(Karlstad University, Sweden and
Institute of Applied Mathematics, Moscow)
Boltzmann equation and hydrodynamics at the Burnett level
present a review of some results on Burnett-type hydrodynamic
equations derived from the Boltzmann equation. The well-known
problem here is connected with regularization of classical
(ill-posed) Burnett equations[1-5]. There are several ways to deal
with this problem. We discuss in detail one of the approaches,
proposed in  and further developed in [2-4]. Our approach is
based on infinitesimal changes of variables, it shows that the way
of truncation of the Chapman-Enskog series is not unique. It is
the only approach which does not use any information beyond the
classical Burnett equations. We show how to derive a
two-parameter family of stable Generalized Burnett Equations
(GBEs)  and discuss the optimal choice of the parameters.
Surprisingly the resulting well-posed equations are simpler than
the original Burnett equations. The equations are derived for
arbitrary intermolecular forces. Some special properties of (a)
stationary problems and (b) linear non-stationary problems are
discussed in more detail. Finally we present some recent
results on the shock-wave structure , which show that GBEs
yield certain improvement of the Navier-Stokes results for
moderate Mach numbers. Some open questions are also discussed.
 A.V.Bobylev, Instabilities in the
Chapman-Enskog expansion and hyperbolic Burnett equations,
J.Stat.Phys. 124, 371 (2006).
 A.V. Bobylev, Generalized
Burnett hydrodynamics, J.Stat.Phys. 132, 569 (2008).
M.Bisi, M.P.Cassinari and M.Groppi, Qualitative analysis of the
Generalized Burnett Equations and applications to half-space
problems, Kinet. Relat. Models 1, 295 (2008).
M.Bisi, M.P.Cassinari and G.Spiga, Shock wave structure for
generalized Burnett equations, Phys. of Fluids 23, 1 (2011).
A.V.Bobylev and A.Windfall, Boltzmann equation and hydrodynamics
at the Burnett level, Kinet. Relat. Models 5, No.2 (2012).
Seong Jun Kim
(Mathematics, Georgia Tech)
Optimal path for a scan of the entire environment under limited
Abstract: We propose a computational strategy
to find the optimal path for a mobile sensor with limited range to
traverse a cluttered region and achieve complete coverage of the
environment. We first pose the problem in the level set framework,
and consider a related question of placing multiple stationary
sensors to obtain the full surveillance of the environment. The
locations of the stationary sensors are then used for the
initialization of the path for the moving sensor. The path is
optimized by following the gradient flow of the connecting points,
which is a system of ODEs, to shrink its length while maintaining
the complete coverage of the environment. Furthermore, we use
intermittent diffusion, which converts the ODEs into SDEs, to find
the global optimal solution. In addition, we introduce two
techniques, disentanglement and removing redundant connecting
points in the system of SDEs, to reduce the dimension of the
system and improve the efficiency of the computation. This is a
joint work with Haomin Zhou and Sung Ha Kang.
Krein-Gelfand-Levitan algorithm for inverse hyperbolic problems
via spectrally matched finite-difference grids.
a method for the numerical solution of inverse problems for
coefficients of hyperbolic PDEs based on the spectrally matched
finite-difference grids (a.k.a. Gaussian quadrature rules or
optimal grids). The method is built around an algorithm for
interpolation of the measured time domain data. Once an
interpolant is obtained, it can be expressed in terms of Stieltjes
continued fraction or its matrix generalization. The use of
S-fraction coefficients in inversion is twofold. First, they can
be used to reformulate the traditional optimization-based
approaches to drastically improve the objective functional, which
addresses issues such as local minima and slow convergence.
Second, the coefficients provide a way to obtain direct,
non-iterative reconstructions on the spectrally matched grids.
supplement the theoretical considerations with numerical
results. This is a joint with V. Druskin and M. Zaslavsky.
University of Catania,
Deterministic and stochastic simulations of electron transport in graphene
Graphene is a gapless semiconductor made of a sheet composed of a single layer of carbon atoms
arranged into a honeycomb hexagonal lattice. In view of application in
graphene-based electron devices, it is crucial to understand the basic
transport properties of this material. A physically accurate model is
given by a semiclassical transport equation whose scattering terms have
been deeply analyzed recently. Due to the computational difficulties,
the most part of the available solutions have been obtained by direct
Monte Carlo simulations. The aim of this work is to use a numerical
scheme based on the discontinuous Galerkin method for finding
deterministic (non stochastic) solutions of the electron Boltzmann
transport equation in graphene. The same method has been already
successfully applied to a more conventional semiconductor material like
Silicon or Gallium Arsenide. A n-type doping or equivalently a
high value of the Fermi potential is considered. Therefore we neglect
the inter band scatterings but retain all the main
electron-phonon scatterings. Simulations in graphene nano-ribbons
are presented and a comparison with stochastic simulations are
presented. We will show that standard techniques for direct Monte Carlo
simulations require important adjustments, and we propose a new
stochastic scheme, which can be also validated by means of
deterministic numerical solutions. Work in collaboration with Vittorio
(Department of Computing and Mathematical Sciences, Caltech)
Spectral method for linear
half-space kinetic equations
Understanding the coupling
of physical models at different scales is important and
challenging. In this talk, we focus on the issue of kinetic-fluid
coupling, in particular, the half-space problems for kinetic
equations coming from the boundary layer. We will present some
recent progress in algorithm development and analysis for the
linear half-space kinetic equations, and its application in
coupling of neutron transport equations with diffusion equations.
This is a joint work with Jianfeng Lu and Weiran Sun.