A rational construction of discrete velocity models of the Boltzmann equation

A general and effective construction of the collisional operator for the discrete velocity models of the Boltzmann equation is presented. First, definitions of collisions and space of collisional invariants are given, and next a collisional operator with the space of collisional invariants given in advance is constructed. A theorem asserting that every bilinear operator in Euclidean spaces, which has the basic properties of collisional operators, is necessarily of the well known classical form, is demonstrated.