Newton-Krylov methods and Krylov-Schwarz (domain decomposition) methods have begun to become established in computational fluid dynamics (CFD) over the past decade. The former employ a Krylov method, such as the generalized minimal residual method, inside of Newton's method in a Jacobian-free manner, through directional differencing. The latter employ overlapping Schwarz-type decomposition to derive a preconditioner for the Krylov accelerator that relies primarily on local information, for parallelism. They may be composed as Newton-Krylov-Schwarz methods, which seem particularly well suited for solving nonlinear elliptic systems in high-latency distributed-memory environments.
We describe recent numerical simulations with Newton-Krylov-Schwarz methods in CFD carried out at ICASE/NASA Langley, emphasizing trade-offs in convergence rate and concurrency in implicit algorithms, the preconditioning of a higher-order discrete operator with a lower-order discrete operator, and comparisons with multigrid and standard defect-correction approaches. We also present recent results on globalization through pseudo-transient continuation and additive Schwarz for convectively dominated problems.