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Abstract
The Boltzmann equation is an important mathematical model in the study of gas dynamics, offering a kinetic description of particles that undergo transport and collisions. The Bhatnagar-Gross-Krook (BGK) collision operator provides a simplification of the full Boltzmann collision operator. The resulting Boltzmann-BGK equation has been used in a variety of applications, including the space shuttle reentry problem and traffic flow modeling. The Knudsen number, $\varepsilon > 0$, is a non-dimensional parameter that measures the strength of the collision operator. A key feature that efficient numerical discretizations of the Boltzmann equation should be endowed with is the {\it asymptotic-preserving} property. This property allows a numerical method to be stable at fixed mesh parameters for any value of the Knudsen number $\varepsilon > 0$. In other words, the numerical scheme can be used efficiently in the fluid ($0 < \varepsilon < 0.1$), transition $(0.1 \le \varepsilon \le 10)$, and free molecular flow ($\varepsilon > 10$) regimes. The limit $\varepsilon \rightarrow 0^+$ is particularly challenging, since this is a singular limit. In this work, we develop a novel spectral element moment closure (SEC) method for numerically solving the Boltzmann-BGK model. The SEC method divides phase space into discrete velocity bands, inside of which a local linear moment closure is applied; an advantage of the SEC method is that it is high-order accurate in velocity. Additionally, we discretize the SEC method in space using a discontinuous Galerkin (DG) finite element method. We show how to make the resulting SEC/DG method asymptotic-preserving using an operator splitting approach, which results in two separate equations: (1) the collision term and (2) the transport term. We solve these two systems in time using a backward Euler method for the collision term and a locally-implicit method for the transport term. The resulting method is implemented in the open-source C++ software package {\sc dogpack}. For comparison purposes, we also implement in Python an implicit-explicit (IMEX) scheme for the Boltzmann-BGK equation from the literature. The resulting codes are validated on standard test cases.
