Semi-consistent diffusion synthetic acceleration for discontinuous discretizations of transport problems
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We have developed a diffusion synthetic acceleration (DSA) method for discontinuous finite element methods (DFEMs) in which the diffusion operator is constructed from the matrices that appear in the transport discretization. This allows the diffusion operation to be coded once (in terms of single-cell transport matrices) and then used by any DFEM or related method that is implemented in the code. The diffusion operation first solves a global system for a continuous function and then performs local (cell by cell) calculations to obtain the desired discontinuous function. We cast the DSA scheme as a preconditioner to set the stage for use with Krylov methods. Using Fourier analysis we determine the spectral radius of the iteration operator for a one-group infinite-homogeneous medium transport problem. The results of simple test problems agree with the Fourier analysis results. The method behaves similarly to other partially consistent methods: a peak in spectral radius is obtained for square cells on the order of one mean free path (mfp), and this peak increases in value and broadens in range as the cell aspect ratio increases. The peak spectral radius stays bounded well below unity except for large aspect ratios. Based on these results and results from previous researchers, we conclude that the method's preconditioner should work quite well within Krylov solvers. Further, its definition in terms of single-cell transport methods offers an attractive implementation advantage (a single block of coding to handle all DFEM and related methods).
