Space-angle-energy multigrid methods for Sn discretizations of the multi-energetic Boltzmann equation


B. Lee, Computational Sciences & Mathematics Division, Pacific Northwest National Laboratory, Richland, WA.



In a recent article, the author presented several improved multiple-coarsening/semi-coarsening schemes for Sn discretizations of the Boltzmann transport equation, improved over the original multiple-coarsening/semi-coarsening schemes. These improvements were derived from detailed space-angle descriptions of the near-nullspace components of the integral equation operator. In this paper, we use the techniques of this article to derive a description of the near-nullspace components of the multi-energetic Boltzmann equation, and use this description to develop a space-angle-energy multigrid method for this equation. This multigrid method is a scheme for solving a high-dimensional equation: for spatial 3-d, the equation is 6-d; for spatial 2-d, the equation is 5-d. This method is more robust and efficient than both the commonly used block Gauss-Seidel iteration that requires solving mono-energetic Boltzmann equations, and the improved multiple-coarsening/semi-coarsening schemes simultaneously applied to all the energy groups. Numerical experiments applied to multi-energetic equations with isotropic scattering cross-sections that simulate Compton-like scattering and fission, as well as anisotropic scattering cross-sections, are performed to demonstrate the effectiveness of the new scheme. Copyright © 2011 John Wiley & Sons, Ltd.