Curious Convergence Properties of Lattice Boltzmann Schemes for Diffusion with Acoustic Scaling (Communications in Computational Physics, 2020)
We consider the D1Q3 lattice Boltzmann scheme with an acoustic scale forthe simulation of diffusive processes. When the mesh is refined while holding the diffusivityconstant, we first obtain asymptotic convergence. When the mesh size tendsto zero, however, this convergence breaks down in a curious fashion, and we observequalitative discrepancies from analytical solutions of the heat equation. In this work,a new asymptotic analysis is derived to explain this phenomenon using the Taylor expansionmethod, and a partial differential equation of acoustic type is obtained in theasymptotic limit. We show that the error between the D1Q3 numerical solution and afinite-difference approximation of this acoustic-type partial differential equation tendsto zero in the asymptotic limit. In addition, a wave vector analysis of this asymptoticregime demonstrates that the dispersion equation has nontrivial complex eigenvalues,a sign of underlying propagation phenomena, and a portent of the unusual convergenceproperties mentioned above.