An HLL Riemann solver for the hybridised discontinuous Galerkin formulation of compressible flows

TL;DR

This paper introduces a high-order HDG method incorporating the HLL Riemann solver for compressible flows, demonstrating improved robustness and positivity preservation in supersonic regimes compared to traditional fluxes.

Contribution

It presents a unified high-order HDG framework for multiple Riemann solvers, emphasizing the robustness and positivity-preserving capabilities of the HLL solver in complex flow regimes.

Findings

HLL provides superior robustness in supersonic flows.

The HDG method achieves accuracy comparable to existing fluxes in subsonic and transonic flows.

HLL-based HDG method preserves positivity without entropy fixes.

Abstract

This work proposes a high-order hybridised discontinuous Galerkin (HDG) formulation of the Harten-Lax-Van Leer (HLL) Riemann solver for compressible flows. A unified framework is introduced to present Lax-Friedrichs, Roe and HLL Riemann solvers via appropriate definitions of the HDG numerical fluxes. The resulting high-order HDG method with HLL Riemann solver is evaluated through a set of numerical simulations of inviscid compressible flows in different regimes, from subsonic isentropic flows to transonic and supersonic problems with shocks. The accuracy of the proposed method is comparable with the one of Lax-Friedrichs and Roe numerical fluxes in subsonic and transonic flows. The superior performance of HLL is highlighted in supersonic cases, where the method provides extra robustness, being able to produce positivity preserving approximations without the need of any user-defined entropy fix.