Fully Consistent Diffusion Synthetic Acceleration of Linear Discontinuous SN Transport Discretizations on Unstructured Tetrahedral Meshes

Abstract We recently presented a method for efficiently solving linear discontinuous discretizations of the two-dimensional P1 equations on rectangular meshes. The linear system was efficiently solved with Krylov iterative methods and a novel two-level preconditioner based on a linear continuous finite element discretization of the diffusion equation. Here, we extend the preconditioned solution method to three-dimensional, unstructured tetrahedral meshes. Solution of the P1 equations forms the basis of a diffusion synthetic acceleration (DSA) scheme for three-dimensional SN transport calculations with isotropic scattering. The P1 equations and the transport equation are both discretized with isoparametric linear discontinuous finite elements so that the DSA method is fully consistent. Fourier analysis in three dimensions and computational results show that this DSA scheme is stable and very effective. The fully consistent method is compared to other “partially consistent” DSA schemes. Results show that the effectiveness of the partially consistent schemes can degrade for skewed or optically thick mesh cells. In fact, one such scheme can degrade to the extent of being unstable even though it is both unconditionally stable and effective on rectangular grids. Results for a model application show that our fully consistent DSA method can outperform the partially consistent DSA schemes under certain circumstances.

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