1. DSMC Collision Frequency Traditional & Sophisticated
Alejandro L. Garcia
Dept. Physics, San Jose State Univ.
Center for Comp. Sci. & Eng., LBNL

2. Lucky Number 7
Graeme’s notes on sophisticated DSMC say that accuracy of collision rate depends on number of particles per cell.
Lucky 7
No such dependence in traditional DSMC. Why?

3. Collision Frequency
From basic kinetic theory, collision frequency (number of collisions per particle per unit time in a volume V) is So the total number of collisions in a time step is

4. DSMC Collisions
DSMC uses this result to determine the number of attempted collisions in a cell as Attempted collisions are accepted with probability,

5. Traditional DSMC Collisions
In traditional DSMC, the average number of collisions is This gives the correct result since for Poisson,

6. Alternative Formulation
In Graeme’s 1994 book he uses This also gives the correct result since, As mentioned in his notes for this meeting, the approach is now obsolete.

7. Nearest Particle Selection
In traditional DSMC, collisions partners are drawn at random in a cell. In sophisticated DSMC, the nearest particle in the cell is used as the collision partner (unless those two particles recently collided). Does this introduce a bias in average relative velocity if number of particles in a cell is small? Preliminary 1D runs indicated that it does not bias the acceptance rate or collision frequency.

8. Sophisticated DSMC
In sophisticated DSMC the time step and cell size vary dynamically so now Dt and V are also random variables.

9. Sophisticated DSMC Collisions
In sophisticated DSMC, the average number of collisions is
If N, V, and Dt are correlated then equality does not hold.

10. Simple Example
Suppose we dynamically make the cell sizes such that the number of particles in a cell is exactly N0
This simple example is not sophisticated DSMC yet it illustrates the effect of a dynamically variable cell volume.

11. Collisions in Simple Example
Since the number of particles in a cell is exactly N0 the average number of collisions is Two problems:

12. Results Simple Example
Quick calculation estimates that number of collisions will be lower by a factor of
<N> Prediction Simulation

13. Quick “Fix” in Simple Example
Since the number of particles in a cell is exactly N0 we might think that instead we should compute the number of attempted collisions as so that

14. Results for Quick “Fix”
Quick calculation estimates that number of collisions will be higher by a factor of
<N> Prediction Simulation

15. Conclusion
Sophisticated DSMC is a powerful and useful extension to traditional DSMC. For many reasons we SHOULD NOT be thinking of returning to traditional DSMC. The development of traditional DSMC benefitted from theoretical analysis. Sophisticated DSMC is more complex so this analysis will be more difficult, but still needed.