Numerical methods for nuclear fuel burnup calculations

The material composition of nuclear fuel changes constantly due to nuclides transforming to other nuclides via neutron-induced transmutation reactions and spontaneous radioactive decay. The objective of burnup calculations is to simulate these changes over time. This thesis considers two essential topics of burnup calculations: the numerical solution of burnup equations based on computing the burnup matrix exponential, and the uncertainty analysis of neutron transport criticality equation based on perturbation theory. The burnup equations govern the changes in nuclide concentrations over time. They form a system of first order differential equations that can be formally solved by computing the matrix exponential of the burnup matrix. Due to the dramatic variation in the half-lives of different nuclides, the system is extremely stiff and the problem is complicated by vast variations in the time steps used in burnup calculations. In this thesis, the mathematical properties of burnup matrices are studied. It is deduced that their eigenvalues are generally confined to a region near the negative real axis. Rational approximations that are accurate near the negative real axis, and the Chebyshev rational approximation method (CRAM) in particular, are proposed as a novel method for solving the burnup equations. The results suggest that the proposed approach is capable of providing a robust and accurate solution to the burnup equations with a very short computation time. When a mathematical model contains uncertain parameters, this uncertainty is propagated to responses dependent on the model. This thesis studies the propagation of neutron interaction data uncertainty through the criticality equation on a fuel assembly level. The considered approach is based on perturbation theory, which allows computing the sensitivity profiles of a response with respect to any number of parameters in an efficient manner by solving an adjoint system in addition to the original forward problem. The uncertainty related to these parameters can then be propagated deterministically to the response by linearizing the response.