“Virtual density” and traditional boundary perturbation theories: Analytic equivalence and numeric comparison

We compare and contrast “virtual density” perturbation theory with the traditional boundary perturbation theory developed by Pomraning, Larsen, and Rahnema in the context of diffusion theory. First, after reviewing that literature, we mathematically prove that virtual density perturbations and traditional boundary perturbations are precisely equivalent for arbitrary 1-D problems, which constitute non-uniform isotropic expansions. We also mathematically prove that these two perturbation theories are equivalent for 2-D boundary shift problems, which constitute non-uniform anisotropic expansions. Extension of this proof to swellings or 3-D problems is straightforward. We compare the two theories numerically for a series of alternating uranium and sodium 1-D slabs in finite difference diffusion, and we show that virtual density theory predicts reactivities much more accurately and efficiently than traditional boundary perturbation theory. Boundary perturbation theory is often very inaccurate on a coarse mesh but converges to the virtual density solution as the mesh becomes finer. We also compare the two theories for axial assembly swelling in an abbreviated FFTF benchmark with a coarse mesh. Here we find that reactivity coefficients obtained via virtual density perturbation theory agree with reference solutions to within 0.1%, while those obtained via boundary perturbation theory exhibit sporadic accuracy - sometimes in the range of 1-5% error, more frequently in the range 5-20% error, and occasionally well over 100% error in control rod assemblies. We conclude that although virtual density perturbation theory and boundary perturbation theory are analytically equivalent, boundary perturbations in diffusion theory are often thwarted in coarse mesh finite difference solutions due to inaccurate flux gradients along mesh cell surfaces in heterogeneous cores.