Partial Spectral Information from Linear Systems to Speed-Up Numerical Simulations in Computational Fluid Dynamics

Abstract

It was observed that all the different linear systems arising in an iterative fluid flow simulation algorithm have approximately constant invariant subspaces associated with their smallest eigenvalues. For this reason, we propose to perform one single computation of the eigenspace associated with the smallest eigenvalues, at the beginning of the iterative process, to improve the convergence of the Krylov method used in subsequent iterations of the fluid flow algorithm by means of this pre-computed partial spectral information. The Subspace Inverse Iteration Method with Stabilized Block Conjugate Gradient is our choice for computing the spectral information, which is then used to remove the effect of the smallest eigenvalues in two different ways: either building a spectral preconditioner that shifts these eigenvalues from almost zero close to the unit value, or performing a deflation of the initial residual in order to remove parts of the solution corresponding to the smallest eigenvalues. Under certain conditions, both techniques yield a reduction of the number of iterations in each subsequent runs of the Conjugate Gradient algorithm.