Derivation of Optimum Polar Angle Quadrature Set for the Method of Characteristics Based on Approximation Error for the Bickley Function

Abstract

In this paper, dedicated polar angle quadrature sets for the method of characteristics (MOC) are developed, based on the equivalence between MOC and the collision probability method. The discretization error of polar angle in MOC can be considered as an approximation error of the Bickley function used in the collision probability method; the Bickley function is numerically integrated in MOC using a quadrature set for polar direction (i.e., a set of polar angles and weights). Therefore, by choosing an appropriate quadrature set, the approximation error of the Bickley function which appears in MOC can be reduced, thus the calculation accuracy of MOC increases. Quadrature sets from one to three polar angle divisions are derived by minimizing the maximum approximation error of the Bickley function. The newly derived polar angle quadrature set (Tabuchi-Yamamoto or the TY quadrature set) is tested in the C5G7 and 4-loop PWR whole core problems and its accuracy is compared with other quadrature sets, e.g., Gauss-Legendre. The calculation results indicate that the TY quadrature set that is newly developed in the present paper provides better accuracy than the other methods. Since the number of polar angle divisions is proportional to computation time of MOC, utilization of the TY quadrature set will be computationally efficient.

Preservation of transmission probabilities in the method of characteristics

“…Therefore, the sign of β k does not affect the form of Equation (8). Since α i, p is always positive and only absolute value is considered for β k , L (x) should be evaluated only for positive x.…”

“…Geometries for the verification calculations. Number of polar angle division (for π /2, TY-optimum quadrature set) [8] 3…”

Aegis: An Advanced Lattice Physics Code for Light Water Reactor Analyses

“…When isotropic scattering is assumed, the polar integrated angular flux calculation on a particular ray trace using MOC is equivalent to that using the collision probability method [30]. In the collision probability method, the polar angle integration is analytically carried out; thus, the collision probability is expressed with the Bickely-Naylor function in generalized two-dimensional geometry.…”

“…Since the angular flux distribution for the azimuthal direction is very bumpy in heterogeneous geometry, a uniform angular division is applied for the azimuthal direction. However, for the polar direction, angular flux is rather smooth; thus, a dedicated quadrature set can reduce the discretization error for the polar direction [28][29] [30] .…”

Treatment of Staggered Mesh for BWR Pin-by-Pin Core Analysis

“…9) For the cell-homogeneous pin-by-pin calculations, the inhouse SP3 theory code is used. In the SP3 theory code, the relative convergence criterion of neutron flux and k-infinity was set to be 1:0 Â 10…”