QUARTERLY OF APPLIED MATHEMATICS VOLUME LVII, NUMBER 4 DECEMBER 1999, PAGES 699-721 CONSERVATIVE ENERGY DISCRETIZATION OF BOLTZMANN COLLISION OPERATOR By L. PREZIOSI and L. RONDONI Dipartimento di Matematica, Politecnico di Torino, Corso Duca degli Abruzzi 24, Torino, Italy Abstract. The paper deduces a kinetic model obtained introducing a discretization of the Boltzmann equation based on an equispaced distribution of allowed particle energies. The model obtained is a system of integro-differential equations with integration over suitable angular variables: one over the portion of the unit sphere between two parallels symmetric with respect to the equatorial plane perpendicular to the velocity of the field particle, and one over a unit circle. The model preserves mass, momentum and energy. Furthermore, there exists an ff-functional describing trend toward an equilibrium state described by a Gaussian distribution. Particular attention is paid to the identification of a criterion which indicates the values of the discretization parameters. 1. Introduction. The Boltzmann equation is the classical model used to describe the dynamics of rarefied gases. In principle, it is able to provide a quite accurate description of a physical system that is characterized by a huge number of collisions governed by classical mechanics. Unfortunately, from a numerical point of view, it is very hard to exploit all the capability of the model and all the physical information it contains. This is essentially related to the structure of the collision term, a five-fold integral that is very expensive to be computed. Furthermore, this term needs to be evaluated with great precision, because all the mechanical properties characterizing particle collisions, e.g., conservation of mass, momentum, and energy, are included in it. In order to tackle the multiple integration with a not too large number of operations, most of the numerical codes that integrate Boltzmann-like models make use of Monte- Carlo procedures. Then, usually, in order to preserve mass, momentum, and energy, the solution is suitably corrected at each time step [1,2]. On the other hand, very recently, several authors [3]—[8]focused their attention on the possibility of handling the integration in a different way. They aimed at identifying discretization procedures that lead to models possessing most of the properties char- acteristic of the Boltzmann equation, namely, conservation of mass, momentum, and Received December 12, 1997. 1991 Mathematics Subject Classification. Primary 76P05. Key words and phrases. Rarefied gas dynamics, Boltzmann equation. ©1999 Brown University 699