Stability limits of the single relaxation-time advection–diffusion lattice Boltzmann scheme

In many cases, multi-species and/or thermal flows involve large discrepancies between the different diffusion coefficients involved — momentum, heat and species diffusion. In the context of classical passive scalar lattice Boltzmann (LB) simulations, the scheme is quite sensitive to such discrepancies, as relaxation coefficients of the flow and passive scalar fields are tied together through their common lattice spacing and time-step size. This in turn leads to at least one relaxation coefficient, τ being either very close to 0.5 or much larger than unity which, in the case of the former (small relaxation coefficient), has been shown to cause instability. The present work first establishes the stability boundaries of the passive scalar LB method in the sense of von Neumann and as a result shows that the scheme is unconditionally stable, even for τ=0.5, provided that the nondimensional velocity does not exceed a certain threshold. Effects of different parameters such as the distribution function and lattice speed of sound on the stability area are also investigated. It is found that the simulations diverge for small relaxation coefficients regardless of the nondimensional velocity. Numerical applications and a study of the dispersion–dissipation relations show that this behavior is due to numerical noise appearing at high wave numbers and caused by the inconsistent behavior of the dispersion relation along with low dissipation. This numerical noise, known as Gibbs oscillations, can be controlled using spatial filters. Considering that noise is limited to high wave numbers, local filters can be used to control it. In order to stabilize the scheme with minimal impact on the solution even for cases involving high wave number components, a local Total Variation Diminishing (TVD) filter is implemented as an additional step in the classical LB algorithm. Finally, numerical applications show that this filter eliminates the unwanted oscillations while closely reproducing the reference solution.