Node Numbering for Stabilizing Preconditioners Based on Incomplete LU Decomposition

Matrix-reordering strategies that reduce the occurrence of instabilities often associated with preconditioners based on incomplete LU decomposition are presented. Several example matrices, extracted from turbulent-flow simulations, and from structural mechanics, are used to demonstrate the occurrence of instabilities and to investigate the underlying cause. The current work demonstrates that appropriate schemes for reordering unknowns can provide a very effective strategy for stabilizing the preconditioner, thereby enabling a Krylov-subspace method to solve linear systems that were heretofore unsolvable. Several new numbering strategies are presented, one of which provides a tunable parameter to control the bandwidth of the renumbered matrix. Using the discussed methodology greatly increases the robustness and reliability of the simulations.

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