We establish the notion of limit consistency as a modular part in proving the consistency of lattice Boltzmann equations (LBEs) with respect to a given partial differential equation (PDE) system. The incompressible Navier--Stokes equations (NSE) are used as paragon. Based upon the hydrodynamic limit of the Bhatnagar--Gross--Krook (BGK) Boltzmann equation towards the NSE, we provide a successive discretization by nesting conventional Taylor expansions and finite differences. We track the discretization state of the domain for the particle distribution functions and measure truncation errors at all levels within the derivation procedure. Via parametrizing equations and proving the limit consistency of the respective families of equations, we retain the path towards the targeted PDE at each step of discretization, i.e. for the discrete velocity BGK Boltzmann equations and the space-time discretized LBEs. As a direct result, we unfold the discretization technique of lattice Boltzmann methods as chaining finite differences and provide a generic top-down derivation of the numerical scheme which upholds the continuous limit.

翻译：我们建立了极限一致性概念，将其作为证明格子玻尔兹曼方程（LBE）相对于给定偏微分方程（PDE）系统一致性的模块化组成部分。以不可压缩纳维-斯托克斯方程（NSE）为范例。基于Bhatnagar-Gross-Krook（BGK）玻尔兹曼方程向NSE的流体力学极限，我们通过嵌套常规泰勒展开和有限差分法进行逐步离散化。我们追踪粒子分布函数所在域的离散化状态，并在推导过程中测量各层级的截断误差。通过参数化方程并证明各自方程族的极限一致性，我们在离散化的每一步（即离散速度BGK玻尔兹曼方程和时空离散化的LBE）都保持了通向目标PDE的路径。直接结果是，我们揭示了将格子玻尔兹曼方法的离散化技术视为有限差分的链式组合，并提供了保持连续极限的数值方案的通用自上而下推导。