Commun. Comput. Phys., 14 (2013), pp. 242-264.

A Direct Solver for Initial Value Problems of Rarefied Gas Flows of Arbitrary Statistics

Jaw-Yen Yang 1*, Bagus Putra Muljadi 2, Zhi-Hui Li 3, Han-Xin Zhang 3

1 Institute of Applied Mechanics, National Taiwan University,Taipei 10764, Taiwan.
2 Institute of Applied Mechanics, National Taiwan University,Taipei 10764, Taiwan; and FCS STAE, Universite Paul Sabatier, Institut de mathematiques de Toulouse, Toulouse 31400, France.
3 National Laboratory for Computational Fluid Dynamics, Beijing 100191, China; and China Aerodynamics Research and Development Center, Mianyang, 621000, China.

Received 29 January 2012; Accepted (in revised version) 3 August 2012
Available online 15 November 2012


An accurate and direct algorithm for solving the semiclassical Boltzmann equation with relaxation time approximation in phase space is presented for parallel treatment of rarefied gas flows of particles of three statistics. The discrete ordinate method is first applied to discretize the velocity space of the distribution function to render a set of scalar conservation laws with source term. The high order weighted essentially non-oscillatory scheme is then implemented to capture the time evolution of the discretized velocity distribution function in physical space and time. The method is developed for two space dimensions and implemented on gas particles that obey the Maxwell-Boltzmann, Bose-Einstein and Fermi-Dirac statistics. Computational examples in one- and two-dimensional initial value problems of rarefied gas flows are presented and the results indicating good resolution of the main flow features can be achieved. Flows of wide range of relaxation times and Knudsen numbers covering different flow regimes are computed to validate the robustness of the method. The recovery of quantum statistics to the classical limit is also tested for small fugacity values.

AMS subject classifications: 82-08, 82B30, 82B40, 35Q20

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Key words: Semiclassical Boltzmann-BGK equation, weighted essentially non-oscillatory, discrete ordinate method.

*Corresponding author.
Email: (J.-Y. Yang), (B. P. Muljadi), (Z.-H. Li), (H.-X. Zhang)

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