Numerical Examples

Numerical examples are provided here to demonstrate that straightforward subtraction of two independent Monte Carlo simulations is problematical for calculating small perturbation effects. All the test problems correspond to a homogeneous, one energy group, slab, with thickness of 16 cm and vacuum boundary conditions on both ends. Table 3.1 shows the unperturbed and perturbed cross sections and $\Delta$K results calculated by two independent Monte Carlo simulations and the reference TWODANT simulation. Monte Carlo results are for 30 inactive batches, 70 active batches and 2000 neutrons per batch. The TWODANT results were generated using S₃₂ quadrature sets and inner and outer iteration convergence criteria of 10^-12.

 

1|cUnperturbed 1ccross sections: 1c 1c|

$$\Sigma_t \nu\Sigma_f \Sigma_s$$ 1c|

$$\Delta \Delta$$ 1|cProblem # 1|c|Perturbed cross sections

$$\Delta\Sigma_t \Delta\Sigma_a \mp$$ 1 -.008807

$$\Delta\Sigma_t \Delta\Sigma_a \mp$$ 2 -.000888

$$\Delta\Sigma_f {\pm}$$ 3 .000903

Results of table 3.1 show, with the exception of the first test problem, that subtracting two independent Monte Carlo runs to calculate small perturbation effects can yield significant errors. Similar observations can be found in other references [Wal94, Gal95] also. Later in this chapter (in section 3.3.4), test problems 1, 2 and 3 are solved using the combined correlated sampling fission matrix (CSFM) approach (with the same number of batches and neutrons per batch as in the cases of table 3.1) and the results are significantly more accurate than in table 3.1. Also, later in this chapter, it is shown that when the correlated sampling technique is applied with the source iteration method, it encounters severe difficulties; and it is found that the combined CSFM method yields far superior results in Monte Carlo calculations of perturbation effects.