An Efficient Low-order Integral Transport Matrix Method for Optically Thick Problems Implemented in Griffin

In this study, a new low-order integral transport matrix method (LITMM) based on the truncation of small-magnitude elements in the integral transport matrix (ITM) for DFEM-SN discretizations of the neutron transport equation has been developed. The integral transport matrix method (ITMM) enables solving fixed source problems by constructing and solving a global linear system without requiring source iterations. However, direct solution of the linear system is expensive since it involves a dense matrix, and the cost of constructing the ITM is immense. In the current work, we developed a dynamic truncation scheme that truncates ITM elements smaller than a user-specified threshold on the fly, thus reducing the construction cost of the full ITM, and leading to a sparse low-order operator. The sparse structure of the new matrix allows for the use of Krylov subspace solvers, thereby reducing the expense of solving the global linear system. We implemented the LITMM in the particle transport code Griffin for solving one-speed, fixed source problems as a solver and as a preconditioner for the Richardson iterations with isotropic scattering. We utilized a three-dimensional cubic geometry problem comprised of two regions for testing the method by altering the optical thickness and truncation threshold for constructing the LITM. We compared the LITMM-based results with Griffin's DFEM-SN solutions on the same mesh. The results showed that the LITMM is an accurate low-order method that generates high-order equivalent solutions, and it is a computationally efficient preconditioner for Richardson iterations for optically thick problems.