The Bhatnagar–Gross–Krook (BGK) single-relaxation-time collision model for the Boltzmann equation serves as the foundation of the lattice BGK (LBGK) method developed in recent years. The description of the collision as a uniform relaxation process of the distribution function towards its equilibrium is, in many scenarios, simplistic. Based on a previous series of papers, we present a collision model formulated as independent relaxations of the irreducible components of the Hermite coefficients in the reference frame moving with the fluid. These components, corresponding to the irreducible representation of the rotation group, are the minimum tensor components that can be separately relaxed without violating rotation symmetry. For the 2nd, 3rd and 4th moments, respectively, two, two and three independent relaxation rates can exist, giving rise to the shear and bulk viscosity, thermal diffusivity and some high-order relaxation process not explicitly manifested in the Navier–Stokes-Fourier equations. Using the binomial transform, the Hermite coefficients are evaluated in the absolute frame to avoid the numerical dissipation introduced by interpolation. Extensive numerical verification is also provided.

This article is part of the theme issue ‘Progress in mesoscale methods for fluid dynamics simulation’.

玻尔兹曼方程的 Bhatnagar-Gross-Krook (BGK) 单松弛时间碰撞模型是近年来开发的晶格 BGK (LBGK) 方法的基础。在许多情况下，将碰撞描述为分布函数向其平衡的均匀松弛过程是简单化的。基于之前的一系列论文，我们提出了一个碰撞模型，该模型表示为随流体移动的参考系中 Hermite 系数的不可约分量的独立松弛。这些分量对应于旋转群的不可约表示，是可以在不违反旋转对称性的情况下单独松弛的最小张量分量。对于二阶、三阶和四阶矩，分别可以存在两个、二个和三个独立的松弛率，引起剪切和体积粘度、热扩散率和一些在 Navier-Stokes-Fourier 方程中未明确显示的高阶弛豫过程。使用二项式变换，在绝对坐标系中计算 Hermite 系数以避免由插值引入的数值耗散。还提供了广泛的数值验证。

本文是主题问题“流体动力学模拟中尺度方法的进展”的一部分。