We theoretically explore boundary conditions for lattice Boltzmann methods, focusing on a toy two-velocities scheme to tackle a linear one-dimensional advection equation. 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，重点研究一个用于求解线性一维平流方程的简易双速度格式。通过将格子玻尔兹曼格式映射至有限差分格式，我们得以进行严格的一致性和稳定性分析。我们针对流入和流出边界发展了动力学边界条件，揭示了精度与稳定性之间的权衡关系，并成功克服了这一矛盾。一致性分析基于修正方程进行，而稳定性则通过GKS（Gustafsson、Kreiss和Sundström）理论进行评估——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。