Computational Methods in Science and Engineering

Diffusion Synthetic Acceleration Preconditioning for Discontinuous Galerkin Discretizations of $S_N$ Transport on High-Order Curved Meshes

Abstract

This paper derives and analyzes new diffusion synthetic acceleration (DSA) preconditioners for the $S_N$ transport equation when discretized with a high-order (HO) discontinuous Galerkin (DG) discretization. DSA preconditioners address the need to accelerate the $S_N$ transport equation when the mean free path $\varepsilon$ of particles is small and the condition number of the $S_N$ transport equation scales like $\mathcal{O}( \varepsilon^{-2} )$. By expanding the $S_N$ transport operator in $\varepsilon$ and employing a rigorous singular matrix perturbation analysis, we derive a DSA matrix that reduces to the symmetric interior penalty (SIP) DG discretization of the standard continuum diffusion equation when the mesh is first-order and the total opacity is constant. We prove that preconditioning the HO DG $S_N$ transport equation with the SIP DSA matrix results in an $\mathcal{O}( \varepsilon )$ perturbation of the identity, and fixed-point iteration therefore converges rapidly for optically thick problems. However, the SIP DSA matrix is conditioned like $\mathcal{O}( \varepsilon^{-1} )$, making it difficult to invert for small $\varepsilon$. We further derive a new two-part, additive DSA preconditioner based on a continuous Galerkin 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 (for a given discrete photon direction). In particular, we prove that performing two additional transport sweeps, with fixed scalar flux, in 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 a HO, highly curved two- and three-dimensional meshes that are generated from an arbitrary Lagrangian--Eulerian hydrodynamics code, where the additional inner sweeps between DSA steps offer up to a 4 x reduction in total number of sweeps required for convergence.

Keywords

  1. $S_N$ transport
  2. diffusion synthetic acceleration
  3. high-order DG

MSC codes

  1. 65Fxx
  2. 65Bxx
  3. 65Mxx
  4. 65Zxx

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Information & Authors

Information

Published In

cover image SIAM Journal on Scientific Computing
SIAM Journal on Scientific Computing
Pages: B1271 - B1301
ISSN (online): 1095-7197

History

Submitted: 14 March 2019
Accepted: 2 July 2020
Published online: 27 October 2020

Keywords

  1. $S_N$ transport
  2. diffusion synthetic acceleration
  3. high-order DG

MSC codes

  1. 65Fxx
  2. 65Bxx
  3. 65Mxx
  4. 65Zxx

Authors

Affiliations

Funding Information

Lawrence Livermore National Laboratory https://doi.org/10.13039/100006227 : DE-AC52-07NA27344, B614452, B627942
U.S. Department of Energy https://doi.org/10.13039/100000015 : DE-NA0002376

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