The Asymptotic Diffusion Limit of Numerical Schemes for the S_N Transport Equation

Abstract

The S_N transport equation asymptotically tends to an equivalent diffusion equation in the limit of optically thick systems with small absorption and sources. A spatial discretization of the S_N equation is of practical interest if it possesses the optically thick diffusion limit. Such a numerical scheme will yield accurate solutions for diffusive problems if the spatial mesh size is thin with respect to a diffusion length, whereas the mesh cells are thick in terms of a mean free path. Many spatial discretization methods have been developed for the S_N transport equation, but only a few of them can obtain the thick diffusion limit under certain conditions. This paper presents a theoretical result that simply states that the mesh size required for a finite difference scheme to attain the diffusion limit is ${ }_{}{\epsilon }^{1/k}h$, where $k$ is the order of accuracy of spatial discretization, $h$ is the “diffusion” mesh size that can be many mean free paths thick, and $\epsilon$ is a small positive scaling parameter that can be defined as the ratio of a particle mean free path to a characteristic scale length of the system. Numerical results for schemes such as the Diamond Difference method, Step Characteristic method, Step Difference method, Second-Order Upwind method, and Lax-Friedrichs Weighted Essentially Non-Oscillatory method of the third order (LF-WENO3) are presented that demonstrate the validity and accuracy of our analysis.

Keywords:

• S_N transport equation
• diffusion limit
• asymptotic analysis

Acknowledgments

The author would like to thank Yulong Xing and Zeyun Wu for helpful discussions during the preparation of this work and anonymous reviewers for helpful comments and suggestions.