Axial Expansion Methods for Solution of the Multi-Dimensional Neutron Diffusion Equation.

Abstract

The feasibility and practical implementation of axial expansion methods for the solution of the multi-dimensional multigroup neutron diffusion (MGD) equations is investigated. The theoretical examination which is applicable to the general MGD equations in arbitrary geometry includes the derivation of a new weak (reduced) form of the MGD equations by exp and ing the axial component of the neutron flux in a series of known trial functions and utilizing the Galerkin weighting. A general two-group albedo boundary condition is included in the weak form as a natural boundary condition. The application of different types of trial functions is presented. The practical implementation of the axial expansion method has involved two major tasks: (1) the development of a computer code for solving the MGD equations in two dimensions and (2) the unique implementation of two versions of the three-dimensional (3-D) method within a production level two-dimensional (2-D) MGD code 2DB-UM. Both the 2-D and 3-D computer codes solve the reduced weak form of the diffusion equation with conventional finite difference methods. The 3-D code is intended as a practical engineering tool which can be used for realistic reactor analysis. Both the Fourier sine series and the Legendre polynomial expansion were used as expansion functions for the 3-D application. Numerical results using the 2-D version of the code have indicated the advantage of the Fourier expansion method versus a conventional finite difference method due to an improved convergence rate, which was observed to be quadratic with the number of trial functions. The 3-D implementation has been constrained by the existing iteration methodology contained in the 2DB-UM code and optimum convergence rates similar to the 2-D method have not been obtained. Recommendations for further improvements of the iteration scheme and the use of alternative axial trial functions are made.