General fourth order Chapman-Enskog expansion of lattice Boltzmann schemes

TL;DR

This paper proves that the Chapman-Enskog and Taylor expansion methods yield identical fourth-order accurate partial differential equations for a broad class of lattice Boltzmann schemes under acoustic scaling.

Contribution

It establishes the mathematical equivalence of Chapman-Enskog and Taylor expansions up to fourth order for lattice Boltzmann schemes, clarifying their relationship.

Findings

Both methods produce identical results at fourth order.

The equivalence holds for a very general family of schemes.

Examples with scalar conservation illustrate the results.

Abstract

In order to derive the equivalent partial differential equations of a lattice Boltzmann scheme,the Chapman Enskog expansion is very popular in the lattive Boltzmann community. A maindrawback of this approach is the fact that multiscale expansions are used without any clearmathematical signification of the various variables and operators. Independently of thisframework, the Taylor expansion method allows to obtain formally the equivalent partialdifferential equations. In this contribution, we prove that both approaches give identicalresults with acoustic scaling for a very general family of lattice Boltzmann schemes and upto fourth order accuracy. Examples with a single scalar conservation illustrate our purpose.