TY - JOUR
T1 - High Order Conservative Semi-Lagrangian Scheme for the BGK Model of the Boltzmann Equation
AU - Boscarino , Sebastiano
AU - Cho , Seung-Yeon
AU - Russo , Giovanni
AU - Yun , Seok-Bae
JO - Communications in Computational Physics
VL - 1
SP - 1
EP - 56
PY - 2020
DA - 2020/11
SN - 29
DO - http://doi.org/10.4208/cicp.OA-2020-0050
UR - https://global-sci.org/intro/article_detail/cicp/18421.html
KW - BGK model, Boltzmann equation, semi-Lagrangian scheme, conservative correction,
discrete Maxwellian.
AB - <p style="text-align: justify;">In this paper, we present a conservative semi-Lagrangian finite-difference
scheme for the BGK model. Classical semi-Lagrangian finite difference schemes, coupled with an L-stable treatment of the collision term, allow large time steps, for all the
range of Knudsen number [17, 27, 30]. Unfortunately, however, such schemes are not
conservative. Lack of conservation is analyzed in detail, and two main sources are
identified as its cause. Firstly, when using classical continuous Maxwellian, conservation error is negligible only if velocity space is resolved with sufficiently large number
of grid points. However, for a small number of grid points in velocity space such error
is not negligible, because the parameters of the Maxwellian do not coincide with the
discrete moments. Secondly, the non-linear reconstruction used to prevent oscillations
destroys the translation invariance which is at the basis of the conservation properties
of the scheme. As a consequence, the schemes show a wrong shock speed in the limit
of small Knudsen number. To treat the first problem and ensure machine precision
conservation of mass, momentum and energy with a relatively small number of velocity grid points, we replace the continuous Maxwellian with the discrete Maxwellian
introduced in [22]. The second problem is treated by implementing a conservative correction procedure based on the flux difference form as in [26]. In this way we can construct conservative semi-Lagrangian schemes which are Asymptotic Preserving (AP)
for the underlying Euler limit, as the Knudsen number vanishes. The effectiveness of
the proposed scheme is demonstrated by extensive numerical tests.</p>