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

TL;DR

This paper develops a high-performance, matrix-free discontinuous Galerkin solver for compressible Navier-Stokes equations and compares its efficiency with an incompressible formulation for turbulent flow simulations.

Contribution

It introduces a highly efficient matrix-free DG solver for compressible flows and provides a detailed performance comparison with an incompressible DG solver on modern multicore architectures.

Findings

Compressible DG solver shows significant potential for performance improvements.

Performance comparison reveals differences in efficiency between compressible and incompressible formulations.

Numerical tests demonstrate the solver's effectiveness on turbulent flow benchmarks.

Abstract

Both compressible and incompressible Navier-Stokes solvers can be used and are used to solve incompressible turbulent flow problems. In the compressible case, the Mach number is then considered as a solver parameter that is set to a small value, $\mathrm{M}\approx 0.1$, in order to mimic incompressible flows. This strategy is widely used for high-order discontinuous Galerkin discretizations of the compressible Navier-Stokes equations. The present work raises the question regarding the computational efficiency of compressible DG solvers as compared to a genuinely incompressible formulation. Our contributions to the state-of-the-art are twofold: Firstly, we present a high-performance discontinuous Galerkin solver for the compressible Navier-Stokes equations based on a highly efficient matrix-free implementation that targets modern cache-based multicore architectures. The performance results presented in this work focus on the node-level performance and our results suggest that there is great potential for further performance improvements for current state-of-the-art discontinuous Galerkin implementations of the compressible Navier-Stokes equations. Secondly, this compressible Navier-Stokes solver is put into perspective by comparing it to an incompressible DG solver that uses the same matrix-free implementation. We discuss algorithmic differences between both solution strategies and present an in-depth numerical investigation of the performance. The considered benchmark test cases are the three-dimensional Taylor-Green vortex problem as a representative of transitional flows and the turbulent channel flow problem as a representative of wall-bounded turbulent flows.