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.
