On Some Criticality Benchmarks in Spherical Geometry

In a paper published in 2003, Sood et al. reported numerical results for a set of problems that are useful as benchmarks for criticality code verification. Here, we reconsider a subset of those problems, namely the one-group problems for homogeneous spheres with linearly anisotropic angular distribution of secondary neutrons, and we use an improved version of the P_N method to obtain very accurate numerical results for them. The standard way of implementing the P_N method for spheres has been shown to be numerically unstable by Aronson. Recently, in a joint work with C. E. Siewert of NCSU and J. R. Thomas, Jr. of VPI, we have developed a new version of the PN method for spheres that overcomes the difficulties pointed out by Aronson. In our implementation, we use singular-value decomposition (SVD) to solve, in a least squares sense, the system of equations obtained from the boundary condition for the coefficients of the approximation. In doing so, we can go to extremely high orders of the P_N approximation (tens of thousands) without losing accuracy. (authors)