Space-angle-energy multigrid methods for Sn discretizations of the multi-energetic Boltzmann equation
Article first published online: 18 OCT 2011
Copyright © 2011 John Wiley & Sons, Ltd.
Numerical Linear Algebra with Applications
Volume 19, Issue 4, pages 773–795, August 2012
How to Cite
Lee, B. (2012), Space-angle-energy multigrid methods for Sn discretizations of the multi-energetic Boltzmann equation. Numer. Linear Algebra Appl., 19: 773–795. doi: 10.1002/nla.808
- Issue published online: 10 JUL 2012
- Article first published online: 18 OCT 2011
- Manuscript Accepted: 31 JUL 2011
- Manuscript Revised: 28 JUN 2011
- Manuscript Received: 14 MAY 2010
- Boltzmann equation;
- multigrid method;
- Sn discretizations;
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.