The diffusion synthetic acceleration (DSA) method has emerged as a powerful tool for accelerating the iterative convergence rate of discrete-ordinate transport calculations. However, in multi-dimensional geometries, only the diamond-differenced scheme has been efficiently solved by the DSA procedure. More advanced and accurate schemes, such as the discontinuous finite element schemes, have not been efficiently solved by DSA because applying the standard DSA procedure results in a large, complicated system of equations that cannot be collapsed into an efficiently solvable discrete diffusion equation. Here we present a new procedure for diffusion-accelerating certain discontinuous finite element schemes for slab and x-y geometry discrete-ordinates problems. The novel aspect of this procedure is that the discretized diffusion problem is derived from an asymptotic expansion of the discrete transport problem. The motivation for this procedure is that the resulting diffusion problem is relatively "simple" and easily solvable. The asymptotic expansion also shows that these discontinuous finite element schemes are highly accurate for diffusive problems with optically thick spatial meshes. Therefore, these schemes possess two very desirable properties: they are very accurate for all problems with optically thin meshes and diffusive problems with optically thick meshes, and they are efficiently solved by a diffusion-synthetic acceleration procedure. Specifically, we consider the conventional and lumped linear discontinuous schemes in slab geometry and a certain lumped bilinear discontinuous scheme in x-y geometry. In slab geometry, the new "asymptotic" DSA procedure is very efficient for problems that contain either isotropic scattering or linearly-anisotropic scattering. In x-y geometry, this procedure is very efficient provided the spatial cells do not have large aspect ratios and the system does not have highly anisotropic scattering. Also, the resulting discrete diffusion equation has a very simple five-point stencil with a one-point removal term and is very efficiently solved by the multigrid method. We provide numerical results that demonstrate the high level of accuracy and rapid convergence of the new methods.Ph.D.Nuclear Engineering and Scientific ComputingUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttp://deepblue.lib.umich.edu/bitstream/2027.42/105813/1/9208688.pdfDescription of 9208688.pdf : Restricted to UM users only
