A parallel monolithic multilevel Schwarz preconditioner for the neutron transport criticality calculations with a nonlinear diffusion acceleration method
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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 an efficient parallel algorithm for the transport criticality calculations is a challenging task especially when the number of processor cores is large and the unstructured mesh is adopted since both the compute time and the memory usage need to be taken into consideration. In this paper, we study a monolithic multilevel Schwarz preconditioner for the transport criticality calculations using the nonlinear diffusion acceleration (NDA). In NDA, the linear systems of equations arising from the discretizations of the nonlinear diffusion equations and the transport equations need to be efficiently solved. To achieve this goal, we propose a monolithically coupled approach equipped with several important ingredients; e.g., subspace-based coarsening, aggressive coarsening and strength matrix thresholding. The proposed monolithic multilevel method is capable of efficiently handling the linear systems of equations for both the transport system and the diffusion system. In the multilevel method, the construction of coarse spaces is nontrivial and expensive. We propose a subspace-based coarsening algorithm to resolve this issue 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 a full space based coarsening approach on thousands of processor cores for an unstructured mesh neutron transport problem with billions of unknowns.
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