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.

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