Highly scalable solution of incompressible Navier-Stokes equations using the spectral element method with overlapping grids

Abstract

We present a highly-flexible Schwarz overlapping framework for simulating turbulent fluid/thermal transport in complex domains. The approach is based on a variant of the Schwarz alternating method in which the solution is advanced in parallel in separate overlapping subdomains. In each domain, the governing equations are discretized with an efficient high-order spectral element method (SEM). At each step, subdomain boundary data are determined by interpolating from the overlapping region of adjacent subdomains. The data are either lagged in time or extrapolated to higher-order temporal accuracy using a novel stabilized predictor-corrector algorithm. Matrix stability analysis is used to determine the optimal number of corrector iterations. Stability and accuracy are further improved with an optimal mass flux correction to guarantee mass conservation throughout the domain. The method supports an arbitrary number of subdomains. A new multirate time-stepping scheme is developed (a first for incompressible flow simulations) that allows the underlying equations to be advanced with time-step sizes varying as much as an order-of-magnitude between adjacent domains. All the developments maintain the third-order temporal convergence and exponential convergence of the originating SEM framework. This dissertation also presents a mesh optimizer that has been specifically designed for meshes generated for turbulent flow problems. The optimizer supports surface mesh improvement, which minimizes geometrical approximation errors. The smoother is shown to reduce the computational cost of numerical calculations by as much as 40%. Numerous examples illustrate the effectiveness of these new technologies for analyzing challenging turbulence problems that were previously infeasible.