Mitigating Spatial Error in the Iterative Quasi–Monte Carlo (iQMC) Method for Neutron Transport Simulations with Linear Discontinuous Source Tilting and Effective Scattering and Fission Rate Tallies

Abstract

The iterative Quasi–Monte Carlo (iQMC) method is a recently proposed method for neutron transport simulations. iQMC can be viewed as a hybrid between deterministic iterative techniques, Monte Carlo simulation, and Quasi–Monte Carlo techniques. iQMC holds several algorithmic characteristics that make it desirable for high-performance computing environments, including an $\mathit{O}(N^{-1})$ convergence scheme, a ray-tracing transport sweep, and a highly parallelizable nature similar to analog Monte Carlo. While there are many potential advantages of using iQMC, there are also inherent disadvantages, namely, the spatial discretization error introduced from the use of a mesh across the domain.

This work introduces two significant modifications to iQMC to help reduce the spatial discretization error. The first is an effective source transport sweep, whereby the source strength is updated on the fly via an additional tally. This version of the transport sweep is essentially agnostic to the mesh, material, and geometry. The second is the addition of a history-based linear discontinuous source tilting method. Traditionally, iQMC utilizes a piecewise constant source in each cell of the mesh. However, through the proposed source tilting technique, iQMC can utilize a piecewise linear source in each cell and reduce spatial error without refining the mesh.

Numerical results are presented from the two-dimensional (2-D) C5G7 and Takeda-1 k-eigenvalue benchmark problems. The results show that the history-based source tilting significantly reduces error in global tallies and the eigenvalue solution in both benchmarks. Through the effective source transport sweep and linear source tilting, iQMC was able to converge the eigenvalue from the 2-D C5G7 problem to less than 0.04% error on a uniform Cartesian mesh with only 204 × 204 cells.

Keywords:

• Neutron transport
• Monte Carlo methods
• Quasi–Monte Carlo
• Krylov linear solvers
• source tilting

Acknowledgments

A special thank you is extended to John Tramm for his consultation in developing k-eigenvalue capabilities for iQMC. We also gratefully acknowledge the computing resources provided by Livermore Computing at Lawrence Livermore National Laboratory. Finally, thanks to the entire MC/DC development team for providing an excellent MC test bed environment.

Disclosure Statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

This work was funded by CEMeNT, a PSAAP-III project funded by the U.S. Department of Energy (DE-NA003967), and supported by National Science Foundation grants DMS-1745654 and DMS-1906446.