Multigroup diffusion preconditioners for multiplying fixed-source transport problems

Several preconditioners based on multigroup diffusion are developed for application to multiplying fixed-source transport problems using the discrete ordinates method. By starting from standard, one-group, diffusion synthetic acceleration (DSA), a multigroup diffusion preconditioner is constructed that shares the same fine mesh as the transport problem. As a cheaper but effective alternative, a two-grid, coarse-mesh, multigroup diffusion preconditioner is examined, for which a variety of homogenization schemes are studied to generate the coarse mesh operator. Finally, a transport-corrected diffusion preconditioner based on application of the Newton–Shulz algorithm is developed. The results of several numerical studies indicate the coarse-mesh, diffusion preconditioners work very well. In particular, a coarse-mesh, transport-corrected, diffusion preconditioner reduced the computational time of multigroup GMRES by up to a factor of 17 and outperformed best-case Gauss–Seidel results by over an order of magnitude for all problems studied.