Diffusion Synthetic Acceleration Coupled with Fine Mesh Rebalance for Multigroup Transport Calculations with Tetrahedral Meshes

A diffusion synthetic acceleration (DSA) technique for the SN transport equation discretized with the linear discontinuous expansion method with subcell balance (LDEM-SCB) on unstructured tetrahedral meshes is presented. The LDEM-SCB scheme solves the transport equation with the discrete ordinates method by using the subcell balance and linear discontinuous expansion of the flux. Discretized DSA equations are developed by consistently discretizing the continuous diffusion equation with the LDEM-SCB method. In addition, a fine mesh rebalance (FMR) method is devised to accelerate the discretized diffusion equation coupled with the preconditioned conjugate gradient (CG) method. The DSA method is applied to various test problems to show its effectiveness in speeding up the iterative convergence of the transport equation. The results show that the DSA method gives small spectral radii for the tetrahedral meshes having various aspect ratios even in highly scattering dominant mediums (scattering to total cross-section ratio of 0.9999) for the homogeneous test problems. The numerical tests for the homogeneous and heterogeneous problems show that DSA with FMR (with preconditioned CG) gives significantly higher speedups and robustness than the one with the Gauss-Seidel (GS)-like iteration. In this work, a three-dimensional Fourier analysis of the FMR acceleration of the LDEMSCB discretization method for solving the neutron diffusion equation on three-dimensional tetrahedral meshes is also presented in order to theoretically understand the convergence characteristics. The Fourier analysis is performed on one and two cubes called the basic elements with the translation and reflective boundary conditions on the external boundaries. The Fourier analysis shows that the FMR method significantly accelerates the GS-like iteration to solve the LDEM-SCB discretized diffusion equation. The analytical results of the homogeneous test problem taking different aspect ratios show that the convergence of FMR with the translation boundary conditions is less sensitive on the aspect ratio than the one with the reflective boundary conditions. In addition, the convergence characteristics of the GS-like iteration and its FMR acceleration of the heterogeneous test problems are also identified