  Lattice Boltzmann schemes are efficient numerical methods to solve a broad
range of problems under the form of conservation laws. However, they suffer
from a chronic lack of clear theoretical foundations. In particular, the
consistency analysis and the derivation of the modified equations are still
open issues. This has prevented, until today, to have an analogous of the Lax
equivalence theorem for Lattice Boltzmann schemes. We propose a rigorous
consistency study and the derivation of the modified equations for any lattice
Boltzmann scheme under acoustic and diffusive scalings. This is done by passing
from a kinetic (lattice Boltzmann) to a macroscopic (Finite Difference) point
of view at a fully discrete level in order to eliminate the non-conserved
moments relaxing away from the equilibrium. We rewrite the lattice Boltzmann
scheme as a multi-step Finite Difference scheme on the conserved variables, as
introduced in our previous contribution. We then perform the usual analyses for
Finite Difference by exploiting its precise characterization using matrices of
Finite Difference operators. Though we present the derivation of the modified
equations until second-order underacoustic scaling, we provide all the elements
to extend it to higher orders, since the kinetic-macroscopic connection is
conducted at the fully discrete level. Finally, we show that our strategy
yields, in a more rigorous setting, the same results as previous works in the
literature.
