Nonuniqueness of lattice Boltzmann schemes derived from finite difference methods
Eliane Kummer, Stephan Simonis
TL;DR
The paper addresses whether a multi-step finite difference scheme for conserved variables uniquely determines a lattice Boltzmann scheme. It uses explicit counterexamples to show nonuniqueness of the LBS formulation given an FDS, including a $D_{1}Q_{3}$ (one dimension, three velocities) case and discussions of smaller stencils, demonstrating that different LBSs can map to the same FDS. It introduces equivalence classes of relaxation systems defined by similarity transformations and equal characteristic polynomials to explain how distinct LBSs can yield identical FDSs. The results reveal intrinsic nonuniqueness in deriving LBSs from FDSs and motivate future work to classify these equivalence classes and study implications for the mesoscopic-to-mmacroscopic transition in kinetic theory.
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.