Prediction and Analysis of Stochastic Convergence in the Standard and Extrapolated Power Methods Applied to Monte Carlo Fission Source Iterations

Abstract

In Monte Carlo (MC) eigenvalue calculations, numerous cycles of the fission source iterative procedure may need to be discarded to sufficiently converge an initial guess toward the steady-state fission distribution in a system. Reducing the number of discarded cycles is desirable to save computer time, and knowing when one may stop discarding is important to avoid introducing a bias in desired quantities from simulations. In this work, approximate prescriptions are derived for the evolution of eigenmode components of the fission source at each cycle using the standard and extrapolated power methods adapted to MC neutron transport. The approximate prescriptions may be used to diagnose and predict convergence of the standard and extrapolated fission source iterations. Despite the lack of success by previous researchers, the approximate prescriptions reveal that the extrapolated power method can reduce the number of discarded cycles compared with the standard power method for some cases.

The probability density functions (PDFs) of eigenmode components are shown to be approximately normally distributed and thus may be wholly described by their means and variances. Two methods for approximating the mean of single-cycle variances of eigenmode components are constructed and validated. The mean of single-cycle variances of eigenmode components may be used to predict the variance of eigenmode components after multiple cycles.

A novel convergence diagnostic that focuses on convergence of the probability density functions (PDFs) of eigenmode components of the fission source distribution is developed. The Bhattacharyya coefficient and relative entropy are examined as measures of similarity between eigenmode component PDFs, and relative entropy is shown to be preferable. A predictive convergence diagnostic is constructed using a threshold maximum value for the relative entropy between eigenmode PDFs at each cycle and their steady-state PDFs. Prescriptions for the optimal numbers of extrapolated and standard cycles to discard are derived using the new diagnostic such that the number of discarded cycles is minimized for a chosen relative entropy threshold. The new diagnostic is significant because it may be applied before or during a running simulation, and it provides the user with tangible confidence intervals for the eigenmode components when given appropriate input.