W. D. Gropp, D. E. Keyes, L. C. McInnes, and M. D. Tidriri, "Globalized Newton-Krylov-Schwarz Algorithms and Software for Parallel Implicit CFD," Preprint ANL/MCS-P788-0100, January 2000. [pdf]
Implicit solution methods are important in applications modeled by PDEs with disparate temporal and spatial scales. Because such applications require high resolution with reasonable turnaround, parallelization is essential. The pseudo-transient matrix-free Newton-Krylov-Schwarz algorithmic framework is presented as a widely applicable answer. This article shows that, for the classical problem of three-dimensional transonic Euler flow about an M6 wing, NKS can simultaneously deliver
* globalized, asymptotically rapid convergence through adaptive pseudo-transient continuation and Newton's method;
* reasonable parallelizability for an implicit method through deferred synchronization and favorable communication-to-communication scaling in the Krylov linear solver; and
* high per-processor performance through attention to distributed memory and cache locality, especially through the Schwarz preconditioner.
Two discouraging features of NKS methods are their sensitivity to the coding of the underlying PDE discretization and the large number of parameters that must be selected to govern convergence. We therefore distill several recommendations from our experience and from our reading of the literature on various algorithmic components of NKS, and we describe a freely available, MPI-based portable parallel software implementation of the solver employed here.