Moment Analysis of Angular APproximation Methods for Time-Dependent Radiation Transport

Abstract

We extend moment analysis, a technique developed for investigating the accuracy of discrete-ordinates spatial discretization schemes, to time-dependent radiation transport and apply it to several angular approximation methods. Specifically, we examine the diffusion approximation, the P _1/3 approximation, and three time-dependent generalizations of the simplified P_Napproximation: the SP ₂ , SP ₃ , and SSP ₃ approximations. We show that all of the these methods preserve the correct flux-weighted average of x but not the correct flux-weighted average of (x-x_a)², where x is the spatial variable and x_ais an arbitrary point. We also demonstrate that, for general cross sections and large elapsed time, the error in the flux-weighted average of (x-x_a)² is smallest in magnitude for the SP ₂ and approximations. In addition, we present a simple improvement to the SP ₂ approximation that allows this method to produce the correct flux-weighted average of (x-x_a)². We present numerical results that test this analysis. From these results, we find that the angular approximation methods with the most accurate solutions also have the most accurate values for the flux-weighted average of (x-x_a)². In particular, the SP ₂ and SP ₃ approximations are two of the most accurate methods at large elapsed times, while the improved SP ₂ approximation is one of the most accurate methods at all times. We also observe, however, that an accurate value for the flux-weighted average of (x-x_a)² is not always accompanied by an accurate solution. Consequently, we conclude that an accurate flux-weighted average of (x-x_a)² is a necessary rather than sufficient condition for an overall accurate angular approximation method.

Keywords:

• Time-dependent radiation transport
• Moment analysis
• Diffusion approximation
• P _1/3 approximation
• Simplified P_N approximation

Acknowledgments

The work of the first author (J.D.D.) was performed under U.S. government contract DE-AC52-06NA25396 for Los Alamos National Laboratory, which is operated by Los Alamos National Security, LLC, for the U.S. Department of Energy.