Velocity Quadratures
The velocity quadrature used in a simulation is a critical factor to determine the accuracy of the results obtained. With the form of the equilibrium function defined above, it is necessary to use a Gauss–Hermite quadrature for the velocity lattice, with the appropriate basis function. This guarantees that summing over the velocity basis, to compute quantities such as the density, results in the most accurate value for the quantity of interest when compared to an integral over the continuum velocity space (which is what the lattice Boltzmann method approximates).
COMSOL Multiphysics uses velocity quadratures based on the formalism described in Ref. 1. Consider first the generation of a one-dimensional quadrature. The pth order Hermite polynomial is generated from the function w(ξ)=exp(−ξ2/2)/√(2π) by the equation
To generate a one-dimensional quadrature of order p, the roots of the equation H(p)(ξ)=0 need to be evaluated. These roots represent the velocity coordinates of the quadrature. The pth order polynomial, H(p)(ξ), has p roots: ξq, where q = 1, …, p. To compute the weights associated with the quadrature the following formula is used:
Two-dimensional and three-dimensional quadratures can be generated from a one-dimensional quadrature through the product rule. The product rule constructs a two-dimensional quadrature by placing the one-dimensional quadrature along both the x- and y-axes and generating a grid that passes through all the points on the axes. The points at the intersection of the grid lines represent the velocity coordinates of the two-dimensional quadrature. The weights corresponding to each velocity coordinate are also multiplied together. Three-dimensional quadratures are generated in an analogous manner.
The following quadratures have been generated as described above using the product rule: D2Q9, D2Q16, D2Q25, D2Q36, D2Q64, D2Q100, D3Q27, D3Q64, D3Q216, and D3Q512. COMSOL Multiphysics also includes some lower-order quadratures that are not generated directly from the product rule. These are listed in Table 3-1.
A correct transitional flow solution should be independent of the quadrature when the number of velocity directions is increased. It is therefore necessary to demonstrate that solutions are both quadrature and mesh independent as part of the validation procedure.
 
Table 3-1: Two-Dimensional velocity lattices available in the transitional flow interface.
Quadrature
w
Notes
D2Q12
D2Q12
(continued)
D2Q12
(continued)
Note that ξa is the velocity associated with the given lattice point (cs is the characteristic speed) whilst w is the corresponding weight.