Simulating Incompressible Flows at Petascale and Beyond

Accuracy and stability have long been essential to 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 component. 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 methods in large-scale incompressible and low-Mach number flow simulations on tens and hundreds of thousands of processors. Discretization advances 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 components, we touch upon a few technical details required to exceed processor counts of one hundred thousand. We present simulation results from several application areas, including reactor hydrodynamics, MHD, turbulent autoignition, and transitional flow in arteriovenous grafts, and conclude with a brief discussion of modeling considerations as we move beyond petascale.