On acceleration of Krylov-subspace-based Newton and Arnoldi iterations for incompressible CFD: replacing time steppers and generation of initial guess

We propose two techniques aimed at improving the convergence rate of steady state and eigenvalue solvers preconditioned by the inverse Stokes operator and realized via time-stepping. First, we suggest a generalization of the Stokes operator so that the resulting preconditioner operator depends on several parameters and whose action preserves zero divergence and boundary conditions. The parameters can be tuned for each problem to speed up the convergence of a Krylov-subspace-based linear algebra solver. This operator can be inverted by the Uzawa-like algorithm, and does not need a time-stepping. Second, we propose to generate an initial guess of steady flow, leading eigenvalue and eigenvector using orthogonal projection on a divergence-free basis satisfying all boundary conditions. The approach, including the two proposed techniques, is illustrated on the solution of the linear stability problem for laterally heated square and cubic cavities

Order-of-magnitude speedup for steady states and traveling waves via Stokes preconditioning in Channelflow and Openpipeflow

Steady states and traveling waves play a fundamental role in understanding hydrodynamic problems. Even when unstable, these states provide the bifurcation-theoretic explanation for the origin of the observed states. In turbulent wall-bounded shear flows, these states have been hypothesized to be saddle points organizing the trajectories within a chaotic attractor. These states must be computed with Newton's method or one of its generalizations, since time-integration cannot converge to unstable equilibria. The bottleneck is the solution of linear systems involving the Jacobian of the Navier-Stokes or Boussinesq equations. Originally such computations were carried out by constructing and directly inverting the Jacobian, but this is unfeasible for the matrices arising from three-dimensional hydrodynamic configurations in large domains. A popular method is to seek states that are invariant under numerical time integration. Surprisingly, equilibria may also be found by seeking flows that are invariant under a single very large Backwards-Euler Forwards-Euler timestep. We show that this method, called Stokes preconditioning, is 10 to 50 times faster at computing steady states in plane Couette flow and traveling waves in pipe flow. Moreover, it can be carried out using Channelflow (by Gibson) and Openpipeflow (by Willis) without any changes to these popular spectral codes. We explain the convergence rate as a function of the integration period and Reynolds number by computing the full spectra of the operators corresponding to the Jacobians of both methods.Comment: in Computational Modelling of Bifurcations and Instabilities in Fluid Dynamics, ed. Alexander Gelfgat (Springer, 2018