# Neutron transport criticality calculations using a parallel monolithic multilevel Schwarz preconditioner together with a nonlinear diffusion acceleration method

## TL;DR

This paper introduces a parallel monolithic multilevel Schwarz preconditioner combined with a nonlinear diffusion acceleration method to efficiently solve large-scale neutron transport criticality problems on supercomputers, significantly reducing computation time.

## Contribution

It develops a scalable monolithic multilevel Schwarz preconditioner with a subspace-based coarsening algorithm for neutron transport calculations, improving efficiency on large unstructured meshes.

## Key findings

- Preconditioner is twice as fast as traditional methods on thousands of cores.
- Effective handling of unstructured meshes with billions of unknowns.
- Demonstrates scalability and efficiency in large-scale neutron transport simulations.

## Abstract

The multigroup neutron transport criticality calculations using modern supercomputers have been widely employed in a nuclear reactor analysis for studying whether or not a system is self-sustaining. However, the design and development of efficient parallel algorithms for the transport criticality calculations is challenging especially when the number of processor cores is large and an unstructured mesh is adopted. In particular, both the compute time and memory usage have to be carefully taken into consideration due to the dimensionality of the neutron transport equations. In this paper, we study a monolithic multilevel Schwarz preconditioner for the transport criticality calculations based on a nonlinear diffusion acceleration (NDA) method. We propose a monolithic multilevel Schwarz method that is capable of efficiently handling the systems of linear equations for both the transport system and the diffusion system. However, in the multilevel method, algebraically constructing coarse spaces is expensive and often unscalable. We study a subspace-based coarsening algorithm to address such a challenge by exploring the matrix structures of the transport equations and the nonlinear diffusion equations. We numerically demonstrate that the monolithic multilevel preconditioner with the subspace-based coarsening algorithm is twice as fast as that equipped with an unmodified coarsening approach on thousands of processor cores for an unstructured mesh neutron transport problem with billions of unknowns.