Brendan Sheehan AbstractPhD, May 2007
Multigrid Methods for Isotropic Neutron Transport
Advisor: Thoma A. Manteuffel
Two multigrid algorithms for isotropic neutron transport are presented in x-y geometry. Both methods discretize the problem with discrete ordinates in angle and corner balance finite differencing in space.
One algorithm is a spatial multigrid routine, with spatial smoothing accomplished by four color block Jacobi relaxation, where the diagonal blocks correspond to four-cell blocks on the spatial grid. A bi-linear interpolation operator and its transpose are used for the grid transfer operators. This method is designed to give fast convergence on diffusive domains without using transport sweeps. Numerical results are presented for homogeneous and heterogeneous materials.
The other algorithm coarsens the problem in angle before coarsening in space. Like Diffusion Synthetic Acceleration (DSA), this algorithm employs transport sweeps, followed by a correction from a coarse angular subspace. While DSA uses a Galerkin P1 closure to form the coarse-grid system, the method presented here uses a scaled least squares minimization principle. The goal of the minimization principle is to avoid the problems of consistent differencing associated with DSA. The algorithm is viewed as a two-grid scheme in angle, where sweeps constitute relaxation, and the coarse-grid system is obtained from the scaled least squares minimization principle. The coarse-grid system is subsequently solved with standard spatial multigrid. Numerical results are presented.