131 research outputs found
Kinetic models of BGK type and their numerical integration
This minicourse contains a description of recent results on the modelling of
rarefied gases in weakly non equilibrium regimes, and the numerical methods
used to approximate the resulting equations. Therefore this work focuses on BGK
type approximations, rather than on full Boltzmann models. Within this
framework, models for polyatomic gases and for mixtures will be considered. We
will also address numerical issues characteristic of the difficulties one
encounters when integrating kinetic equations. In particular, we will consider
asymptotic preserving schemes, which are designed to approximate equilibrium
solutions, without resolving the fast scales of the approach to equilibrium.Comment: Lecture notes for the 9th summer school Methods And Models Of Kinetic
Theory, M&MKT 201
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
Analysis of a heterogeneous kinetic model for traffic flow
In this work we extend a recent kinetic traffic model to the case of more
than one class of vehicles, each of which is characterized by few different
microscopic features. We consider a Boltzmann-like framework with only binary
interactions, which take place among vehicles belonging to the various classes.
Our approach differs from the multi-population kinetic model based on a lattice
of speeds because here we assume continuous velocity spaces and we introduce a
parameter describing the physical velocity jump performed by a vehicle that
increases its speed after an interaction. The model is discretized in order to
investigate numerically the structure of the resulting fundamental diagrams and
the system of equations is analyzed by studying well posedness. Moreover, we
compute the equilibria of the discretized model and we show that the exact
asymptotic kinetic distributions can be obtained with a small number of
velocities in the grid. Finally, we introduce a new probability law in order to
attenuate the sharp capacity drop occurring in the diagrams of traffic.Comment: 31 page
Derivation and stability analysis of macroscopic multi-lane models for vehicular traffic flow
The mathematical modeling and the stability analysis of multi-lane traffic in
the macroscopic scale is considered. We propose a new first order model derived
from microscopic dynamics with lane changing, leading to a coupled system of
hyperbolic balance laws. The macroscopic limit is derived without assuming ad
hoc space and time scalings. The analysis of the stability of the equilibria of
the model is discussed. The proposed numerical tests confirm the theoretical
findings between the macroscopic and microscopic modeling, and the results of
the stability analysis
A consistent kinetic model for a two-component mixture of polyatomic molecules
We consider a multi component gas mixture with translational and internal
energy degrees of freedom assuming that the number of particles of each species
remains constant. We will illustrate the derived model in the case of two
species, but the model can be easily generalized to multiple species. The two
species are allowed to have different degrees of freedom in internal energy and
are modelled by a system of kinetic ES-BGK equations featuring two interaction
terms to account for momentum and energy transfer between the species. We prove
consistency of our model: conservation properties, positivity of the
temperature, H-theorem and convergence to a global equilibrium in the form of a
global Maxwell distribution. Thus, we are able to derive the usual macroscopic
conservation laws. For numerical purposes we apply the Chu reduction to the
developed model for polyatomic gases and give an application for a gas
consisting of a mono atomic and a diatomic species.Comment: arXiv admin note: text overlap with arXiv:1806.1050
Fundamental diagrams in traffic flow: the case of heterogeneous kinetic models
Experimental studies on vehicular traffic provide data on quantities like
density, flux, and mean speed of the vehicles. However, the diagrams relating
these variables (the fundamental and speed diagrams) show some peculiarities
not yet fully reproduced nor explained by mathematical models. In this paper,
resting on the methods of kinetic theory, we introduce a new traffic model
which takes into account the heterogeneous nature of the flow of vehicles along
a road. In more detail, the model considers traffic as a mixture of two or more
populations of vehicles (e.g., cars and trucks) with different microscopic
characteristics, in particular different lengths and/or maximum speeds. With
this approach we gain some insights into the scattering of the data in the
regime of congested traffic clearly shown by actual measurements.Comment: 26 pages, 11 figure
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