Commun. Comput. Phys., 11 (2012), pp. 1397-1414.

A Novel Dynamic Quadrature Scheme for Solving Boltzmann Equation with Discrete Ordinate and Lattice Boltzmann Methods

C. T. Hsu 1*, S. W. Chiang 1, K. F. Sin 1

1 Department of Mechanical Engineering, Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong.

Received 15 May 2010; Accepted (in revised version) 15 May 2011
Available online 30 November 2011


The Boltzmann equation (BE) for gas flows is a time-dependent nonlinear differential-integral equation in 6 dimensions. The current simplified practice is to linearize the collision integral in BE by the BGK model using Maxwellian equilibrium distribution and to approximate the moment integrals by the discrete ordinate method (DOM) using a finite set of velocity quadrature points. Such simplification reduces the dimensions from 6 to 3, and leads to a set of linearized discrete BEs. The main difficulty of the currently used (conventional) numerical procedures occurs when the mean velocity and the variation of temperature are large that requires an extremely large number of quadrature points. In this paper, a novel dynamic scheme that requires only a small number of quadrature points is proposed. This is achieved by a velocity-coordinate transformation consisting of Galilean translation and thermal normalization so that the transformed velocity space is independent of mean velocity and temperature. This enables the efficient implementation of Gaussian-Hermite quadrature. The velocity quadrature points in the new velocity space are fixed while the correspondent quadrature points in the physical space change from time to time and from position to position. By this dynamic nature in the physical space, this new quadrature scheme is termed as the dynamic quadrature scheme (DQS). The DQS was implemented to the DOM and the lattice Boltzmann method (LBM). These new methods with DQS are therefore termed as the dynamic discrete ordinate method (DDOM) and the dynamic lattice Boltzmann method (DLBM), respectively. The new DDOM and DLBM have been tested and validated with several testing problems. Of the same accuracy in numerical results, the proposed schemes are much faster than the conventional schemes. Furthermore, the new DLBM have effectively removed the incompressible and isothermal restrictions encountered by the conventional LBM.

AMS subject classifications: 35Q20, 76N15

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Key words: Dynamics quadrature, dynamic discrete ordinate method, dynamic lattice Boltzmann method, thermal lattice Boltzmann method.

*Corresponding author.
Email: (C. T. Hsu), (S. W. Chiang), (K. F. Sin)

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