2:15 pm Thursday, May 23, 2002
Special TICAM Seminar: Stochastic and Deterministic Numerics for the Boltzmann Equation by Prof. Sergej Rjasanow (Universitaet des Saarlandes, Saarbrucken, Germany) in ACES 6.304
In the first part of the talk we introduce the Boltzmann equation, discuss its properties and briefly describe the Direct Simulation Monte Carlo (DSMC) method (see [1]) which is widely applied in numerics. Then, in the second part of the talk, we present the Stochastic Weighted Particle Method (SWPM) which was introduced in [2]. We apply this method to the numerical solution of the spatially two-dimensional Boltzmann equation. The computation of macroscopic quantities in regions with low particle density is of special interest. The numerical solution of the Boltzmann equation using deterministic methods is difficult for two reasons. First of all the evaluation of the five-fold collision integral is extremely expensive from the numerical point of view. Thus a naive approximation of this integral leads to the amount of numerical work of the order O(n^8), where n denotes the number of discrete velocities in one direction. On the other hand, uniform discretisation of the velocity space does not match exactly the unit sphere in the collision integral and leads to low accuracy of the approximation. In the third 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 recent numerical experiments obtained by a new deterministic approximation of the Boltzmann equation using Fast FourierTransform. [1] Bird G.A."Molecular Gas Dynamics and the Direct Simulation of Gas Flows",Clarendon Press, Oxford, 1994.[2]Rjasanow S., Wagner W."A Stochastic Weighted Particle Method for the Boltzmann Equation", Journal of Computational Physics, Vol. 124, 243-253, 1996. Submitted by
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