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Arbeitsgruppe Numerische Mathematik
und Informationsverarbeitung

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Prof. Dr. rer. nat. habil. Hans Babovsky

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INHALTE

### 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