Volume 29, Issue 1
High Order Conservative Semi-Lagrangian Scheme for the BGK Model of the Boltzmann Equation

Sebastiano Boscarino, Seung-Yeon Cho, Giovanni RussoSeok-Bae Yun

Commun. Comput. Phys., 29 (2021), pp. 1-56.

Published online: 2020-11

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  • Abstract

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.

  • Keywords

BGK model, Boltzmann equation, semi-Lagrangian scheme, conservative correction, discrete Maxwellian.

  • AMS Subject Headings

65L06, 65M25, 76P05

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{CiCP-29-1, author = {Boscarino , Sebastiano and Cho , Seung-Yeon and Russo , Giovanni and Yun , Seok-Bae}, title = {High Order Conservative Semi-Lagrangian Scheme for the BGK Model of the Boltzmann Equation}, journal = {Communications in Computational Physics}, year = {2020}, volume = {29}, number = {1}, pages = {1--56}, abstract = {

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.

}, issn = {1991-7120}, doi = {https://doi.org/10.4208/cicp.OA-2020-0050}, url = {http://global-sci.org/intro/article_detail/cicp/18421.html} }
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 -

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.

Sebastiano Boscarino, Seung-Yeon Cho, Giovanni Russo & Seok-Bae Yun. (2020). High Order Conservative Semi-Lagrangian Scheme for the BGK Model of the Boltzmann Equation. Communications in Computational Physics. 29 (1). 1-56. doi:10.4208/cicp.OA-2020-0050
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