Investigating mixed FEM SP_n as a synthetic acceleration for the DFE transport equation

This paper presents an implementation of a synthetic acceleration based on the SP_n method and a mixed finite element method - the Raviart-Thomas discretization - for the discrete ordinates transport equation. Similar work was previously done using a diffusion solver for the high-order diamond differencing method in Cartesian geometry. However, this research takes it further by investigating the compatibility with the discontinuous finite element (DFE) method; the hexagonal geometry; and the anisotropy of the source and higher angular orders that are possible with the SP_n method. Discussion on the correction of the flux moments other than the zeroth moment flux is included, as well as correction of the boundary fluxes, i.e., the angular surface fluxes, in the case where reflective or albedo boundary conditions are used on the boundary. Results are presented for two two-dimensional test cases: the AIC Assembly (Cartesian geometry) and the Monju 2D (hexagonal geometry) benchmarks. It was found that the Diffusion Synthetic Acceleration (DSA) performed best when an SP₁ method with a flat order Raviart-Thomas spatial discretization is used. It also generally performed better for the DFE method. A reduction of at least 45% (and up to 90%) in the number of inner iterations and at least 25% (and up to 80%) in the computational times was observed. There were, however, some instabilities - especially in the case of the diamond difference method - which were more present when reflective boundary conditions are present. Instabilities also arose in the case of insufficient sub-meshing. (authors)