TY - JOUR T1 - Curious Convergence Properties of Lattice Boltzmann Schemes for Diffusion with Acoustic Scaling AU - Bruce M. Boghosian, François Dubois, Benjamin Graille, Pierre Lallemand & Mohamed-Mahdi Tekitek JO - Communications in Computational Physics VL - 4 SP - 1263 EP - 1278 PY - 2018 DA - 2018/04 SN - 23 DO - http://doi.org/10.4208/cicp.OA-2016-0257 UR - https://global-sci.org/intro/article_detail/cicp/11215.html KW - Artificial compressibility method, Taylor expansion method. AB -

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.