A Novel Dynamic Quadrature Scheme for Solving Boltzmann Equation with Discrete Ordinate and Lattice Boltzmann Methods (Communications in Computational Physics, 2020)
The Boltzmann equation (BE) for gas flows is a time-dependent nonlineardifferential-integral equation in 6 dimensions. The current simplified practice is to linearize the collision integral in BE by the BGK model using Maxwellian equilibriumdistribution and to approximate the moment integrals by the discrete ordinate method(DOM) using a finite set of velocity quadrature points. Such simplification reducesthe dimensions from 6 to 3, and leads to a set of linearized discrete BEs. The maindifficulty of the currently used (conventional) numerical procedures occurs when themean velocity and the variation of temperature are large that requires an extremelylarge 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 bya velocity-coordinate transformation consisting of Galilean translation and thermalnormalization so that the transformed velocity space is independent of mean velocity and temperature. This enables the efficient implementation of Gaussian-Hermitequadrature. The velocity quadrature points in the new velocity space are fixed whilethe correspondent quadrature points in the physical space change from time to timeand from position to position. By this dynamic nature in the physical space, this newquadrature scheme is termed as the dynamic quadrature scheme (DQS). The DQS wasimplemented 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 DDOMand DLBM have been tested and validated with several testing problems. Of the sameaccuracy 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.