Consistency study of Lattice-Boltzmann schemes macroscopic limit

Owing to the lack of consensus about the way Chapman-Enskog should be performed, a new Taylor-expansion of lattice-Boltzmann models is proposed. In contrast to the Chapman-Enskog expansion, recalled in this manuscript, the method only assumes a sufficiently small time step. Based on the Taylor expansion, the collision kernel is reinterpreted as a closure for the stress-tensor equation. Numerical coupling of lattice-Boltzmann models with other numerical schemes, also encompassed by the method, is shown to create error terms whose scalings are more complex than those obtained via Chapman-Enskog. An athermal model and two compressible models are carefully analyzed through this new scope, casting a new light on each model's consistency with the Navier-Stokes equations.