Lattice Boltzmann schemes are efficient numerical methods to solve a broad range of problems under the form of conservation laws. However, they suffer from a chronic lack of clear theoretical foundations. In particular, the consistency analysis is still an open issue. We propose a rigorous derivation of the macroscopic equations for any lattice Boltzmann scheme under acoustic scaling. This is done by passing from a kinetic (lattice Boltzmann) to a macroscopic (Finite Difference) point of view at a fully discrete level in order to eliminate the non-conserved moments relaxing away from the equilibrium. We rewrite the lattice Boltzmann scheme as a multi-step Finite Difference scheme on the conserved variables, as introduced in our previous contribution. We then perform the usual consistency analysis for Finite Difference by exploiting its precise characterization using matrices of Finite Difference operators. Though we present the derivation until second-order under acoustic scaling, we provide all the elements to extend it to higher orders and to other scalings, since the kinetic-macroscopic connection is conducted at the fully discrete level. Finally, we show that our strategy yields, in a mathematically rigorous setting, the same results as previous works in the literature.

翻译：Lattice Boltzmann 计划是解决保护法形式下一系列广泛问题的有效数字方法,但长期缺乏明确的理论基础。 特别是, 一致性分析仍是一个未决问题。 我们提议在声学规模下为任何 Lattice Boltzmann 计划严格推算宏观方程式的公式。 这是通过从动性( lattice Boltzmann ) 转向宏观分层( 绝对差异) 的观点实现的, 以便在完全离散的层次上将其扩展到完全离散的层次, 以便消除无法保持的从平衡中解脱出来的时刻。 我们重写 lattice Boltzmann 计划, 把它作为我们以前投入的关于受保护变量的多步骤的“ 有限差异” 计划。 我们随后通过利用“ 最小差异操作者矩阵” 的精确特征进行常规的一致分析。 尽管我们在声学规模下的第二顺序下展示了推算结果, 我们提供所有要素将其扩展至更高的顺序和其他尺度, 因为动性- 矩阵连接是在完全离散的数学水平上进行, 最后, 我们展示了我们之前的数学水平上的一项成果。