DSA Preconditioning for DG discretizations of $S_{N}$ transport and High-Order curved meshes

This paper derives and analyzes new diffusion synthetic acceleration (DSA) preconditioners for the SN transport equation when discretized with a high-order (HO) discontinuous Galerkin (DG) discretization. DSA preconditioners address the need to accelerate solving SN transport when the mean free path $\varepsilon$ of particles is small and the condition number of the SN transport equation scales like $O(\varepsilon^{-2})$. By expanding the SN transport operator in $\varepsilon$ and employing a singular matrix perturbation analysis, we derive a DSA matrix that reduces to the symmetric interior penalty (SIP) DG discretization of diffusion when the mesh is first-order and the total opacity is constant. We prove that preconditioning HO DG SN transport with the SIP DSA matrix results in an $O(\varepsilon)$ perturbation of the identity, and fixed-point iteration therefore converges rapidly for optically thick problems. However, the SIP DSA matrix is conditioned like $O(\varepsilon^{-1})$, making it difficult to invert for small $\varepsilon$. We further derive a new two-part, additive DSA preconditioner based on a continuous discretization of diffusion-reaction, which has a condition number independent of $\varepsilon$, and prove that this DSA variant has the same theoretical efficiency as the SIP DSA preconditioner in the optically thick limit. The analysis is extended to the case of HO (curved) meshes, where so-called mesh cycles can result from elements both being upwind of each other. In particular, we prove that performing two additional transport sweeps between DSA steps yields the same theoretical conditioning of fixed-point iterations as in the cycle-free case. Theoretical results are validated by numerical experiments on HO, highly curved meshes generated from an ALE hydrodynamics code, where the additional inner sweeps between DSA steps offer up to a 4x reduction in total sweeps.
Comment: 26 pages, Accepted to SISC