Hybridisable discontinuous Galerkin formulation of compressible flows

TL;DR

This paper reviews high-order hybridisable discontinuous Galerkin methods for compressible flows and introduces a unified framework for Riemann solvers, demonstrating improved accuracy and stability in various flow simulations.

Contribution

It presents the first unified framework for deriving Riemann solvers within the HDG context, including HLL and HLLEM, and evaluates their performance in compressible flow benchmarks.

Findings

HLL Riemann solvers outperform Roe in supersonic flows due to positivity preservation.

HLLEM excels in boundary layer approximation because of shear preservation.

The HDG scheme with the proposed Riemann solvers is competitive in viscous and inviscid flow simulations.

Abstract

This work presents a review of high-order hybridisable discontinuous Galerkin (HDG) methods in the context of compressible flows. Moreover, an original unified framework for the derivation of Riemann solvers in hybridised formulations is proposed. This framework includes, for the first time in an HDG context, the HLL and HLLEM Riemann solvers as well as the traditional Lax-Friedrichs and Roe solvers. HLL-type Riemann solvers demonstrate their superiority with respect to Roe in supersonic cases due to their positivity preserving properties. In addition, HLLEM specifically outstands in the approximation of boundary layers because of its shear preservation, which confers it an increased accuracy with respect to HLL and Lax-Friedrichs. A comprehensive set of relevant numerical benchmarks of viscous and inviscid compressible flows is presented. The test cases are used to evaluate the competitiveness of the resulting high-order HDG scheme with the aforementioned Riemann solvers and equipped with a shock treatment technique based on artificial viscosity.