Analysis of Inherent Oscillations in Multidimensional S_N Solutions of the Neutron Transport Equation

Recent pressure vessel fast fluence calculations have revealed numerical difficulties (spatial oscillations) in the S_N solutions, which have persisted in spite of mesh refinement. It is demonstrated that other shielding/deep-penetration applications may be affected; in fact, any S_N solution in which the uncollided flux component is significant is likely to exhibit such difficulties. Test problems have been designed to characterize and understand numerical difficulties. Main analyses are performed using the diamond-difference (DD) scheme, which is linear and forms the basis for other (more complex) low-order differencing schemes. The genesis of oscillations is shown to be related to several effects specific for multidimensional geometries as follows: ambiguity in the interpretation of boundary conditions, discontinuities, and different directions of particle streaming and differencing. It has further been explained why the mesh refinement does not produce the intuitively expected results. Other low-order differencing schemes (e.g., the DD with negative flux fixup and the θ-weighted) may partly remedy the situation by reducing the oscillations or by eliminating the oscillations at a cost of “oversmoothing” the results everywhere (e.g., the zero-weighted scheme). These schemes provide more robust solutions, but the inherent difficulties (although reduced) still remain. Types of discontinuities that trigger the oscillations are also examined; it is difficult to envisage an actual practical application free of such discontinuities. The magnitude of numerical difficulties (oscillations) and their practical relevance will depend on all S_N model features, the differencing scheme being used, and the application requirements, but this study has shown that they are inherent to multidimensional finite-difference S_N algorithms.