Spatial Moments of Continuous Transport Problems Computed on Grids

Abstract

The method of moments is a well-known technique for determining exact expressions for spatial and angular moments of radiation distributions in infinite, homogeneous media. These moments can further be used to calculate other quantities of interest. We examine how the method of moments is altered when the underlying transport problem is spatially continuous but involves a grid and moments are computed on this grid instead of through integration over the entire domain. For the problem we consider, we employ both singular-eigenfunction and Fourier-transform approaches to show that when moments are evaluated in this manner (i) the flux-weighted average of x remains equal to the source-weighted average of x, but (ii) the flux-weighted average of (x−xa )2 is greater than the source-weighted average of (x−xa )2 by an additional error term, where x is the spatial variable and xa is an arbitrary point. We also demonstrate that the two resulting expressions for this error term are equivalent.