Efficiency of high-performance discontinuous Galerkin spectral element methods for under-resolved turbulent incompressible flows

The present paper addresses the numerical solution of turbulent flows with high-order discontinuous Galerkin methods for discretizing the incompressible Navier-Stokes equations. The efficiency of high-order methods when applied to under-resolved problems is an open issue in the literature. This topic is carefully investigated in the present work by the example of the three-dimensional Taylor-Green vortex problem. Our implementation is based on a generic high-performance framework for matrix-free evaluation of finite element operators with one of the best realizations currently known. We present a methodology to systematically analyze the efficiency of the incompressible Navier-Stokes solver for high polynomial degrees. Due to the absence of optimal rates of convergence in the under-resolved regime, our results reveal that demonstrating improved efficiency of high-order methods is a challenging task and that optimal computational complexity of solvers and preconditioners as well as matrix-free implementations are necessary ingredients in achieving the goal of better solution quality at the same computational costs already for a geometrically simple problem such as the Taylor-Green vortex. Although the analysis is performed for a Cartesian geometry, our approach is generic and can be applied to arbitrary geometries. We present excellent performance numbers on modern cache-based computer architectures achieving a throughput for operator evaluation of 3.10(8) up to1.10(9) DoFs/s (degrees of freedom per second) on one Intel Haswell node with 28 cores. Compared to performance results published within the last five years for high-order discontinuous Galerkin discretizations of the compressible Navier-Stokes equations, our approach reduces computational costs by more than one order of magnitude for the same setup.