Abstract
Effective preconditioning of neutron diffusion problems is necessary for the development of efficient DSA schemes for neutron transport problems. This paper uses P-multigrid techniques to expand two preconditioners designed to solve the MIP diffusion neutron diffusion equation with a discontinuous Galerkin (DG-FEM) framework using first-order elements. These preconditioners are based on projecting the first-order DG-FEM formulation to either a linear continuous or a constant discontinuous FEM system. The P-multigrid expansion allows the preconditioners to be applied to problems discretised with second and higher-order elements. The preconditioning algorithms are defined in the form of both a V-cycle and W-cycle and applied to solve challenging neutron diffusion problems. In addition a hybrid preconditioner using P-multigrid and AMG without a constant or continuous coarsening is used. Their performance is measured against a computationally efficient standard algebraic multigrid preconditioner. The results obtained demonstrate that all preconditioners studied in this paper provide good convergence with the continuous method generally being the most computationally efficient. In terms of memory requirements the preconditioners studied significantly outperform the AMG.
Highlights
A major focus in the development of efficient computational methods to solve SN neutron transport equations is that of diffusion synthetic acceleration (DSA) (Larsen, 1984)
This paper studies the preconditioning of a discontinuous Galerkin (DG) diffusion scheme developed by Wang and Ragusa, the modified interior penalty (MIP) formulation, which has been shown to be effective for use within DSA (Wang and Ragusa, 2010)
For scalar neutron flux f, macroscopic removal cross-section Sr, diffusion coefficient D and neutron source S the steady state mono-energetic form of the neutron diffusion equation at position r is: VDðrÞVfðrÞ À SrðrÞfðrÞ þ SðrÞ 1⁄4 0 (1). This equation is discretised for DG-FEM using the modified interior penalty (MIP) scheme (Wang and Ragusa, 2010), which is a variation of the symmetric interior penalty (SIP) (Arnold et al, 2002; Di Pietro and Ern, 2012)
Summary
A major focus in the development of efficient computational methods to solve SN neutron transport equations is that of diffusion synthetic acceleration (DSA) (Larsen, 1984). In (O'Malley et al, 2017) two hybrid multilevel preconditioning methods based on methods developed in (Dobrev, 2007) and (Van. Slingerland and Vuik, 2012) are presented which were shown to effectively accelerate the solution of discontinuous neutron diffusion problems. Slingerland and Vuik, 2012) are presented which were shown to effectively accelerate the solution of discontinuous neutron diffusion problems These preconditioners worked by creating a coarse space of either linear continuous or constant discontinuous finite elements. From this coarse space a preconditioning step of an algebraic multigrid (AMG) preconditioner was used to provide a coarse correction, leading to a hybrid multilevel scheme
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