Reduced-order modeling of neutron transport separated in axial and radial space by Proper Generalized Decomposition with applications to nuclear reactor physics

In this article, we demonstrate the novel use of Proper Generalized Decomposition (PGD) to separate the axial and, optionally, polar dimensions of neutron transport. Doing so, the resulting Reduced-Order Models (ROMs) can exploit the fact that nuclear reactors tend to be tall, but geometrically simple, in the axial direction $z$, and so the 3D neutron flux distribution often admits a low-rank "2D/1D" approximation. Through PGD, this approximation is computed by alternately solving 2D and 1D sub-models, like in existing 2D/1D models of reactor physics. However, the present methodology is more general in that the decomposition is arbitrary-rank, rather than rank-one, and no simplifying approximations of the transverse leakage are made. To begin, we derive two original models: that of axial PGD -- which separates only $z$ and the sign of the polar angle $\alpha\in\{-1,+1\}$ -- and axial-polar PGD -- which separates both $z$ and the full polar angle $\mu$ from the radial domain. Additionally, we grant that the energy dependence $E$ may be ascribed to either radial or axial modes, or both, bringing the total number of candidate 2D/1D ROMs to six. To assess performance, these PGD ROMs are applied to two few-group benchmarks characteristic of Light Water Reactors. Therein, we find both the axial and axial-polar ROMs are convergent and that the latter are often more economical than the former. Ultimately, given the popularity of 2D/1D methods in reactor physics, we expect a PGD ROM which achieves a similar effect, but perhaps with superior accuracy, a quicker runtime, and/or broader applicability, would be eminently useful, especially for full-core problems.