Commun. Comput. Phys., 15 (2014), pp. 1368-1406.


Dimension-Reduced Hyperbolic Moment Method for the Boltzmann Equation with BGK-Type Collision

Zhenning Cai 1, Yuwei Fan 1, Ruo Li 2, Zhonghua Qiao 3*

1 School of Mathematical Sciences, Peking University, Beijing 100871, China.
2 CAPT, LMAM & School of Mathematical Sciences, Peking University, Beijing 100871, China.
3 Department of Applied Mathematics, The Hong Kong Polytechnic University, Hung Hom, Hong Kong.

Received 22 March 2013; Accepted (in revised version) 28 October 2013
Available online 5 March 2014
doi:10.4208/cicp.220313.281013a

Abstract

We develop the dimension-reduced hyperbolic moment method for the Boltzmann equation, to improve solution efficiency using a numerical regularized moment method for problems with low-dimensional macroscopic variables and high-dimensional microscopic variables. In the present work, we deduce the globally hyperbolic moment equations for the dimension-reduced Boltzmann equation based on the Hermite expansion and a globally hyperbolic regularization. The numbers of Maxwell boundary condition required for well-posedness are studied. The numerical scheme is then developed and an improved projection algorithm between two different Hermite expansion spaces is developed. By solving several benchmark problems, we validate the method developed and demonstrate the significant efficiency improvement by dimension-reduction.

AMS subject classifications: 35L25, 65M08, 76P05, 78M05

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Key words: Dimension-reduction, moment system, NRxx, global hyperbolicity, boundary condition, microflow.

*Corresponding author.
Email: caizn@pku.edu.cn (Z. Cai), ywfan@pku.edu.cn (Y. Fan), rli@math.pku.edu.cn (R. Li), zqiao@polyu.edu.hk (Z. Qiao)
 

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