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Research articles

ScienceAsia 35 (2009): 70-79 |doi: 10.2306/scienceasia1513-1874.2009.35.070

A deterministic spectral method for solving the Boltzmann equation for one-dimensional flows

Chatchawan Watchararuangwita,*, Yurii N. Grigorievb, Sergey V. Meleshkoa

ABSTRACT:     A new deterministic numerical method for solving the kinetic Boltzmann equation for Maxwellian molecules with cylindrical symmetry in velocity space is developed. Using the splitting method with respect to physical processes, the Boltzmann equation is decomposed into the space-homogeneous Boltzmann equation and the transport equation. The transport equation is solved by either Lax-Wendroff or upwind schemes. For Maxwell's model, the space-homogeneous Boltzmann equation is simplified by taking the Fourier transform with respect to velocity. Because of the cylindrical symmetry in velocity space, the three-dimensional Fourier transform is equivalent to a one-dimensional Fourier transform and a Hankel transform. An exponential grid in velocity space allows the application a fast Fourier transform algorithm to compute the Hankel transform. The space homogeneous Boltzmann equation in Fourier space is solved by the Runge-Kutta scheme. The new method is applied to solving the heat transfer problem between parallel plates.

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a School of Mathematics, Suranaree University of Technology, Nakorn Ratchasima 30000, Thailand
b Institute of Computational Technologies, Novosibirsk, Russia

* Corresponding author, E-mail: chat@math.sut.ac.th

Received 30 Apr 2008, Accepted 5 Jan 2009