# The Boltzmann equation. Theory and Numerics

## Univ.-Prof. Dr. Sergej Rjasanow

**March 1, 2005, 2:30 p.m. HF 136**

In the first part of the talk we introduce the Boltzmann equation

$ f_t + (v, grad_x f) = \int_{\mathbb{R}^3} \int_{S^2} B(v,w,e) \Big(f(v')f(w') - f(v)f(w) \Big)~de~dw$,

discuss its properties and give an overview on existence and uniqueness of solutions. Especially our new existence results in Sobolev spaces will be presented (see [1]). Then the Direct Simulation Monte Carlo method (DSMC) (see [2]) 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 [3]. The complete theory of this method which is a generalisation of the DSMC can be found in [4]. 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 (see [5]).

References:

[1] Duduchava R., Kirsch R., Rjasanow S. Sobolev space solutions to the Boltzmann equation, University of Saarland, Preprint 127, 2005

[2] Bird G. A. Molecular Gas Dynamics and the Direct Simulation of Gas Flows. Clarendon Press, Oxford, 1994.

[3] Rjasanow S., Wagner W. A Stochastic Weighted Particle Method for the Boltzmann Equation, Journal of Computational Physics, Vol. 124, 243-253, 1996.

[4] Rjasanow S., Wagner W. Stochastic Numerics for the Boltzmann equation, Springer, 2005.

[5] Ibragimov I., Rjasanow S. Numerical solution of the Boltzmann equation on the uniform grid. Computing, Vol. 69 (2002), no. 2, 163-186