Noise term modeling of dynamic Monte Carlo using stochastic differential equations

Dynamic Monte Carlo (DMC) method recently became a relevant tool to perform kinetic calculations even on the scale of power plants and further to simulate coupled neutronics - thermal-hydraulics problems. DMC offers time-dependent, high fidelity calculations in very detailed and complex geometries in return for the high computational need. DMC provides stochastic results which is an indescribable property by the mathematics of the deterministic neutron transport. The stochasticity of the results also rises the unsolved problem of stability and convergence during the nonlinear, coupled DMC and thermal-hydraulics simulation. This paper aims to introduce the Stochastic Differential Equations (SDEs) in the field of Monte Carlo neutron transport for connecting the DMC method and differential equation formalism. The derivation of a Non-Analog Monte Carlo (NAMC) model is shown for the Stochastic Point-Kinetics equation (SPKe) to determine a noise model capable to approximate one trajectory of DMC. The theoretical framework allows the reader to observe analytic solutions for a random variate, the expectation and the variance of the reactor power. Also, analytic formulas are given for a simple system of coupled reactor physics and thermal-hydraulics without feedback. The analytic variance of the Monte Carlo simulated reactor power were compared to the calculations of GUARDYAN (GpU Assisted Reactor DYnamic ANalysis) code in a subcritical, a critical and a supercritical reactor models.

Automated summary

The Dynamic Monte Carlo (DMC) method is a relevant tool for kinetic calculations and simulations of coupled neutronics-thermal-hydraulics problems. It provides high fidelity calculations in detailed and complex geometries, but requires high computational resources. The stochastic nature of DMC results poses challenges in terms of stability and convergence in coupled DMC and thermal-hydraulics simulations. This paper introduces a Stochastic Differential Equations (SDEs) approach to connect the DMC method with differential equation formalism and presents a Non-Analog Monte Carlo (NAMC) model for the Stochastic Point-Kinetics equation (SPKe). Analytic solutions for random variates, reactor power expectation, and variance are given. A comparison is made between the analytic variance obtained from Monte Carlo simulations and the calculations from GUARDYAN code in different reactor models.