The Legendre polynomial axial expansion method

This work seeks to extend efforts of a previously described method for solving the 3D neutron transport equation. Namely, the method of Legendre polynomial axial expansion as an alternative to traditional 2D/1D for reactor applications. The intent is that this method specifically be used in situations where traditional 2D/1D breaks down, such as problems that need thin 2D slices or problems with large void-like regions. Our previous work presented only the final base system of equations without 2D/1D MOC or explanation of the derivation, and was limited to 1D/1D systems for demonstrative purposes. The full derivation for the method is shown in this work, albeit only for the upward moving equations for the sake of brevity. Furthermore, an alternative method of computing matrix exponentials for these systems is derived and presented. The method is also implemented into the MPACT code with full coupling to CMFD and shielding calculation capabilities of the code. The method is first tested in MPACT for three separate single pin cell systems of increasing complexity. These tests demonstrate the correctness of the implementation for simple systems with comparisons made against very fine 2D/1D runs. The results of these tests indicate that the axial expansion method is roughly comparable to 2D/1D in terms of accuracy for the global eigenvalue, with results averaging around 100 pcm off the reference calculation, compared to traditional 2D/1D averaging about 400 pcm off the reference calculation. It is then tested on a small pin cell array for three separate control rod insertions with similar results showing comparable accuracy as traditional 2D/1D. Finally, one of the models is tested to demonstrate efficacy of the matrix exponential tables, which show a speedup of between 20% and 80%. (authors)