An efficient space-angle subgrid scale discretisation of the neutron transport equation. (August 2016)

• Record Type: Journal Article
• Authors: Buchan, A.G.; Pain, C.C.
• Is Part Of: Annals of nuclear energy. Volume 94(2016:Aug.)
• Page Start: 440
• Page End: 450
• Publication Date: 2016-08
• Subjects: Finite elements -- Subgrid scale -- Boltzmann transport equation -- Spherical harmonics; Nuclear energy -- Periodicals; Nuclear engineering -- Periodicals; 621.4805
• DOI: 10.1016/j.anucene.2016.03.012
• Languages: English
• ISSNs: 0306-4549

Abstract

Highlights: A new space-angle subgrid scale discretisation of the Boltzmann transport equation. Efficient integration of the spherical harmonic approximation for neutron transport. Accurate solutions obtained for transport within a range of material types. Abstract: This article presents a new efficient space-angle subgrid scale formulation (SGS) for the space-angle phase-space discretisation of the Boltzmann transport equation (BTE). It applies both coarse and fine scale discretisations to the space-angle phase-spaces, for which the fine scale solutions yield more accurate but more expensive solutions compared to those of the coarse scale approach. Accuracy and efficiency are maintained by integrating the fine scale components within the coarse scale formulation. This results in a method that generates linear systems the same size as those that would be generated by the coarse scale system. The formulation is based on using finite elements (FEs) to represent the spatial variable of the BTE, whilst spherical harmonics are employed to represent the particles' directions. In space, the fine and coarse scales are represented through discontinuous and continuous linear finite elements. In angle, the scales are represented by the full spherical harmonics (coarse scale), and a partitioning of these basis functions into subsets that couple only between themselves (fine scale). Whilst the spatial discretisation has been previously documented, it is the angular treatment that is critical for efficient solutions. This is particularly the case for large angular expansion sizes involving angular discretisations that couple moments through the streaming term (i.e. not discrete ordinate schemes), where it is shown here that the computing costs are reduced to O ( M ) from O ( M 3 ), where M denotes the angular expansion size. It is shown how the method increases efficiency of the solving process through the improved efficiency of matrix vector multiplication which forms the most expensive component of a Krylov based solver. This matrix vector product is also computed without the matrix being stored in memory which would be a requirement for large scale radiation transport calculations. The method's capabilities are demonstrated by solving fixed source problems involving a range of material types – from optically thin to opaque. In all cases the new method yields accurate and stable solutions.