Accurate, Stable, and Scalable Algorithms for Convection-Dominated Flows

Accuracy and stability have long been essential pillars of numerical algorithms for the simulation of fluid flow. With the advent of tera- and petascale parallel computers comprising thousands and hundreds of thousands of processors, scalability is emergent as another essential pillar. To first order, scalability implies that the solution time be only weakly dependent on the number of processors P, with n/P fixed, where n is the number of degrees of
freedom in the problem. Time-dependent transport problems having minimal dissipation, such as electromagnetics and convection-dominated flow, face an additional scalability challenge, namely, that dispersion errors accumulated at small scales may become dominant when propagated through the large domains that are afforded by petaflops computers.

This talk will cover several critical developments that make it possible to use
spectral element simulations in large-scale convection-dominated incompressible flow simulations on tens and hundreds of thousands of processors.
Discretization advances that have made the spectral element viable for these
problems include stabilizing filters and spectral element dealiasing. Solver advances include spectral element multigrid methods that employ robust Schwarz-based smoothers and scalable parallel coarse-grid solvers. In addition
to these fundamental elements, we touch upon a few technical details required to exceed processor counts of ten thousand. We present the results of several spectral element simulations, including heat transfer in reactor core
subchannels, turbulent MHD, and a detailed discussion of transitional flow in arteriovenous grafts. We conclude with some perspectives on the future of high-order methods and high performance computing applied to computational
fluid dynamics.