78 research outputs found
The BGK approximation of kinetic models for traffic
We study spatially non-homogeneous kinetic models for vehicular traffic flow.
Classical formulations, as for instance the BGK equation, lead to
unconditionally unstable solutions in the congested regime of traffic. We
address this issue by deriving a modified formulation of the BGK-type equation.
The new kinetic model allows to reproduce conditionally stable non-equilibrium
phenomena in traffic flow. In particular, stop and go waves appear as bounded
backward propagating signals occurring in bounded regimes of the density where
the model is unstable. The BGK-type model introduced here also offers the
mesoscopic description between the microscopic follow-the-leader model and the
macroscopic Aw-Rascle and Zhang model
A statistical mechanics approach to macroscopic limits of car-following traffic dynamics
We study the derivation of macroscopic traffic models from car-following
vehicle dynamics by means of hydrodynamic limits of an Enskog-type kinetic
description. We consider the superposition of Follow-the-Leader (FTL)
interactions and relaxation towards a traffic-dependent Optimal Velocity (OV)
and we show that the resulting macroscopic models depend on the relative
frequency between these two microscopic processes. If FTL interactions dominate
then one gets an inhomogeneous Aw-Rascle-Zhang model, whose (pseudo) pressure
and stability of the uniform flow are precisely defined by some features of the
microscopic FTL and OV dynamics. Conversely, if the rate of OV relaxation is
comparable to that of FTL interactions then one gets a
Lighthill-Whitham-Richards model ruled only by the OV function. We further
confirm these findings by means of numerical simulations of the particle system
and the macroscopic models. Unlike other formally analogous results, our
approach builds the macroscopic models as physical limits of particle dynamics
rather than assessing the convergence of microscopic to macroscopic solutions
under suitable numerical discretisations.Comment: 21 pages, 6 figure
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
Steady-state traffic flow on a ring road with up- and down- slopes
This paper studies steady-state traffic flow on a ring road with up- and
down- slopes using a semi-discrete model. By exploiting the relations between
the semi-discrete and the continuum models, a steady-state solution is uniquely
determined for a given total number of vehicles on the ring road. The solution
is exact and always stable with respect to the first-order continuum model,
whereas it is a good approximation with respect to the semi-discrete model
provided that the involved equilibrium constant states are linearly stable. In
an otherwise case, the instability of one or more equilibria could trigger
stop-and-go waves propagating in certain road sections or throughout the ring
road. The indicated results are reasonable and thus physically significant for
a better understanding of real traffic flow on an inhomogeneous road
Derivation and Empirical Validation of a Refined Traffic Flow Model
The gas-kinetic foundation of fluid-dynamic traffic equations suggested in
previous papers [Physica A 219, 375 and 391 (1995)] is further refined by
applying the theory of dense gases and granular materials to the Boltzmann-like
traffic model by Paveri-Fontana. It is shown that, despite the
phenomenologically similar behavior of ordinary and granular fluids, the
relations for these cannot directly be transferred to vehicular traffic. The
dissipative and anisotropic interactions of vehicles as well as their
velocity-dependent space requirements lead to a considerably different
structure of the macroscopic traffic equations, also in comparison with the
previously suggested traffic flow models. As a consequence, the instability
mechanisms of emergent density waves are different. Crucial assumptions are
validated by empirical traffic data and essential results are illustrated by
figures.Comment: For related work see
http://www.theo2.physik.uni-stuttgart.de/helbing.htm
Micro-Macro limit of a non-local generalized Aw-Rascle type model
International audienceWe introduce a Follow-the-Leader approximation of a non-local generalized Aw-Rascle-Zhang (GARZ) model for traffic flow. We prove the convergence to weak solutions of the corresponding macroscopic equations deriving and BV estimates. We also provide numerical simulations illustrating the micro-macro convergence and we investigate numerically the non-local to local limit for both the microscopic and macroscopic models
- …