Hybridisable Discontinuous Galerkin Formulation of Compressible Flows
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ORIGINAL PAPER
Hybridisable Discontinuous Galerkin Formulation of Compressible Flows Jordi Vila-Pérez1,2
· Matteo Giacomini1,3
· Ruben Sevilla4
· Antonio Huerta1,3
Received: 20 September 2020 / Accepted: 30 September 2020 © CIMNE, Barcelona, Spain 2020
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. Keywords hybridisable discontinuous Galerkin · compressible flows · Riemann solvers · HLL-type numerical fluxes · high-order · numerical benchmarks Mathematics Subject Classification 65M12 · 65M22 · 65M60 · 76N15 · 76N17
1 Introduction This work was partially supported by the Spanish Ministry of Economy and Competitiveness (Grant Number: DPI2017-85139-C2-2-R). J.V.P. was supported by the Spanish Ministry of Economy and Competitiveness, through the María de Maeztu programme for units of excellence in R&D (Grant Number: MDM-2014-0445). M.G. and A.H. are also grateful for the support provided by the Spanish Ministry of Economy and Competitiveness through the Severo Ochoa programme for centres of excellence in RTD (Grant Number: CEX2018-000797-S) and the Generalitat de Catalunya (Grant Number: 2017-SGR-1278). R.S. also acknowledges the support of the Engineering and Physical Sciences Research Council (Grant Number: EP/P033997/1).
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Jordi Vila-Pérez [email protected]
1
Laboratori de Càlcul Numèric (LaCàN), ETS de Ingenieros de Caminos, Canales y Puertos, Universitat Politècnica de Catalunya, Barcelona, Spain
2
Barcelona Graduate School of Mathematics (BGSMath), Barcelona, Spain
3
International Centre for Numerical Methods in Engineering (CIMNE), Barcelona, Spain
High-order methods have experienced a growing interest within the computational fluid dynamics (CFD) community because of their increased accuracy when compared to loworder methods [80,135]. However, low-order finite volume (FV) or stabilised finite element (FE) methods are still the most employed strategies in commercial, industrial and open source CFD solvers [14,55,70,128]. Low-order FV and FE methods are robust, easy to implement and p
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