Numerical stability of Monte Carlo neutron transport and isotopic depletion for nuclear reactor analysis
Coupling Monte Carlo neutron transport with isotopic depletion is known to produce non-physical results for large reactor geometries. This thesis begins with a survey of the stable methods used to avoid this and highlights some of their drawbacks.
Chapter 2 introduces the known phenomenon of neutron clustering in Monte Carlo as a factor strongly contributing to previous reports of instability. It is shown that attempting to minimise clustering effects produces more stable burn-up calculations. Furthermore, accounting for clustering is shown to allow for the accurate simulation of xenon transients in simple systems which have been noted to pose a challenge for burn-up simulations. Finally, it is demonstrated that neutron clustering can also affect burn-up simulations substantially even when xenon equilibrium is enforced, namely by way of the previously hypothesised `gadolinium instabilities'.
Chapter 3 begins by highlighting how implicit burn-up schemes may be viewed as root-finding schemes for a discrete map. It is shown that, for reasonably long time-steps, the corrector step of predictor-corrector schemes does not succeed in locating the root (or the stable solution) of this map. Hence, relaxation schemes are introduced; relaxation schemes have been applied to neutronics/depletion coupling previously in the form of the stochastic approximation, although this is relatively inefficient. The relaxation scheme proposed here uses a fixed relaxation factor (rather than the variable factor previously used) and demonstrates its improved stability and computational efficiency compared to the stochastic approximation. The same investigations are applied to a depletion problem where equilibrium xenon is enforced -- it is seen that this, too, can be unstable, but is resolvable through relaxation.
Chapter 4 performs a Von Neumann stability analysis of a simple coupled neutron diffusion-depletion system. Extending a previous analysis, this chapter provides a justification for applying a relaxation to predictor-corrector schemes, shows the possibility of obtaining symmetric burn-up instabilities, and demonstrates that no neutronics-depletion coupling scheme, without relaxation, is assuredly more stable than another, depending on the depletion system in question.
Finally, Chapter 5 summarises the findings, proposes an explanation regarding general cases of burn-up instability, and suggests future work.