Nonlinear Analytic and Semi-Analytic Nodal Methods for Multigroup Neutron Diffusion Calculations

Abstract

Two advanced nodal methods for the solution of the multigroup neutron diffusion equations are developed, using the nonlinear coarse-mesh finite difference (CMFD) scheme. Based on the analytic and semi-analytic methods, the relationships between the flux and current on the nodal surface are derived, by which the two-node problem is formulated in an efficient way. The issue of the complex eigenmodes and the instability problem inherent in the analytic solution are considered in order to have an efficient and stable algorithm. The numerical results demonstrate the effectiveness of the two methods.

Development of a parallel analytic coarse mesh finite difference code for two group diffusion equation solution in two dimensional rectangular geometries

“…All nodal methods are categorized to two general concepts: Nodal Expansion Method (Finnemann and Bennewitz, 1977) and Analytical Nodal Method (Smith, 1979). Other nodal methods are a combination of two above methods such as Semi Analytic Nodal Method (Fu and Cho, 2002), Analytic Function Expansion Nodal (Cho and Noh, 1995), Flux Expansion Nodal Method (Bangyang and Zhongsheng, 2005), Nodal Green's Function Method (Lawrence, 1977), Exponential Flux Expansion Nodal (Yunzhao et al, 2010).…”

Two-Level Coarse Mesh Finite Difference Formulation with Multigroup Source Expansion Nodal Kernels

“…and it is not possible to further reduce Eqs. (36), (39) and (40) without using the inverse of a G Â G full matrix, we end up with the following 4G Â 4G linear system: 6) used the interface current vector as the final unknown vector and eliminated all the expansion coefficient vectors using the correlations between the current, flux and expansion coefficients so that, finally, a G Â G linear system is realized. But this approach is not pursued here because this reduction involves numerous inverses of G Â G matrices.…”

Two-Level Coarse Mesh Finite Difference Formulation with Multigroup Source Expansion Nodal Kernels

“…This approach, however, requires a simultaneous solution of the expansion coefficients of all groups. The derivation to determine the group-coupled expansion coefficients in a two-node configuration was done by Zimin 2) and Fu 6) for the case employing a second-order polynomial and two exponential functions as the basis functions. The derivation below is to increase the order of the polynomial to four.…”

“…On the other hand, Fu and Cho 6) also worked on the flux expansion form of SANM. The form of the flux expansion of Fu and Cho is essentially the same as that of Zimin and Ninokata in that they used a quadratic polynomial in the polynomial part.…”