We theoretically explore boundary conditions for lattice Boltzmann methods, focusing on a toy two-velocities scheme. By mapping lattice Boltzmann schemes to Finite Difference schemes, we facilitate rigorous consistency and stability analyses. We develop kinetic boundary conditions for inflows and outflows, highlighting the trade-off between accuracy and stability, which we successfully overcome. Consistency analysis relies on modified equations, whereas stability is assessed using GKS (Gustafsson, Kreiss, and Sundstr{\"o}m) theory and -- when this approach fails on coarse meshes -- spectral and pseudo-spectral analyses of the scheme's matrix that explain effects germane to low resolutions.

翻译：本文从理论上探讨格子玻尔兹曼方法的边界条件，聚焦于一个简化的双速度格式。通过将格子玻尔兹曼格式映射至有限差分格式，我们为严格的一致性及稳定性分析提供了便利。我们针对流入与流出情形发展了动力学边界条件，重点阐述了精度与稳定性之间的权衡关系，并成功克服了该问题。一致性分析基于修正方程进行，而稳定性评估则采用GKS（Gustafsson、Kreiss与Sundström）理论——当该方法在粗网格上失效时，我们通过对格式矩阵进行谱分析与伪谱分析，揭示了低分辨率所特有的效应。