We consider an approximation of the solution of the Boltzmann equation be means of the three-way decomposition [1] with respect to the single
components of the velocity vector, i.e. by a degenerated function of small rank. The motivation for this kind of approximation is based on the fact that many important distribution functions in kinetic theory are of low rank. The Maxwell distribution is of rank one, the Bobylev-Krook- and Wu solution
has a rank equal to three, and, furthermore, any final part of the Chapman-Enskog series is also of low rank. After discretisation on an appropriate uniform grid (cf. [2]), a given
density function f will be represented in a tensor-like form which requires O(n^3) words of computer memory. However, its three-way decomposition will require at most 3r(n+1)+3 words of memory, i.e. a linear amount for increasing n. We will refer to the number r in as the rank of the function f. As we will see, numerical work can also be significantly reduced if the distribution function is degenerate. It is clear that the majority of realistic distribution functions is not degenerate. However, some of them, related to the Boltzmann equation can be approximated up to the given accuracy by a degenerate function.
References
[1] I. Ibragimov and S. Rjasanow. Three way decomposition for the Boltzmann equation. J. Comp. Math., 26, 2009. To appear.
[2] I. Ibragimov and S. Rjasanow. Numerical solution of the Boltzmann equation on the uniform grid. Computing, 69(2):163--186, 2003.
Copyright. All Rights Reserved.