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 and the derivation of the modified equations are still open issues. This has prevented, until today, to have an analogous of the Lax equivalence theorem for Lattice Boltzmann schemes. We propose a rigorous consistency study and the derivation of the modified equations for any lattice Boltzmann scheme under acoustic and diffusive scalings. 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 analyses for Finite Difference by exploiting its precise characterization using matrices of Finite Difference operators. Though we present the derivation of the modified equations until second-order underacoustic scaling, we provide all the elements to extend it to higher orders, since the kinetic-macroscopic connection is conducted at the fully discrete level. Finally, we show that our strategy yields, in a more rigorous setting, the same results as previous works in the literature.

翻译：格子玻尔兹曼格式是求解多种守恒律形式问题的有效数值方法，然而其理论根基长期存在不足。特别地，相容性分析与修正方程的推导仍是悬而未决的问题，这导致至今无法建立类似于Lax等价定理的格子玻尔兹曼格式理论框架。本文针对声学尺度与扩散尺度下的任意格子玻尔兹曼格式，提出严格的相容性研究，并给出修正方程的推导。我们通过在全离散层面从动能（格子玻尔兹曼）视角过渡到宏观（有限差分）视角，以消除偏离平衡态的非守恒矩。将格子玻尔兹曼格式改写为关于守恒变量的多步有限差分格式（这延续了我们先前工作的思路），进而利用有限差分算子矩阵的精确表征，对有限差分进行常规分析。尽管本文仅推导了声学尺度下二阶精度的修正方程，但提供了扩展至高阶的全部要素——因为动能-宏观的关联是在全离散层面建立的。最后，我们证明该策略在更严格的框架下能得到与现有文献相同的结果。