Deterministic Numerical Methods for the Boltzmann Equation

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.