Hydrodynamic behavior of lattice Boltzmann and lattice Bhatnagar-Gross-Krook models

Incompressible limits of lattice Boltzmann equations using multiple relaxation times

“…The behavior of a fully discrete lattice Boltzmann equation such as (25) is more commonly analysed as an eigenvalue problem [12,24,25,26,2,10,31], instead of as an initial value problem for specific initial conditions as above. Assuming an exponential dependence in time for the h i with growth rate σ, (35) becomes a 9×9 matrix eigenvalue problem with eigenvalue λ = e σ∆t ,…”

Duality in matrix lattice Boltzmann models

“…This motivates the introduction of projection operators [17] which project the distribution function onto the hydrodynamic, transport, and ghost subspaces,…”

“…For a general athermal DdQn LB model with n velocities in d space dimensions, the n × n collision matrix L ij has d + 1 null eigenvectors corresponding to the density and d components of the conserved momentum, d(d + 1)/2 eigenvectors corresponding to the stress modes, and n − (d + 1) − d(d + 1)/2 eigenvectors corresponding to the ghost modes [15,17]. The choice of the null and stress eigenvectors {1, c iα , Q iαβ } follows directly from the physical definition of the densities associated with them.…”

“…We then introduce and illustrate the idea of duality with concrete examples. We show how a model in which all the ghost degrees of freedom are relaxed at the same rate [17,18] may provide the best compromise between full MTR models where every mode has a separate relaxation time and the LBGK model. We end with a discussion on how the present work is relevant to the algorithmic improvement of the LB method.…”

“…In the next section we follow the notation in [17] to highlight the structure of the kinetic space spanned by the eigenvectors and how a change of basis from populations to the moments reveals the dynamics of the various modes. We then introduce and illustrate the idea of duality with concrete examples.…”