@Article{CiCP-23-1263,
author = {Bruce M. Boghosian, François Dubois, Benjamin Graille, Pierre Lallemand and Mohamed-Mahdi Tekitek},
title = {Curious Convergence Properties of Lattice Boltzmann Schemes for Diffusion with Acoustic Scaling},
journal = {Communications in Computational Physics},
year = {2018},
volume = {23},
number = {4},
pages = {1263--1278},
abstract = {<p style="text-align: justify;">We consider the D1Q3 lattice Boltzmann scheme with an acoustic scale for
the simulation of diffusive processes. When the mesh is refined while holding the diffusivity
constant, we first obtain asymptotic convergence. When the mesh size tends
to zero, however, this convergence breaks down in a curious fashion, and we observe
qualitative discrepancies from analytical solutions of the heat equation. In this work,
a new asymptotic analysis is derived to explain this phenomenon using the Taylor expansion
method, and a partial differential equation of acoustic type is obtained in the
asymptotic limit. We show that the error between the D1Q3 numerical solution and a
finite-difference approximation of this acoustic-type partial differential equation tends
to zero in the asymptotic limit. In addition, a wave vector analysis of this asymptotic
regime demonstrates that the dispersion equation has nontrivial complex eigenvalues,
a sign of underlying propagation phenomena, and a portent of the unusual convergence
properties mentioned above.</p>},
issn = {1991-7120},
doi = {https://doi.org/10.4208/cicp.OA-2016-0257},
url = {http://global-sci.org/intro/article_detail/cicp/11215.html}
}