  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.
