Nonuniqueness of lattice Boltzmann schemes derived from finite difference methods

TL;DR

This paper demonstrates that multiple lattice Boltzmann schemes can correspond to the same finite difference scheme, challenging the idea of a unique lattice Boltzmann formulation for a given finite difference scheme.

Contribution

It provides counterexamples showing nonuniqueness and introduces the concept of equivalence classes for discretized relaxation systems.

Findings

Counterexamples of nonuniqueness in lattice Boltzmann schemes

Existence of equivalence classes for discretized relaxation systems

Challenges the conjecture of unique lattice Boltzmann formulations

Abstract

Recently, the construction of finite difference schemes from lattice Boltzmann schemes has been rigorously analyzed [Bellotti et al. (2022), Numer. Math. 152, pp. 1-40]. It is thus known that any lattice Boltzmann scheme can be expressed in terms of a corresponding multi-step finite difference scheme on the conserved variables. In the present work, we provide counterexamples for the conjecture that any multi-step finite difference scheme has a unique lattice Boltzmann formulation. Based on that, we indicate the existence of equivalence classes for discretized relaxation systems.