General solution of lattices for Cartesian lattice Bhatanagar-Gross-Krook models

We give the general solutions of lattices, i.e., velocity sets and weights, for the lattice Bhatanagar-Gross-Krook (LBGK) models on two-and three-dimensional Cartesian grids. The solutions define the necessary and sufficient conditions so that the resulting LBGK model can accurately capture the dynamics of the moments retained in the distribution function. In the parameter space of the weights, the general solutions form low-dimensional linear spaces from which minimal velocity sets are identified for the degrees of precision that are most relevant to the construction of high-order LBGK models. All well-known LBGK lattices are found to be special cases of the given general solutions.