Numerical transport theory; Final report

Abstract

The basic problem addressed in the project was that of accelerating the iterative convergence of Discrete Ordinates (S{sub N}) problems. Important previous work on this problem, much of which was done at LANL, has shown that the Diffusion Synthetic Acceleration (DSA) method can be a very effective acceleration procedure. However, in two-dimensional geometries, only the diamond differenced S{sub N} equations have been efficiently solved using DSA. This is because, for the 2-D diamond-differenced S{sub N} equations, the standard DSA procedure leads to a relatively simple discretized low-order diffusion equation that for many problems can be efficiently solved by a multigrid method. For other discretized versions of the S{sub N} equations, the standard DSA procedure leads to much more complicated discretizations of the low-order diffusion equation that have not been efficiently solved by multigrid (or other) methods. In this project, we have developed a new procedure to obtain discretized diffusion equations for DSA-accelerating the convergence of the S{sub N} equations using certain lumped discontinuous finite element spatial differencing methods. The idea is to use an asymptotic analysis for the derivation of the discretized diffusion equation. This is based on the fact that diffusion theory is an asymptotic limit of transport theory. The asymptotic analysis also shows that the schemes considered in this project are highly accurate for diffusive problems with spatial meshes that are optically thick. Specifically, we apply this DSA procedure to a lumped Linear Discontinuous (LD) scheme for slab geometry and a lumped Bilinear Discontinuous (BLD) scheme for x,y-geometry. Our theoretical and numerical results indicate that these schemes are very accurate and can be solved efficiently using the new method. We describe the concept that underlies the DSA method. We describe the basic asymptotic relationship between transport and diffusion theory.