Higher-order difference methods for the solutions of neutron diffusion and transport equations

Abstract

Many people have tried to simulate the behavior of neutrons in a medium which may not be solved analytically, and many code systems have been constructed to obtain more accurate and faster solutions for nuclear reactor analysis and radiation shielding design. Currently, the accuracy instead of fastness of numerical solutions has been emphasized due to the advanced computational technologies including softwares as well as hardwares. In this thesis, higher-order difference methods (analytic collocation method and linear multiple balance method) are suggested which expand the existing finite difference method to provide efficient solutions of neutron diffusion and transport equations. Neutron transport equation, which describes the behavior of neutrons for reactor core and blanket design and for personnel and equipment shielding applications, is complicated due to the angular dependency of the neutrons. One of the approximations for angular dependency is the neutron diffusion equation which has been applied to realistic core calculations. Usually, reference calculations are performed by a fine mesh diamond difference (DD) scheme and a finite difference method (FDM) for neutron transport and diffusion equations, respectively. To improve existing finite difference method (FDM), an analytic collocation method (ACM) is suggested as a solver for neutron diffusion equation in this thesis. Especially, the ACM uses an analytic relation for pseudo and real unknowns at the boundaries or interfaces, which results in better solutions of neutron diffusion equation. As an higher-order method, the matrix system of ACM becomes denser than that of FDM. So parallel implementation as well as preconditioned bi-conjugate gradient stabilized method (PBi-CGSTAB) is considered for its efficient solution. ACM is also applied to the simplified even parity neutron transport equation which has an elliptic differential form like the neutron diffusion equation. It is easy to implement A...