We establish the notion of limit consistency as a modular part in proving the consistency of lattice Boltzmann equations (LBE) with respect to a given partial differential equation (PDE) system. The incompressible Navier-Stokes equations (NSE) are used as paragon. Based upon the diffusion limit [L. Saint-Raymond (2003), doi: 10.1016/S0012-9593(03)00010-7] of the Bhatnagar-Gross-Krook (BGK) Boltzmann equation towards the NSE, we provide a successive discretization by nesting conventional Taylor expansions and finite differences. Elaborating the work in [M. J. Krause (2010), doi: 10.5445/IR/1000019768], 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 sequences, we retain the path towards the targeted PDE at each step of discretization, i.e. for the discrete velocity BGK Boltzmann equation and the space-time discretized LBE. 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：我们将限制一致性的概念作为模块的一部分,以证明在特定部分差异方程系统(PDE)中拉蒂斯·博尔茨曼方程(LBE)的一致性。不可压缩的纳维-斯托克斯方程(NSE)用作参数。根据离散限[L. Saint-Raymond(2003), doi:10.1016/S0012-9593(0300010-7),根据Bhatnagar-Gross-Krook(BGK) Boltzmann方程式(LBKK),Boltzmann方程式,我们通过嵌套常规泰勒扩张和有限差异,提供连续离散的离散分解和分解。在[M. J. Krause(2010), doi: 10.5445/IR/10000-1968]中,我们根据离散式分配功能域的离散状态和在衍生程序的所有级别测量电解误差。Via 配方方程和证明各自序列的极限一致性,我们保留在离离散式通用的PDE的每个步骤上一条路径,即为离式通用的离式通用平方平方平方平方平方平方平方平方平方平方平方平方平。