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Author

Date

2018Type

- Doctoral Thesis

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Abstract

The Boltzmann equation offers a mesoscopic description of rarefied gases and is a
typical representative of a class of integro partial differential equations that
model interacting particle systems. The binary particle interactions in
$d$-dimensional
space are modeled by a collision operator which involves a $2d-1$
fold integral. Due to its non-linearity and the high dimension, the evaluation of the
collision operator is computationally challenging. Stochastic simulation methods are
widely used. A well-known example is the direct simulation Monte Carlo (DSMC) method
developed by Bird and Nanbu in~\cite{bird_molecular_1994} and
\cite{nanbu_direct_1980}. Among deterministic approaches Fourier methods are most
popular. In~\cite{pareschi_fourier_1996} Pareschi et al.\ introduced a Fourier based
method, related approaches have been introduced
in~\cite{bobylev_difference_1997, bobylev_fast_1999, gamba_spectral-lagrangian_2009,
wu_deterministic_2013, gamba_fast_2017, filbet_solving_2006}. Fourier methods are
fairly efficient and accurate for short-time simulations, but they suffer from
aliasing errors caused by the periodic truncation of the velocity domain. Another
problem of the Fourier spectral method is that it does not capture the correct
long-time behaviour, unless the steady state preserving
modification~\cite{filbet_steady-state_2015} is employed. The fast spectral method
was recently extended to arbitrary collision kernels~\cite{gamba_fast_2017}, in $3D$
it works with complexity $\mathcal{O}(M N^4 \log (N))$
whereas the original fast spectral method for the hard spheres model by Pareschi, Filbet and
Mouhot~\cite{mouhot_fast_2006,filbet_solving_2006} has $\mathcal{O}(M N^3 \log (N))$.
To overcome these problems a spectral discretization in velocity based on Laguerre
polynomials has been developed in~\cite{fonn_polar_2014} for the spatially
homogeneous Boltzmann equation extending the work done
in~\cite{ender_polynomial_1999}. For this method, no truncation of the velocity
domain is necessary and the natural conserved quantities can be easily preserved by
the numerical scheme. As a consequence, the velocity-spectral method enjoys the
correct long-term behaviour, while the aliasing effects incurred by the plain Fourier
spectral method will in general lead to unphysical solutions (see
Section~\ref{sec:numerical_homogeneous}). Additionally, this approach has the
advantage that no periodic truncation is needed and the collision operator can be
represented as a tensor, which enjoys considerable sparsity and whose entries can be
precomputed with highly accurate quadrature.
%Ender~\cite{ender_polynomial_1999}.
Closely related and conducted parallel to our investigations is the work by Kitzler
and Schöberl~\cite{g._kitzler_efficient_2013,kitzler_high_2015}. These authors also
use a spectral polynomial discretization in velocity, but they rely on a
Petrov-Galerkin discretization. The velocity distribution function (VDF) is
represented by polynomials times a shifted Maxwellian, while the test functions are
polynomials. The complexity for the evaluation of the collision operator is reduced
from $\mathcal{O}(K^{6})$
to $\mathcal{O}(K^{5})$
by exploiting its translation invariance. They locally rescale the basis functions in
velocity to fit macroscopic velocity and temperature. In physical space Kitzler and
Schöberl use a discontinuous Galerkin scheme. On the one hand this offers great
flexibility concerning the local choice of velocity spaces. On the other hand the DG
method involves evaluating interface fluxes and thus requires projection of the
velocity distribution function between adjacent elements. Then stability issues
impose constraints on the temperature differences between neighboring elements.
In this work, we extend this idea to the spatially inhomogeneous Boltzmann equation,
combining a truncation-free spectral Galerkin approximation in velocity with a least
squares stabilized finite element discretization in space. The tensor based local
evaluation of the discrete collision operator involves an asymptotic computational
effort of $O(K^{5})$,
where $K$
is the polynomial degree in one velocity direction, see
Section~\ref{sec:collision_operator}. We also explore ways to ensure discrete
conservation of mass, momentum, and energy, see Section~\ref{sec:cons}. This can be
achieved by direct enforcement of the constraints through Lagrangian multipliers. For
time-stepping we rely on a splitting scheme, which separately treats collisions and
advection. For the former we opt for explicit time-stepping, whereas the latter is
tackled by a time-implicit least squares formulation. This has the advantage, that
for high Knudsen numbers we are not restricted by a CFL condition. However one must
note that for small Knudsen numbers, i.e.\ small mean free path length, the problem
is stiff and the time-step must be chosen sufficiently small. Extensive numerical
tests in various settings typical of flow problems for rarefied gases are reported in
Section~\ref{sec:numerical_experiments}.
The outline of the thesis is as follows: In Section~\ref{sec:boltzmann_equation} we
introduce the Boltzmann equation and its properties that are used in the sequel. The
conservative scheme for the homogeneous Boltzmann equation is discussed in
Section~\ref{sec:collision_operator}, followed by the extension to the spatially
inhomogeneous case in Section~\ref{chap:fem_discretization_in_space}. Numerical
results for a range of benchmark problems are reported in
Section~\ref{sec:numerical_experiments}. The results have been published in~\cite{GHP2017}.
In the second part, the FEM discretization in the spatial coordinate is replaced by
ridgelets. The multi-scale ridgelet transform allows best $N$-term approximations for
functions with line-singularities, essentially what wavelets provide for point-singularities. Show more

Permanent link

https://doi.org/10.3929/ethz-b-000317127Publication status

publishedExternal links

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Publisher

ETH ZurichSubject

Numerical methods; High Performance Computing (HPC); Rarefied gas dynamics; spectral methods; finite elements; Boltzmann equartionOrganisational unit

03632 - Hiptmair, Ralf / Hiptmair, Ralf
Funding

146356 - Sparse Discretization of Kinetic Transport Problems on High-dimensional Phase Spaces (SNF)

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