We study the stability of one-dimensional linear lattice Boltzmann schemes for scalar hyperbolic equations with respect to boundary data. Our approach is based on the original raw algorithm on several unknowns, thereby avoiding the need for a transformation into an equivalent scalar formulation-a challenging process in presence of boundaries. To address different behaviors exhibited by the numerical scheme, we introduce appropriate notions of strong stability. They account for the potential absence of a continuous extension of the stable vector bundle associated with the bulk scheme on the unit circle for certain components. Rather than developing a general theory, complicated by the fact that discrete boundaries in lattice Boltzmann schemes are inherently characteristic, we focus on strong stability-instability for methods whose characteristic equations have stencils of breadth one to the left. In this context, we study three representative schemes. These are endowed with various boundary conditions drawn from the literature, and our theoretical results are supported by numerical simulations.

翻译：我们研究了一维线性格子Boltzmann格式在标量双曲方程中对于边界数据的稳定性。我们的方法基于原始的多未知量算法，从而避免了将其转化为等效标量表述的需求——这一过程在边界存在时尤为困难。为应对数值格式表现出的不同行为，我们引入了适当的强稳定性概念。这些概念考虑了在单位圆上，与主体格式相关的稳定向量丛对于某些分量可能不存在连续延拓的情况。鉴于格子Boltzmann格式中的离散边界本质上是特征性的，这使得一般理论的构建变得复杂，因此我们并未发展一套普适理论，而是聚焦于特征方程具有向左宽度为一的模板方法的强稳定性与不稳定性。在此背景下，我们研究了三种代表性格式。这些格式配备了从文献中选取的多种边界条件，并且我们的理论结果得到了数值模拟的支持。