Volume 12, Issue 2
An Implicit Scheme for Solving Unsteady Boltzmann Model Equation

Xiaowei Li, Chunxin Li, Dan Zhang & Zhihui Li

Numer. Math. Theor. Meth. Appl., 12 (2019), pp. 594-606.

Published online: 2018-12

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  • Abstract

When solving hyperbolic Boltzmann model equation with discrete velocity models (DVM), the strong discontinuity of the velocity distribution function can be captured well by utilizing the non-oscillatory and non-free parameter dissipation (NND) finite difference scheme. However, most NND scheme solvers march in time explicitly, which compromise the computation efficiency due to the limitation of stability condition, especially when solving unsteady problems. In order to improve the efficiency, an implicit scheme based on NND is presented in this paper. Linearization factors are introduced to construct the implicit scheme and to reduce the stencil size. With the help of dual time-stepping method, the convergence rate of unsteady rarefied flow simulation can be massively improved. Numerical tests of steady and unsteady supersonic flow around cylinders are computed in different flow regimes. Results are shown to prove the validity and efficiency of the  implicit scheme.

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COPYRIGHT: © Global Science Press

  • Email address

xwli@staff.shu.edu.cn (Xiaowei Li)

leecx007@gmail.com (Chunxin Li)

dan.zhang@shu.edu.cn (Dan Zhang)

zhli0097@x263.net (Zhihui Li)

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@Article{NMTMA-12-594, author = {Li , Xiaowei and Li , Chunxin and Zhang , Dan and Li , Zhihui}, title = {An Implicit Scheme for Solving Unsteady Boltzmann Model Equation}, journal = {Numerical Mathematics: Theory, Methods and Applications}, year = {2018}, volume = {12}, number = {2}, pages = {594--606}, abstract = {

When solving hyperbolic Boltzmann model equation with discrete velocity models (DVM), the strong discontinuity of the velocity distribution function can be captured well by utilizing the non-oscillatory and non-free parameter dissipation (NND) finite difference scheme. However, most NND scheme solvers march in time explicitly, which compromise the computation efficiency due to the limitation of stability condition, especially when solving unsteady problems. In order to improve the efficiency, an implicit scheme based on NND is presented in this paper. Linearization factors are introduced to construct the implicit scheme and to reduce the stencil size. With the help of dual time-stepping method, the convergence rate of unsteady rarefied flow simulation can be massively improved. Numerical tests of steady and unsteady supersonic flow around cylinders are computed in different flow regimes. Results are shown to prove the validity and efficiency of the  implicit scheme.

}, issn = {2079-7338}, doi = {https://doi.org/10.4208/nmtma.OA-2018-0016}, url = {http://global-sci.org/intro/article_detail/nmtma/12910.html} }
TY - JOUR T1 - An Implicit Scheme for Solving Unsteady Boltzmann Model Equation AU - Li , Xiaowei AU - Li , Chunxin AU - Zhang , Dan AU - Li , Zhihui JO - Numerical Mathematics: Theory, Methods and Applications VL - 2 SP - 594 EP - 606 PY - 2018 DA - 2018/12 SN - 12 DO - http://doi.org/10.4208/nmtma.OA-2018-0016 UR - https://global-sci.org/intro/article_detail/nmtma/12910.html KW - AB -

When solving hyperbolic Boltzmann model equation with discrete velocity models (DVM), the strong discontinuity of the velocity distribution function can be captured well by utilizing the non-oscillatory and non-free parameter dissipation (NND) finite difference scheme. However, most NND scheme solvers march in time explicitly, which compromise the computation efficiency due to the limitation of stability condition, especially when solving unsteady problems. In order to improve the efficiency, an implicit scheme based on NND is presented in this paper. Linearization factors are introduced to construct the implicit scheme and to reduce the stencil size. With the help of dual time-stepping method, the convergence rate of unsteady rarefied flow simulation can be massively improved. Numerical tests of steady and unsteady supersonic flow around cylinders are computed in different flow regimes. Results are shown to prove the validity and efficiency of the  implicit scheme.

Xiaowei Li, Chunxin Li, Dan Zhang & Zhihui Li. (2020). An Implicit Scheme for Solving Unsteady Boltzmann Model Equation. Numerical Mathematics: Theory, Methods and Applications. 12 (2). 594-606. doi:10.4208/nmtma.OA-2018-0016
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