A Highly Parallel Multilevel Newton--Krylov--Schwarz Method with Subspace-Based Coarsening and Partition-Based Balancing for the Multigroup Neutron Transport Equation on Three-Dimensional Unstructured Meshes

Abstract

The multigroup neutron transport equation is crucial for studying the motion of neutrons and their interaction with materials. Numerical simulation of the multigroup neutron transport equation is computationally challenging because the equation is defined on a high-dimensional phase space, the computational spatial domain is complex, and the materials are heterogeneous. A scalable parallel solver is required to address such a challenge. In this paper, we study a highly parallel Newton--Krylov--Schwarz (NKS) method consisting of a Newton-based eigenvalue solver, a Krylov subspace method, and a novel multilevel Schwarz preconditioner. The multilevel method is one of the most popular preconditioners for accelerating neutron transport calculations, but the construction of coarse spaces can be expensive and often unscalable when a large number of processors is used. We propose a novel matrix coarsening algorithm in which a multilevel hierarchy is constructed using a single-component matrix instead of the full matrix of the neutron transport equation. This new coarsening algorithm is referred to as “subspace-based coarsening.” Above 8,000 processors, we show a 13x enhancement in multilevel preconditioner setup time when using the subspace-based coarsening method. A partition-based balancing strategy is studied to enhance the parallel efficiency of the NKS algorithm by equalizing the work for each processor. A hierarchical mesh partitioning algorithm is employed to generate a large number of submeshes while minimizing off-node communication. We demonstrate that the proposed algorithm is scalable with more than 10,000 processors for a realistic application on three-dimensional unstructured meshes with a few billion degrees of freedom. Neutron transport calculations using the improved NKS algorithm are twice as fast as those based on the unmodified NKS solver when over 8,000 processors are employed.