In this contribution we study the formal ability of a multi-resolution-times lattice Boltzmann scheme to approximate isothermal and thermal compressible Navier Stokes equations with a single particle distribution. More precisely, we consider a total of 12 classical square lattice Boltzmann schemes with prescribed sets of conserved and nonconserved moments. The question is to determine the algebraic expressions of the equilibrium functions for the nonconserved moments and the relaxation parameters associated to each scheme. We compare the fluid equations and the result of the Taylor expansion method at second order accuracy for bidimensional examples with a maximum of 17 velocities and three-dimensional schemes with at most 33 velocities. In some cases, it is not possible to fit exactly the physical model. For several examples, we adjust the Navier Stokes equations and propose nontrivial expressions for the equilibria.

翻译：本文研究了多分辨率时间格子玻尔兹曼方案采用单一粒子分布近似等温与热可压缩Navier Stokes方程的形式化能力。具体而言，我们考察了12种经典方形格子玻尔兹曼方案，这些方案均具有预设的守恒矩与非守恒矩集合。核心问题在于确定各方案中非守恒矩的平衡函数代数表达式及对应的弛豫参数。针对二维情形（最多17个速度）和三维情形（最多33个速度）的算例，我们对比了流体方程与二阶精度泰勒展开法的计算结果。在某些情况下，无法精确匹配物理模型。针对若干算例，我们调整了Navier Stokes方程并提出了非平凡的平衡态表达式。