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Oberwolfach Reports
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Published online: 2009-09-30
Mini-Workshop: Numerics for Kinetic Equations
Irene M. Gamba[1], Sergej Rjasanow and Wolfgang Wagner[2] (1) University of Texas at Austin, USA(2) Applied Analysis and Stochastics, Berlin, Germany
The workshop \emph{Numerics for Kinetic Equations},
organized by Irene Gamba (Austin, USA),
Sergej Rjasanow (Saarbr\"ucken, Germany), and
Wolfgang Wagner (Berlin, Germany)
was held November 16th -- November 22nd, 2008. The
meeting was well attended with 15 participants
(11 from Europe, 3 from USA, and 1 from Japan).
This workshop brought together researchers with various
backgrounds: mathematics, physics, chemistry, and engineering science. The
interdisciplinarity of the invited participants was essential because
the numerical solution of kinetic equations is characterized by a wide area of
applications (rarefied gas dynamics, modeling of semiconductors, plasma,
or combustion). Furthermore, the strong coupling between
different areas of science is needed to understand the complicated and highly
nonlinear nature of kinetic equations.
The last decades have shown an amazing development in the design of
effective numerical methods for kinetic equations describing processes in
micro- and nanomachines. A particular example is the Knudsen compressor - an
energy efficient micro-scale pump with no moving parts. Furthermore, the
inelastic Boltzmann equation is more and more used in applications. Finally,
semiconductor transport models are still a challenge for numerics.
In this mini-workshop, we have discussed recent mathematical and computational
issues, mainly related to the so called ``low Mach number'' flows, from
the point of view of numerical efficiency and accuracy. When solving those
problems by the use of the Direct Simulation Monte Carlo method, one has to
deal with a small signal-to-noise ratio. Thus,
due to statistical fluctuations, it is
difficult to compute the physical flow parameters (density, stream velocity,
temperature, and, especially, heat flux) with sufficient accuracy, so variance
reduction methods are desperately needed. On the other hand, deterministic
algorithms do not exhibit statistical fluctuations, and, therefore, may become
competitive for slow flows.
The talks have been devoted to
\begin{itemize}
\item direct simulation Monte Carlo methods for kinetic equations, especially
for very slow flows close to equilibrium (Aoki, Garcia, Hadjiconstantinou, Pareschi),
\item modeling and numerical solution of the kinetic transport in
semiconductors (Degond, Gamba, Muscato),
\item deterministic numerical methods for kinetic equations
(Babovsky, Filbet, Kirsch, Rjasanow, Russo),
\item kinetic equations and numerics for reactive gas flows
and combustion (Kraft, Vikhansky).
\end{itemize}
Furthermore, three extended ``round table'' discussions were devoted to
\begin{itemize}
\item deviational particle methods for kinetic equations,
\item stochastic fluctuations in Monte Carlo methods,
\item benchmark problems for numerical methods.
\end{itemize}
These discussions were extremely useful for facilitating a clear understanding
of the main difficulties
when applying stochastic and/or deterministic methods to
kinetic equations close to the hydrodynamic limit.
The participants appreciated the idea of the mini-workshops -
to have not too many talks, but instead plenty of
time for intensive and detailed discussions.
We would strongly support further such workshops.
Finally, we would like to thank both the
administration and the staff of the MFO
for providing excellent working conditions
and creating a pleasant atmosphere.
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Gamba Irene, Rjasanow Sergej, Wagner Wolfgang: Mini-Workshop: Numerics for Kinetic Equations. Oberwolfach Rep. 5 (2008), 2943-2984. doi: 10.4171/OWR/2008/52