Augmented Lagrangian Preconditioner for Large-Scale Hydrodynamic Stability Analysis
Résumé
Hydrodynamic linear stability analysis of large-scale three-dimensional configurations is usually performed with a "time-stepping" approach, based on the adaptation of existing solvers for the unsteady incompressible Navier-Stokes equations. We propose instead to solve the nonlinear steady equations with the Newton method and to determine the largest growth-rate eigenmodes of the linearized equations using a shift-and-invert spectral transformation and a Krylov-Schur algorithm. The solution of the shifted linearized Navier-Stokes problem, which is the bottleneck of this approach, is computed via an iterative Krylov subspace solver preconditioned by the modified augmented Lagrangian (mAL) preconditioner [12]. The well-known efficiency of this preconditioned iterative strategy for solving the real linearized steady-state equations is assessed here for the complex shifted linearized equations. The effect of various numerical and physical parameters is investigated numerically on a two-dimensional flow configuration, confirming the reduced number of iterations over state-of-the-art steady-state and time-stepping-based preconditioners. A parallel implementation of the steady Navier-Stokes and eigenvalue solvers, developed in the FreeFem++ language , suitably interfaced with the PETSc/SLEPc libraries, is described and made openly available to tackle three-dimensional flow configurations. Its application on a small-scale three-dimensional problem shows the good performance of this iterative approach over a direct LU factorization strategy, in regards of memory and computational time. On a large-scale three-dimensional problem with 75 million unknowns, a 80% parallel efficiency on 256 up to 2,048 processes is obtained.
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