Duality in matrix lattice Boltzmann models

The notion of duality between the hydrodynamic and kinetic (ghost) variables of lattice kinetic formulations of the Boltzmann equation is introduced. It is suggested that this notion can serve as a guideline in the design of matrix versions of the lattice Boltzmann equation in a physically transparent and computationally efficient way.

Figure 1

(Color online) Dispersion relation of the D2Q9 matrix LBE for $\tau =\sigma =1$. This is identical to the BGK model. Note the large overlap of the hydrodynamic (lower curves) and nonhydrodynamic (upper curves) modes.

Figure 2

(Color online) Dispersion relation of the D2Q9 matrix LBE for $\tau =1$, $\sigma =2$. The ghost modes are forcibly made to relax more slowly than the stress modes, leading to poor hydrodynamic behavior.

Figure 3

(Color online) Dispersion relation of the D2Q9 matrix LBE for $\tau =1$, $\sigma =1∕2$. The ghost modes relax twice as fast as the stress modes. there is a clean seperation of time scales and enhanced hydrodynamic behavior compared to the BGK model.