ical interpretation of the scheme as consisting of a particle streaming step followed by a collision results in a very
The lattice Boltzmann equation describes the evolution of the velocity distribution function on a lattice in a manner that macro-simple parallel logic that is well suited for implementation scopic fluid dynamical behavior is recovered. Although the equation on massively parallel computers. The main advantage of is a derivative of lattice gas automata, it may be interpreted as a the LB method is that the particle interpretation allows Lagrangian finite-difference method for the numerical simulation the use of very simple boundary conditions so that the of the discrete-velocity Boltzmann equation that makes use of a parallel implementation may be used even for complex BGK collision operator. As a result, it is not surprising that numerical instability of lattice Boltzmann methods have been frequently en-geometries. For this reason, one of the most successful countered by researchers. We present an analysis of the stability applications of the LB method has been to simulations of of perturbations of the particle populations linearized about equilibflow through porous media [1,2].
rium values corresponding to a constant-density uniform mean
The development of LG models was based on the obserflow. The linear stability depends on the following parameters: the vation that macroscopic behavior of fluid flow is not very distribution of the mass at a site between the different discrete speeds, the BGK relaxation time, the mean velocity, and the wave-sensitive to the underlying microscopic physics. Thus, modnumber of the perturbations. This parameter space is too large els were developed based on the simplest possible particle to compute the complete stability characteristics. We report some microworld that would lead to the incompressible Navierstability results for a subset of the parameter space for a 7-velocity Stokes equation in the limit of small Knudsen number [3].
hexagonal lattice, a 9-velocity square lattice, and a 15-velocity cubic
The methods successfully modeled incompressible fluid lattice. Results common to all three lattices are (1) the BGK relaxation time must be greater than corresponding to positive shear viscos-flow but noise associated with the particle microworld neity, (2) there exists a maximum stable mean velocity for fixed values cessitated the introduction of some type of averaging proof the other parameters, and (3) as is increased from the maximum cedure such as spatial, temporal, or ensemble averaging stable velocity increases monotonically until some fixed velocity is to characterize the macroscopic flow. A second difficulty reached which does not change for larger .