$α$Setup-PCTL: An Adaptive Setup-Based Two-Level Preconditioner for Sequence of Linear Systems of Three-Temperature Energy Equations (Communications in Computational Physics, 2023)
The iterative solution of the sequence of linear systems arising from three-temperature (3-T) energy equations is an essential component in the numerical simulation of radiative hydrodynamic (RHD) problem. However, due to the complicated application features of the RHD problems, solving 3-T linear systems with classical preconditioned iterative techniques is challenging. To address this difficulty, a physical-variable based coarsening two-level (PCTL) preconditioner has been proposed by dividing the fully coupled system into four individual easier-to-solve subsystems. Despite its nearly optimal complexity and robustness, the PCTL algorithm suffers frompoor efficiency because of the overhead associated with the construction of setup phaseand the solution of subsystems. Furthermore, the PCTL algorithm employs a fixedstrategy for solving the sequence of 3-T linear systems, which completely ignores thedynamically and slowly changing features of these linear systems. To address theseproblems and to efficiently solve the sequence of 3-T linear systems, we propose anadaptive two-level preconditioner based on the PCTL algorithm, referred to as $α$Setup-PCTL. The adaptive strategies of the $α$Setup-PCTL algorithm are inspired by those of $α$Setup-AMG algorithm, which is an adaptive-setup-based AMG solver for sequenceof sparse linear systems. The proposed $α$Setup-PCTL algorithm could adaptively employ the appropriate strategies for each linear system, and thus increase the overallefficiency. Numerical results demonstrate that, for 36 linear systems, the $α$Setup-PCTLalgorithm achieves an average speedup of 2.2, and a maximum speedup of 4.2 whencompared to the PCTL algorithm.
Keywords: Sequence of linear systems; sparse linear solver; preconditioning methods; radiation hydrodynamics simulation.