On Cabannes' 32-Velocity Models of the Boltzmann Equation
Shuichi Kawashima, Akemi Watanabe, Machi Maeji, Yasushi Shizuta · Publications of the Research Institute for Mathematical Sciences · 1986
In the preceding papers [8], [9], we studied the general theory of the discrete Boltzmann equation and formulated several conditions under which the solutions in the large for the Gauchy problem exist and approach the Maxwellian corresponding to the initial data as £->oo. (One of these conditions which we shall use in this paper will be called simply the stability condition for the discrete Boltzmann equation.) Also we treated in [9] the 14-velocity model as an application of the results. We continue in this paper the study of concrete discrete models. Our aim is to verify the stability condition for the 32-velocity model introduced by Cabannes [4]. Since the size of the model is relatively large, the computation needed becomes necessarily lengthy. First we recall the definition of the 14-velocity model. We consider a cube centered at the origin of the velocity space. The set of eight vertices of the cube defines an element model Mc. (A discrete model is said to be an element model when the moduli of the velocities are equal.) The centers of six surfaces of the cube form an octahedron and define another element model M0, We introduce here the definition of the similarity. Two discrete models are said to be similar if one equals the other after the multiplication by a suitable positive constant. Let 3F'c and 3F0 be the families of discrete models similar to Me and M03 respectively. Let us consider a