Mathematical Modeling of Neutron Transport Declaration

The subject of this work is computational modeling of neutron transport relevant to economical and safe operation of nuclear facilities. The general mathematical model of neutron transport is provided by the linear Boltzmann’s transport equation and the thesis begins with its precise mathematical formulation and presentation of known conditions for its well-posedness. In the following part, we study approximation methods for the transport equation, starting with the classical discretization of energetic dependence and followed by the review of two most widely used methods for approximating directional dependence (the SN and PN methods). While these methods are usually presented independently of each other, we show that they can be put into a single framework of Hilbert space projection techniques. This fact is then used in conjunction with the results of the first part to rigorously prove rotational invariance of the PN equations and to analyze convergence of the basic iterative scheme for solving the SN equations. This part is concluded by the description of a finite element method for the final discretization of spatial dependence and a discussion of solution of the resulting system of algebraic equations. The main new results are contained in the following two chapters focusing on the simplified PN approximation, which is a computationally more convenient albeit not as mathematically well-founded variant of the PN approximation. We prove well-posedness of the weak form of the SP3−7 equations and present a new way of deriving the equations from an alternative set to the PN equations, obtained from special linear combination of spherical harmonics – the so-called MaxwellCartesian spherical harmonics, hence the abbreviation MCPN . We explicitly show how the MCP3 equations may be transformed to the SP3 equations. The final part of the thesis contains numerical examples of the SN and hpadaptive SPN calculations using a neutronics framework that has been implemented by the author to the hp-adaptive finite element library Hermes2D. The SP1 (or diffusion) model also serves as a basis of a real-world reactor calculation suite co-developed by the author for the purposes of “Project TA01020352 – Increasing utilization of nuclear fuel through optimization of an inner fuel cycle and calculation of neutron-physics characteristics of nuclear reactor cores”. An example benchmark used to test the code concludes the thesis.