NUMERICS OF THE BOLTZMANN EQUATION
Boltzmann equations on lattices
The Boltzmann equation is an equation describing the evolution of a rarefied gas. It consists of the free flow operator and the collision operator. (Details).
The Boltzmann collision operator balances the influence of particle collisions. A collision is the instantaneous transition of a velocity pair (v,w) into another pair (v',w'). Due to physical conservation laws, the velocities v and w as well as v' and w' lie on a sphere with center (v+w)/2. (Details)
Regular lattices: In order to discretize the Boltzmann equation, we restrict to a regular lattice. A special attention has to be paid to ''discrete spheres'' on the lattice. These are defined via the automorphism group of the lattice. This general approach goes back to the work [1] on hexagonal grids.
Discrete kinetic models: Once a discrete kinetic model has been established on the automorphism group, we can extend it to a discrete collision operator on the full lattice.
Numerical experiments
Shock problems:
(a) The shock tube problem
(b) Flow over an obstacle
An evaporation condensation problem
Acoustic waves
Literature
[1] L. S. Andallah, H. Babovsky. A discrete Boltzmann equation based on hexagons. Math. Models Methods Appl. Sci.,13:1537--1563, 2003.
[2] H. Babovsky. Hexagonal kinetic models and the numerical simulation of kinetic boundary layers. In: Analysis and Numerics for Conservation Laws, G. Warnecke (Ed.), Springer, Berlin, pp.47--67, 2005.
[3] H. Babovsky. A numerical scheme for the Boltzmann equation. Proceedings of the 25th International Symposium on Rarefied Gas Dynamics, M. S. Ivanov and A. K. Rebrov (Eds.), pp. 268-273, Novosibirsk, 2007