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

TL;DR

This paper investigates the efficiency of high-order discontinuous Galerkin methods for under-resolved turbulent incompressible flows, demonstrating that optimal solver design is crucial for computational gains, with significant performance improvements over previous methods.

Contribution

The study provides a systematic analysis of high-order DG methods' efficiency for turbulent flows, highlighting the importance of optimal solvers and implementation strategies for under-resolved problems.

Findings

High-order DG methods face efficiency challenges in under-resolved turbulence.

Optimal solvers and preconditioners are essential for computational efficiency.

The proposed approach achieves 3e8 to 1e9 DoFs/sec on modern hardware.

Abstract

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 literature. This topic is carefully investigated in the present work by the example of the 3D 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, preconditioners, and matrix-free implementations are necessary ingredients to achieve 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 3e8 up to 1e9 DoFs/sec on one Intel Haswell node with 28 cores. Compared to performance results published within the last 5 years for high-order DG discretizations of the compressible Navier-Stokes equations, our approach reduces computational costs by more than one order of magnitude for the same setup.