Diffusion Synthetic Acceleration for Heterogeneous Domains, Compatible with Voids

Abstract

A standard approach to solving the S${ }_N$ transport equations is to use source iteration with diffusion synthetic acceleration (DSA). Although this approach is widely used and effective on many problems, there remain some practical issues with DSA preconditioning, particularly on highly heterogeneous domains. For large-scale parallel simulation, it is critical that both (a) preconditioned source iteration converges rapidly and (b) the action of the DSA preconditioner can be applied using fast, scalable solvers, such as algebraic multigrid (AMG). For heterogeneous domains, these two interests can be at odds. In particular, there exist DSA diffusion discretizations that can be solved rapidly using AMG, but they do not always yield robust/fast convergence of the larger source iteration. Conversely, there exist robust DSA discretizations where source iteration converges rapidly on difficult heterogeneous problems, but fast parallel solvers like AMG tend to struggle applying the action of such operators. Moreover, very few current methods for the solution of deterministic transport are compatible with voids. This paper develops a new heterogeneous DSA preconditioner based on only preconditioning the optically thick subdomains. The resulting method proves robust on a variety of heterogeneous transport problems, including a linearized hohlraum mesh related to inertial confinement fusion. Moreover, the action of the preconditioner is easily computed using $\mathcal{O}(1)$ AMG iterations, convergence of the transport iteration typically requires 2 to 5× fewer iterations than current state-of-the-art “full” DSA, and the proposed method is trivially compatible with voids. On the hohlraum problem, rapid convergence is obtained by preconditioning less than 3% of the mesh elements with five to ten AMG iterations.

Keywords:

• Transport
• discrete ordinates
• high order
• sweep, unstructured

Acknowledgments

This material is based upon work supported by the U.S. Department of Energy, National Nuclear Security Administration, under award number DE-NA0002376. This work was performed under the auspices of the U.S. Department of Energy by Lawrence Livermore National Laboratory under contract DE-AC52-07NA27344 (LLNL-JRNL-802400).

Notes

^a Recent work in Ref. 15 has seen success applying similar ideas to precondition the radiation diffusion equations by only solving the diffusion discretization on a physical subdomain.

^b Unfortunately, the only available copy of Ref. 19 appears to be a low-resolution, scanned-in copy, which is difficult to read.

^c Using $\mathrm{\Sigma }\sim {\sigma }_s(\mathbf{x})$ is more common in literature; here, we use $\mathrm{\Sigma }\sim {\sigma }_t(\mathbf{x})$ as derived in Ref. 6. In any case, they can be shown to have the same asymptotic efficiency in the thick limit.

^d Note that for an optimized implementation, it is likely preferable to only discretize and construct the DSA matrix on the thick subdomain, particularly if the thick region is very small (for example, see Sec. IV.B). However, for many finite element libraries or existing code bases, this ends up being more intrusive to implement.