Comparison of analysis techniques for the lattice Boltzmann method

We show that the Chapman–Enskog expansion can be viewed as a special instance of a general expansion procedure which also encompasses other methods like the regular error expansion and multi-scale techniques and that any two expansions which properly describe the lattice Boltzmann solution necessarily coincide up to higher order terms. For a model problem, both the regular error expansion and the Chapman–Enskog expansion are carried out. It turns out that the classical Chapman–Enskog method leads to an unstable equation at super-Burnett order in a parameter regime for which the underlying lattice Boltzmann algorithm is stable. However, our approach naturally allows us to consider variants of the super-Burnett equation which do not suffer from instabilities. The article concludes with a detailed comparison of the Chapman–Enskog and the regular error expansion.