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

Abstract

Summary 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 3·108 up to 1·109 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.

A matrix‐free high‐order discontinuous Galerkin compressible Navier‐Stokes solver: A performance comparison of compressible and incompressible formulations for turbulent incompressible flows

“…A detailed h ‐convergence study for various polynomial degrees k = 2,3,5,7,11,15 is presented in Figure , where the relative L 2 ‐errors are shown as a function of the number of unknowns. The results are in agreement with those published in the works of Fehn et al for the incompressible DG solver. The accuracy (efficiency of the spatial discretization scheme) continuously improves for higher polynomial degrees k .…”

“…Note that an exponent of 1.5 for the convective time step restriction has been found to be appropriate also for the incompressible Navier‐Stokes solver in the work of Fehn et al The following simulations for the Taylor‐Green problem are simulated using Cr = 0.5 and D = 0.02 to ensure stability for all spatial resolutions. For comparison, Cr inc = 0.125 has been used in the work of Fehn et al for all polynomial degrees. Hence, the ratio between the critical Courant number and the Courant number selected for the simulations is approximately the same for the compressible solver and the incompressible solver, allowing a fair comparison between both approaches in terms of computational costs.…”

“…As a consequence of formulating the convective term explicitly, the time step size is restricted according to the CFL condition. Adopting the notation in the work of Fehn et al, we obtain where e = 1.5 was found to be a tight fit for the incompressible Navier‐Stokes DG solver . The Mach number does not occur in the incompressible case, leading to significantly larger time step sizes for the incompressible solver as discussed in Section 4.…”

“…Following the work of Fehn et al, we define the efficiency E of a numerical method as that is, a method is more efficient if it obtains a more accurate solution within a given amount of computational costs or if it allows to reduce computational costs reaching the same level of accuracy. Moreover, the product of wall time and the number of cores used for a simulation is defined as a measure of computational costs.…”

“…In the case of the incompresssible Navier‐Stokes solver and in contrast to the compressible flow solver, each time step is composed of several substeps that can be solved explicitly (convective step) or for which a linear system of equations has to be solved (pressure, projection, and viscous steps). Since the convective term has to be evaluated only once within each time step, this step is negligible in terms of costs, and the efficiency of the incompressible solver depends on how fast one can solve the linear systems of equations in the remaining three substeps, ie, Due to several operators involved in the case of the incompressible flow solver (see Equations , , , and ), the efficiency of the implementation (throughput) cannot be expressed as a single number as for the compressible solver. The factor labeled iterations includes not only evaluations of the discretized operators but also preconditioners of different complexity (inverse mass matrix versus geometric multigrid) so that this quantity should be understood as an effective number of operator evaluations per time step (see also the discussion in the work of Fehn et al).…”