Stationarity Issues in Monte Carlo Simulation for Neutron Transport

Abstract

Monte Carlo (MC) simulation, especially suitable for large and complex nuclear systems, can become computationally expensive due to the large number of neutrons which must be simulated for statistically accurate and precise estimates. It is generally understood that a sample estimate will converge to the population mean when a ‘large’ sample size is taken. The term ‘large’ is usually based on a guess and hence MC simulation is understood to be both an art and a science. Considerable work has been done to analyze convergence of MC results and develop posterior diagnostic tools. This paper addresses the convergence of MC simulation for two problems viz (i) a fixed-source non-multiplying system, and (ii) a critical system represented by Godiva. A traditional approach is used in the first part of the work while a ‘new’ approach essentially following Signals and Systems techniques from Digital Signal Processing gives ‘orginality’ to the analysis as it provides insight into the convergence of didactic problems in neutron transport simulation. The methods used are (i) comparison of MC flux with exact transport and diffusion solutions and relative entropy, with the Kullback-Leibler (KL) divergence, to quantify the convergence of estimates for flux as a function of sample size in Monte Carlo simulations, (ii) the effect of ‘skip cycles’ on the keff estimate, and (iii) a system identification approach based on the ARX (Auto Regressive Exogenous Source) method to determine the correlation between generations. The latter can be incorporated in Monte Carlo codes leading to a priori rather than to a posteriori diagnostic tools for establishment of convergence. The main findings of this work for simple one-group problems are that a Kullback Leibler ε∼10−3 can be specified a priori for the convergence criteria of a fixed source problem while a system-identification approach for a simple Godiva simulation would need a large number of data points to build an accurate ARX model and hence would be more difficult to include as an a priori tool; so it would essentially serve a purpose similar to the FOM which gives a quality metric only after the simulation is completed.