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

TL;DR

This paper introduces two techniques to accelerate Krylov-subspace-based Newton and Arnoldi iterations in incompressible CFD, replacing time-stepping and generating better initial guesses, thereby improving convergence rates.

Contribution

The paper presents a generalized preconditioner operator and a divergence-free initial guess generation method, enhancing convergence without relying on traditional time-stepping.

Findings

Improved convergence rates in steady state and eigenvalue solvers.

Effective preconditioning with parameter tuning for specific problems.

Successful application to stability analysis of heated cavities.

Abstract

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.