A multilevel in space and energy solver for multigroup diffusion eigenvalue problems

In this paper, we present the newly developed Multilevel in Space and Energy Diffusion (MSED) method for solving multigroup diffusion eigenvalue problems. The MSED method can be described as a power iteration scheme with three additional features: (1) a grey (1-group) diffusion equation used to efficiently converge the fission source and eigenvalue, (2) a space-dependent Wielandt shift technique [1] used to reduce the number of power iterations required, and (3) a multigrid in space linear solver for the linear solves required by each power iteration step. It is a method in which the convergence of the solution on the full multigroup diffusion eigenvalue problem is accelerated by performing work on lower-order equations with only 1 group and/or coarser spatial grids. Results from several Fourier analyses and a 1-D Python code are provided to verify the efficiency of the MSED method and to justify the incorporation of the grey diffusion equation and the multigrid linear solver. These results highlight the potential efficiency of the MSED method as a solver for large multigroup diffusion eigenvalue problems, and serve as a proof of principle for our future work going forward. Ultimately, our goal is to implement the MSED method as an efficient solver for the 2-D/3-D Coarse Mesh Finite Difference (CMFD) diffusion system in the Michigan Parallel Characteristics Transport (MPACT) code [2]. The work in this paper represents an important step towards that goal. (authors)