Augmented Lagrangian preconditioner for large-scale hydrodynamic stability analysis

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 (Benzi et al., 2011). 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 2048 processes is obtained. (C) 2019 The Author(s). Published by Elsevier B.V.