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Hybridisable Discontinuous Galerkin Formulation of Compressible Flows

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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.

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Authors and Affiliations

  1. Laboratori de Càlcul Numèric (LaCàN), ETS de Ingenieros de Caminos, Canales y Puertos, Universitat Politècnica de Catalunya, Barcelona, Spain

    Jordi Vila-Pérez, Matteo Giacomini & Antonio Huerta

  2. Barcelona Graduate School of Mathematics (BGSMath), Barcelona, Spain

    Jordi Vila-Pérez

  3. International Centre for Numerical Methods in Engineering (CIMNE), Barcelona, Spain

    Matteo Giacomini & Antonio Huerta

  4. Zienkiewicz Centre for Computational Engineering, College of Engineering, Swansea University, Bay Campus, Wales, SA1 8EN, United Kingdom

    Ruben Sevilla

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  1. Jordi Vila-Pérez
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  2. Matteo Giacomini
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  3. Ruben Sevilla
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  4. Antonio Huerta
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Correspondence to Jordi Vila-Pérez.

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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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Vila-Pérez, J., Giacomini, M., Sevilla, R. et al. Hybridisable Discontinuous Galerkin Formulation of Compressible Flows. Arch Computat Methods Eng 28, 753–784 (2021). https://doi.org/10.1007/s11831-020-09508-z

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