Augmented Lagrangian preconditioner for large-scale hydrodynamic stability analysis

Johann Moulin,Pierre Jolivet,Olivier Marquet

doi:10.1016/j.cma.2019.03.052

Abstract

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.

Highlights

Over the past century, hydrodynamic linear stability theory was developed to understand the early stage of laminar-turbulence transition in parallel flows, such as boundary layers and shear flows [28]
The present results clearly indicate that γ 1 is an optimal value from both the solution accuracy point of view and the preconditioning efficiency point of view when considering steady solutions, as reported before [35], and leading eigenvalues
The stationary base flow as well as the eigenvalue computations involved in hydrodynamic linear stability analysis require multiple solutions of linear systems based on the Jacobian operator of the incompressible steady Navier–Stokes equations

Summary

Introduction

Hydrodynamic linear stability theory was developed to understand the early stage of laminar-turbulence transition in parallel flows, such as boundary layers and shear flows [28]. The first one is the “time-stepping” [47] or “matrix-free” [4] approach based on the use of existing unsteady nonlinear solvers, developed in Computational Fluid Dynamics (CFD). The computation of leading eigenvalues is achieved by noticing that the operations performed at each iteration of the linearized time-stepping solver correspond to an exponential-based transformation of the Jacobian operator [4, 47]. Small time-steps are required for an application of the linearized time-stepper to approximate accurately the exponential transformation [72] This leads to a large number of so-called “outer” iterations (in the 103–104 range) to converge only a few eigenvalues. Note that other strategies for computing matrix exponential allow to relax the small time-step constraint and provide better convergence properties [19, 60]

Objectives

Results

Conclusion