In this thesis several second-order forms of the neutron transport equation (NTE) are spatially
discretised with isogeometric analysis (IGA). IGA allows for the exact representation of geometries
that are produced using computer aided design (CAD) software. Finite element (FE) spatial discretisation methods are incapable of exactly representing all the geometries that are produced by CAD software. Furthermore, the NURBS basis functions allow for high-order continuity within a NURBS patch, whereas FE basis functions are typically C0 continuous between adjacent FEs. The advantages and disadvantages of NURBS based IGA will be investigated by comparisons to FE based spatial discretisations. The second-order forms of the NTE investigated in this thesis are: the self-adjoint angular flux (SAAF) equation, the least-squares (LS) equation, and the weighted least-squares (WLS) equation. The discrete ordinate (SN) method will be used to angularly discretise these equations. A number of verification benchmark problems will be used to determine the numerical accuracy, convergence, and
computational efficiency of both the second-order forms of the NTE, and the IGA spatial discretisation
method.
One major challenge associated with continuous IGA spatial discretisations is performing local refinement of the IGA discretisation. Therefore, a constraint based local adaptive mesh refinement (AMR) algorithm is developed to overcome this challenge. Both heuristic and dual weighted residual (DWR) or goal-based error measures are developed in order to determine where local refinement needs to be performed. The DWR error measures enable the numerical error in both global (Keff) and local (reaction rate) quantities of interest to be determined rigorously. Once again a number of verification benchmark problems are used to analyse the numerical accuracy, convergence and computational efficiency of the IGA AMR algorithm.