Numerical Analysis Seminar

Wednesday, November 28, 2007

Abstract

The transfer of heat by radiation ("radiative transfer") plays an important role in many physical systems, especially those with high temperatures. Some of these systems contain regions that are large compared to the average distance that a photon travels before it is absorbed, and in these regions the photon-matter interactions can be dominated by absorption and emission of photons. In the interior of such regions, the analytic solution of the Boltzmann equation that describes the photon distribution limits to the solution of a (much simpler) diffusion equation. In this talk I will review this asymptotic limit and then address the behavior of discrete transport solutions in the same limit, concentrating mainly on Discontinuous Galerkin Finite Element (DG) discretizations but mentioning other related discretizations as well. The entire family of DG discretizations is amenable to one general analysis in this asymptotic limit. Some DG methods fail dramatically in this limit while others perform quite well. There are two simple properties of the test (or "weighting") functions that distinguish between these groups of methods. Some DG methods perform well even when transport boundary layers are not resolved by the spatial mesh. In every detail, the predictions of the asymptotic theory agree with numerical tests. Most of the material for the talk is taken from the references below.

M. L. Adams, "Discontinuous Finite Element Methods in Thick Diffusive Problems" Nucl. Sci. Eng., 137, 298-333 (2001).

M. L. Adams, T. A. Wareing, and W. F. Walters, "Characteristic Methods in Thick Diffusive Problems" Nucl. Sci. Eng., 130, 18-46 (1998).

M. L. Adams and P. F. Nowak, "Asymptotic Analysis of a Method for Time- and Frequency-Dependent Radiative Transfer" J. Comput. Physics, 146, 366-403 (1998).