The linear source approximation and particle conservation in the Method of Characteristics for isotropic and anisotropic sources

Particle conservation, otherwise known as particle balance, is one of the main properties to examine when considering any transport discretization. In the past, particle conservation has been examined for the 'Flat Source' (FS) approximation used in the Method of Characteristics (MOC) for isotropic and anisotropic sources. The work presented here extends one of the necessary constraints for particle conservation to the recent 'Linear Source' (LS) approximation in the MOC for isotropic and anisotropic sources. As a result, direction-dependent track renormalization and centroids are examined. A new LS MOC set of equations is derived for problems involving anisotropic sources in which the zeroth-order angular moment of the source possesses spatially-linear expansion coefficients and satisfies particle conservation if direction-dependent track renormalization and centroids are employed. Numerical results are presented which verify that the FS and LS MOC satisfy particle conservation for problems involving isotropic and anisotropic sources by considering a simple pin cell test problem. In addition, results from a set of full-core eigenvalue calculations, which make use of the anisotropic source option, are presented and compared to evaluate the impact of using either direction dependent or direction independent track renormalization (and centroids).