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A non-perturbative algebraic theory of lattice Boltzmann method is developedbased on a symmetry of a product. It involves three steps: (i) Derivation ofadmissible lattices in one spatial dimension through a matching condition whichimposes restricted extension of higher-order Gaussian moments, (ii) Specialquasi-equilibrium distribution function found analytically in closed form onthe product-lattice in two and three spatial dimensions, and which provesfactorization of quasi-equilibrium moments, and (iii) Algebraic method ofpruning based on a one-into-one relation between groups of discrete velocitiesand moments. Two routes of constructing lattice Boltzmann equilibria aredistinguished. Present theory includes previously known limiting and specialcases of lattices, and enables automated derivation of lattice Boltzmann modelsfrom two-dimensional tables, by finding roots of one polynomial and solving afew linear systems.

Factorization symmetry in lattice Boltzmann simulations