Functional Analysis and Applications Seminar 

07/04/2006, 15:00 — 16:00 — Room P3.10, Mathematics Building
Sergej Rjasanow, Universität Saarbrücken, Alemanha

The Boltzmann equation. Theory and numerics

In the first part of the talk we introduce the Boltzmann equation, discuss its properties and give an overview on existence and uniqueness of solutions. Especially our new results on mapping properties of the Boltzmann collision operator will be presented. Then the Direct Simulation Monte Carlo method (DSMC) which is widely applied in numerics will be explained. In the third part of the talk we present the Stochastic Weighted Particle Method (SWPM) which was introduced in 90's by Rjasanow and Wagner. We apply this method to the numerical solution of the spatially two-dimensional Boltzmann equation. The numerical solution of the Boltzmann equation using naive deterministic methods leads to the amount of numerical work of the order $O(n^8)$, where $n$ denotes the number of discrete velocities in one direction. In the next part of the talk we give an overview on deterministic numerical methods applied to the Boltzmann equation by a number of authors. Then, in the final part of the talk, we present the results of our numerical experiments obtained by a new deterministic approximation of the Boltzmann equation using Fast Fourier Transform.