Robust and Efficient Multilevel-ILU Preconditioned Newton-GMRES for Incompressible Navier-Stokes
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We introduce a new preconditioned Newton-GMRES method for solving the nonlinear systems arising from incompressible Navier-Stokes equations. When the Reynolds number is relatively high, these systems often involve millions of degrees of freedom (DOFs), and the nonlinear systems are difficult to converge, partially due to their saddle-point structure. In this work, we propose to alleviate these issues by introducing a new preconditioned nonlinear solver called HILUNG, or Hierarchical Incomplete-LU preconditioned Newton-Gmres method. A key novelty of HILUNG is to incorporate an accurate and stable multilevel preconditioner called HILUCSI, which is particularly effective for solving saddle-point problems. HILUCSI enables robust and rapid convergence of the inner iterations in Newton-GMRES. We also introduce physics-aware sparsifying operators, adaptive refactorization and thresholding, and iterative refinement, to improve efficiency without compromising robustness. We show that HILUNG can robustly solve the standard 2D driven-cavity problem with Re 5000, while other nonlinear solvers failed to converge at Re 1000 with a similar configuration. HILUNG also improved the efficiency over another state-of-the-art multilevel ILU preconditioner and a multi-threaded direct solver by more than an order of magnitude for the 3D flow-over-cylinder problem with one million DOFs and enabled the efficient solution with about ten million DOFs using only 60GB of RAM while others fail due to nonrobustness or memory limitation.
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