Efficient iterative algorithms for linear stability analysis of incompressible flows

Linear stability analysis of a dynamical system entails finding the rightmost eigenvalue for a series of eigenvalue problems. For large-scale systems, it is known that conventional iterative eigenvalue solvers are not reliable for computing this eigenvalue. A more robust method recently developed in Elman & Wu (2013, Lyapunov inverse iteration for computing a few rightmost eigenvalues of large generalized eigenvalue problems. SIAM J. Matrix Anal. Appl., 34, 1685-1707) and Meerbergen & Spence (2010, Inverse iteration for purely imaginary eigenvalues with application to the detection of Hopf bifurcation in large-scale problems. SIAM J. Matrix Anal. Appl., 31, 1982-1999), Lyapunov inverse iteration, involves solving large-scale Lyapunov equations, which in turn requires the solution of large, sparse linear systems analogous to those arising from solving the underlying partial differential equations (PDEs). This study explores the efficient implementation of Lyapunov inverse iteration when it is used for linear stability analysis of incompressible flows. Efficiencies are obtained from effective solution strategies for the Lyapunov equations and for the underlying PDEs. Solution strategies based on effective preconditioning methods and on recycling Krylov subspace methods are tested and compared, and a modified version of a Lyapunov solver is proposed that achieves significant savings in computational cost.