Self-consistent Force Scheme in the Discrete Boltzmann Equation

In the work of N. Martys et al. [Nicos S. Martys, Xiaowen Shan, Hudong Chen, Phys. Rev. E, Vol. 58, Num.5, 1998 ], a self-consistent force term to any order in the Boltzmann-BKG equation is derived by the Hermite basis with raw velocity. As an extension, in the present work, the force term is expanded by the Hermite basis with the relative velocity in the comoving coordinate and the Hermite basis with the relative velocity scaled by the local temperature. It is found that the force scheme proposed by He et al. [Xiaoyi He, Xiaowen Shan, Gary D. Doolen, Phys. Rev. E, Vol. 57, Num.1,1998] can be derived by the Hermite basis with the relative velocity. Furthermore, another new force scheme in which the velocity is scaled by the local temperature is obtained.