Numerical comparison of mathematical and computational models for the simulation of stochastic neutron kinetics problems

Abstract

This paper concerns numerical comparisons between five mathematical models capable of modelling the stochastic behaviour of neutrons in low extraneous (extrinsic or fixed) neutron source applications. These models include analog Monte-Carlo (AMC), forward probability balance equations (FPB), generating function form of the forward probability balance equations (FGF), generating function form of the backward probability balance equations (Pál-Bell), and an Itô calculus model using both an explicit and implicit Euler-Maruyama discretization scheme. Results such as the survival probability, extinction probability, neutron population mean and standard deviation, and neutron population cumulative distribution function have all been compared. The least computationally demanding mathematical model has been found to be the use of the Pál-Bell equations which on average take four orders of magnitude less time to compute than the other methods in this study. The accuracy of the AMC and FPB models have been found to be strongly linked to the computational efficiency of the models. The computational efficiency of the models decrease significantly as the maximum allowable neutron population is approached. The Itô calculus methods, utilising explicit and implicit Euler-Maruyama discretization schemes, have been found to be unsuitable for modelling very low neutron populations. However, improved results, using the Itô calculus methods, have been achieved for systems containing a greater number of neutrons.