Discrete-velocity-direction models of BGK-type with minimum entropy. II: Weighted models (Q6536828)

The Bathnagar-Gross-Krook (BGK) model is an approximation of the Boltzmann equation, describing the time evolution of a single monoatomic rarefied gas and satisfying the properties: conservation and entropy inequality. The BGK equation for the density function \(f=f(t,x,\xi ,\zeta )\) has the form \(\partial_tf+\xi\cdot \nabla_xf=\tau^{-1} (\mathcal{E}[f]-f)\), where \((x,\xi )\in \mathbb{R}^D\times \mathbb{R}^D\) with \(D=2\) or 3, \(\zeta\in \mathbb{R}^L\) represents the possible internal molecular degrees of freedom, and \(\tau \) is a characteristic collision time. Here, \(\mathcal{E}[f]\) is the local equilibrium state. In the present paper the authors consider a discrete-velocity-direction model (DVDM) with collisions of BGK-type for simulating gas flows, where the molecular motion is confined to some prescribed directions but the speed is still a continuous variable in each direction. The BGK-DVDM is improved in two aspects. First, the internal molecular degrees of freedom is included so that more realistic fluid properties can be realized. The authors introduce a weighted function in each orientation when recovering the macroscopic parameters, as opposed to the previous treatment. For the new weighted DVDM, the established properties of the well-behaved discrete equilibrium still hold. The DVDM is considered under requirement that the molecule transport is limited to \(N\) prescribed directions \(\{\boldsymbol{l}_m\}_{m=1}^{N}\) with each \(\boldsymbol{l}_m\) located on the unit sphere \(\mathbb{S}^{D-1}\), but the velocity magnitude \(\xi\in \mathbb{R}\) in each direction remains continuous. There is a pair of requirements: 1) \((\boldsymbol{l}_1,\dots, \boldsymbol{l}_N)\in \mathbb{R}^{D\times N}\) is of rank \(D\) and therefore \(N\geq D\); 2) Each direction \(\boldsymbol{l}_m\) and its opposite \(-\boldsymbol{l}_m\) belong to \(S_m\subset\mathbb{S}^{D-1}\), where the quantities \(S_m\) constitute a disjoint partition of the unit sphere \(\mathbb{S}^{D-1}= \bigcup_{m=1}^{N}S_m\). Each \(S_m\) has the same measure. Note that the distribution \(f\) is replaced by \(N\) distributions \(\{f_m(t,x,\xi ,\zeta )\}_{m=1}^{N}\) with \(\xi\in\mathbb{R}\) and \(\zeta\in\mathbb{R}^L\). The transport velocity for \(f_m\) is \(\xi\boldsymbol{l}_m\), and the governing equation for \(f_m\) becomes \(\partial_tf_m+\xi\boldsymbol{l}_m\cdot \nabla_xf_m = \tau^{-1} (\mathcal{E}_m-f_m)\) \((m=1,\dots,N)\) with the local equilibriums \(\mathcal{E}_m\). The authors introduce a weighted function in each orientation when recovering the macroscopic parameters. With the weighted DVDM, the authors consider three submodels by incorporating the discrete velocity method, the Gaussian-extended quadrature method of moments and the Hermite spectral method in each direction. It seems the stated spatial-time submodels are multidimensional versions corresponding to the three approaches. Some numerical tests with a series of 1-D and 2-D flow problems show the efficiency of the weighted DVDM.\N\NFor Part I see [ibid. 95, No. 3, Paper No. 80, 29 p. (2023; Zbl 1515.65235)].