115 research outputs found
Uncertainty damping in kinetic traffic models by driver-assist controls
In this paper, we propose a kinetic model of traffic flow with uncertain
binary interactions, which explains the scattering of the fundamental diagram
in terms of the macroscopic variability of aggregate quantities, such as the
mean speed and the flux of the vehicles, produced by the microscopic
uncertainty. Moreover, we design control strategies at the level of the
microscopic interactions among the vehicles, by which we prove that it is
possible to dampen the propagation of such an uncertainty across the scales.
Our analytical and numerical results suggest that the aggregate traffic flow
may be made more ordered, hence predictable, by implementing such control
protocols in driver-assist vehicles. Remarkably, they also provide a precise
relationship between a measure of the macroscopic damping of the uncertainty
and the penetration rate of the driver-assist technology in the traffic stream
Asymptotic-preserving exponential methods for the quantum Boltzmann equation with high-order accuracy
In this paper we develop high-order asymptotic-preserving methods for the
spatially inhomogeneous quantum Boltzmann equation. We follow the work in Li
and Pareschi, where asymptotic preserving exponential Runge-Kutta methods for
the classical inhomogeneous Boltzmann equation were constructed. A major
difficulty here is related to the non Gaussian steady states characterizing the
quantum kinetic behavior. We show that the proposed schemes work with
high-order accuracy uniformly in time for all Planck constants ranging from
classical regime to quantum regime, and all Knudsen numbers ranging from
kinetic regime to fluid regime. Computational results are presented for both
Bose gas and Fermi gas
Mini-Workshop: Innovative Trends in the Numerical Analysis and Simulation of Kinetic Equations
In multiscale modeling hierarchy, kinetic theory plays a vital role in connecting microscopic Newtonian mechanics and macroscopic continuum mechanics. As computing power grows, numerical simulation of kinetic equations has become possible and undergone rapid development over the past decade. Yet the unique challenges arising in these equations, such as highdimensionality, multiple scales, random inputs, positivity, entropy dissipation, etc., call for new advances of numerical methods. This mini-workshop brought together both senior and junior researchers working on various fastpaced growing numerical aspects of kinetic equations. The topics include, but were not limited to, uncertainty quantification, structure-preserving methods, phase transitions, asymptotic-preserving schemes, and fast methods for kinetic equations
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
A fast iterative scheme for the linearized Boltzmann equation
An iterative scheme can be used to find a steady-state solution to the
Boltzmann equation, however, it is very slow to converge in the near-continuum
flow regime. In this paper, a synthetic iterative scheme is developed to speed
up the solution of the linearized Boltzmann equation. The velocity distribution
function is first solved by the conventional iterative scheme, then it is
corrected such that the macroscopic flow velocity is governed by a diffusion
equation which is asymptotic-preserving in the Navier-Stokes limit. The
efficiency of the new scheme is verified by calculating the eigenvalue of the
iteration, as well as solving for Poiseuille and thermal transpiration flows.
The synthetic iterative scheme is significantly faster than the conventional
iterative scheme in both the transition and the near-continuum flow regimes.
Moreover, due to the asymptotic-preserving properties, the SIS needs less
spatial resolution in the near-continuum flow regimes, which makes it even
faster than the conventional iterative scheme. Using this synthetic iterative
scheme, and the fast spectral approximation of the linearized Boltzmann
collision operator, Poiseuille and thermal transpiration flows between two
parallel plates, through channels of circular/rectangular cross sections, and
various porous media are calculated over the whole range of gas rarefaction.
Finally, the flow of a Ne-Ar gas mixture is solved based on the linearized
Boltzmann equation with the Lennard-Jones potential for the first time, and the
difference between these results and those using hard-sphere intermolecular
potential is discussed.Comment: 13 figs, 5 table
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