Comparative study of inner-outer Krylov solvers for linear systems in structured and high-order unstructured CFD problems

TL;DR

This paper evaluates advanced Krylov subspace methods, especially flexible inner-outer GMRES, for efficiently solving large sparse linear systems in CFD problems with high-order discretizations, demonstrating improved robustness and scalability.

Contribution

It demonstrates the effectiveness of flexible inner-outer Krylov methods with domain decomposition preconditioning for challenging CFD linear systems, including high-order discretizations.

Findings

Flexible Krylov methods overcome preconditioner scalability issues.

Significant improvements in robustness and convergence over standard GMRES.

Satisfactory strong scalability demonstrated in numerical tests.

Abstract

Advanced Krylov subspace methods are investigated for the solution of large sparse linear systems arising from stiff adjoint-based aerodynamic shape optimization problems. A special attention is paid to the flexible inner-outer GMRES strategy combined with most relevant preconditioning and deflation techniques. The choice of this specific class of Krylov solvers for challenging problems is based on its outstanding convergence properties. Typically in our implementation the efficiency of the preconditioner is enhanced with a domain decomposition method with overlapping. However, maintaining the performance of the preconditioner may be challenging since scalability and efficiency of a preconditioning technique are properties often antagonistic to each other. In this paper we demonstrate how flexible inner-outer Krylov methods are able to overcome this critical issue. A numerical study is performed considering either a Finite Volume (FV), or a high-order Discontinuous Galerkin (DG) discretization which affect the arithmetic intensity and memory-bandwith of the algebraic operations. We consider test cases of transonic turbulent flows with RANS modelling over the two-dimensional supercritical OAT15A airfoil and the three-dimensional ONERA M6 wing. Benefits in terms of robustness and convergence compared to standard GMRES solvers are obtained. Strong scalability analysis shows satisfactory results. Based on these representative problems a discussion of the recommended numerical practices is proposed.