Rigorous derivation of the macroscopic equations for the lattice Boltzmann method via the corresponding Finite Difference scheme

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 is still an open issue. We propose a rigorous derivation of the macroscopic equations for any lattice Boltzmann scheme under acoustic scaling. 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 consistency analysis for Finite Difference by exploiting its precise characterization using matrices of Finite Difference operators. Though we present the derivation until second-order under acoustic scaling, we provide all the elements to extend it to higher orders and to other scalings, since the kinetic-macroscopic connection is conducted at the fully discrete level. Finally, we show that our strategy yields, in a mathematically rigorous setting, the same results as previous works in the literature.