General fourth-order Chapman–Enskog expansion of lattice Boltzmann schemes

In order to derive the equivalent partial differential equations of a lattice Boltzmann scheme, the Chapman Enskog expansion is very popular in the lattice Boltzmann community. A main drawback of this approach is the fact that multiscale expansions are used without any clear mathematical signification of the various variables and operators. Independently of this framework, the Taylor expansion method allows to obtain formally the equivalent partial differential equations. The general equivalency of these two approaches remains an open question. In this contribution, we prove that both approaches give identical results with acoustic scaling for a very general family of lattice Boltzmann schemes and up to fourth-order accuracy. Examples with a single scalar conservation illustrate our purpose.