current literature are almost exclusively carried out in the hyperbolic conservation law framework, which requires a good understanding of the delicate and non-trivial Riemann problem for conservation laws. In this paper, we present a novel formulation of certain non-equilibrium traffic flow models based on their isomorphic relation with optimal control problems. This formulation extends the minimum principle observed by the LWR model. We demonstrate that with the new formulation, generic initial-boundary conditions can be conveniently handled and a simplified numerical solution scheme for non-equilibrium models can be devised. Besides deriving the variational formulation, we provide a comprehensive discussion on its mathematical properties and physical implications
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