Summary and Discussion of Numerical Results

The results of table 3.1 show that subtracting the results of two independent Monte Carlo simulations, to compute small perturbation effects can encounter problems. We have shown in section 3.3.3 that the Monte Carlo source iteration method, when combined with the correlated sampling technique, also has difficulty estimating small perturbation results. This led us to combine the correlated sampling technique with the fission matrix approach for eigenvalue calculations. The results of table 3.3 show that the combined CSFM approach can provide significantly improved results compared with direct subtracting or the source iteration method combined with correlated sampling.

In section 3.4 we have shown various numerical results for eigenvalue perturbation problems using the CSFM method. We have compared these Monte Carlo $\Delta$Ks to that of the TWODANT code. For test problems 1, 2a, 2b, 3a, and 3b we observe that $\Delta$Ks calculated with the Monte Carlo and S_N methods always agree within 4% of each other. Even though we have not included the standard deviation of the reference eigenvalue for all the test problems (except for problems of table 3.4 and 3.9), this standard deviation is always approximately an order of magnitude higher than that of the $\Delta$K. Extension to multigroup problems was straightforward and multigroup problems have shown good agreement with the corresponding TWODANT results also.

We notice that the error for test problem 4 is 13.7% which is significantly larger than that of other test problems of section 3.4. If we compare the configuration of test problem 4 with that of test problem 1 we observe that in test problem 4 the middle 0.4 cm region has been replaced by a strong absorber material. In both problems perturbation was done in the middle 0.4 cm region. Test problem 1 shows 2.3% error, whereas test problem 4 shows 13.7% error. Insertion of a strong absorber in the middle region of the slab, converts the test problem 4 into a loosely coupled system and hence there is less communication of neutrons between the two sides of the absorber region. We believe that special variance reduction schemes are needed to account for the lack of neutron communication for loosely coupled systems.

In test problem 5 we have specifically looked into the case where the unperturbed scalar flux differs significantly in shape from the perturbed scalar flux as shown in figure 3.3. The reference system is again chosen as the arithmetic average of the unperturbed and perturbed systems. The purpose of this test problem is to see the effect of fission source convergence between the reference, unperturbed and perturbed systems. Since the scalar flux shapes are significantly different for the unperturbed and perturbed systems, fission sources for the two systems would need to converge to these two different shapes. The fission source of the reference system would converge to a source shape that is in-between the unperturbed and perturbed fission source shapes. From the results of table 3.11 we observe that for this test problem the error in $\Delta$K is approximately 1%. This relatively small error in $\Delta$K implies that the three fission sources for the reference, unperturbed and perturbed systems converged to their respective correct fission sources even though according to figure 3.3 the converged fission source shapes are different for these systems.

In the CSFM method the actual Monte Carlo eigenvalue simulation is done in the reference system. The fission sources of the unperturbed and perturbed systems are correlated to that of the reference system. For every generation the fission neutrons for the reference system are started from the previous generation of fission source distribution of the reference system. No information regarding the fission source distribution of the unperturbed or the perturbed system is carried onto the next generation. It is possible that there would be an inherent bias in $\Delta$K due to this lack of information for the unperturbed and perturbed fission source distributions. Test problems of this chapter (and of chapter 4) show that this effect is not significant. Even though the $\Delta$K results are accurate, it does not imply that the unperturbed and perturbed Ks converge to the correct values. It is possible that the unperturbed and perturbed Ks have biased values, but due to the positive correlation of these Ks the $\Delta$K results are still accurate. We believe that more study might be necessary to clearly define the limits where the CSFM method would encounter difficulties.