A non-negative moment-preserving spatial discretization scheme for the linearized Boltzmann transport equation in 1-D and 2-D Cartesian geometries Peter G. Maginot, Jim E. Morel ⇑ , Jean C. Ragusa Department of Nuclear Engineering, Texas A&M University, College Station, TX 77843, USA article info Article history: Received 24 August 2011 Received in revised form 8 March 2012 Accepted 12 June 2012 Available online 26 June 2012 Keywords: Strictly non-negative closure Discrete ordinates method Radiation transport Discontinuous finite elements abstract We present a new nonlinear spatial finite-element method for the linearized Boltzmann transport equation with S n angular discretization in 1-D and 2-D Cartesian geometries. This method has two central characteristics. First, it is equivalent to the linear-discontinuous (LD) Galerkin method whenever that method yields a strictly non-negative solution. Sec- ond, it always satisfies both the zeroth and first spatial moment equations. Because it yields the LD solution when that solution is non-negative, one might interpret our method as a classical fix-up to the LD scheme. However, fix-up schemes for the LD equations derived in the past have given up solution of the first moment equations when the LD solu- tion is negative in order to satisfy positivity in a simple manner. We present computational results comparing our method in 1-D to the strictly non-negative linear exponential-dis- continuous method and to the LD method. We present computational results in 2-D com- paring our method to a recently developed LD fix-up scheme and to the LD scheme. It is demonstrated that our method is a valuable alternative to existing methods. Ó 2012 Elsevier Inc. All rights reserved. 1. Introduction The linearized Boltzmann transport equation describes the transport of radiation or particles that interact with a back- ground medium but have no self interactions. Examples of such particles include neutrons and gamma rays. The S n or dis- crete-ordinates method is perhaps the most popular and widely used angular discretization for this equation. Spatial discretization methods for the S n equations can produce negative angular flux solutions, which are clearly non-physical. Such negativities can arise in 1-D Cartesian geometry calculations only in optically thick cells but, in multidimensional Cartesian geometries, negativities can also arise in voids. Characteristic methods based upon polynomial scattering source represen- tations of degree greater than zero, such as the Linear Characteristic method (LC) are always non-negative in 1-D geometry as long as the projected scattering source remains non-negative. In 2-D such methods are always positive as long as the pro- jected scattering source and projected outflow boundary fluxes remain positive [1]. The Step Characteristic method (SC), which is based upon a constant scattering source representation [1], is strictly positive and formally second-order accurate, but it can be particularly inaccurate in multidimensional calculations due to a high degree of numerical diffusion resulting from the constant dependence of the angular flux that is assumed on cell faces. Furthermore, unlike most higher-order char- acteristic methods, the step method possesses neither the intermediate [2] nor the thick diffusion limit [3,4]. This can make it require an over-resolved mesh for even mildly diffusive problems. 0021-9991/$ - see front matter Ó 2012 Elsevier Inc. All rights reserved. http://dx.doi.org/10.1016/j.jcp.2012.06.018 ⇑ Corresponding author. E-mail addresses: pmaginot@tamu.edu (P.G. Maginot), morel@tamu.edu (J.E. Morel), jean.ragusa@tamu.edu (J.C. Ragusa). Journal of Computational Physics 231 (2012) 6801–6826 Contents lists available at SciVerse ScienceDirect Journal of Computational Physics journal homepage: www.elsevier.com/locate/jcp