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.
Similar content being viewed by others
References
Aliabadi SK, Ray SE, Tezduyar TE (1993) SUPG finite element computation of viscous compressible flows based on the conservation and entropy variables formulations. Comput Mech 11(5–6):300–312
Arnold DN, Brezzi F, Cockburn B, Marini LD (2002) Unified analysis of discontinuous Galerkin methods for elliptic problems. SIAM J Numer Anal 39(5):1749–1779
Balan A, Woopen M, May G (2015) Hp-adaptivity on anisotropic meshes for hybridized discontinuous Galerkin scheme. AIAA paper
Bassi F, Rebay S (1995) Accurate 2D Euler computations by means of a high order discontinuous finite element method. In: Deshpande SM, Desai SS, Narasimha R (eds) Fourteenth international conference on numerical methods in fluid dynamics. Lecture notes in physics, vol 453, pp 234–240. Springer, Berlin
Bassi F, Rebay S (1997) A high-order accurate discontinuous finite element method for the numerical solution of the compressible Navier–Stokes equations. J Comput Phys 131(2):267–279
Bassi F, Rebay S (1997) High-order accurate discontinuous finite element solution of the 2D Euler equations. J Comput Phys 138(2):251–285
Bassi F, Rebay S (2002) Numerical evaluation of two discontinuous Galerkin methods for the compressible Navier–Stokes equations. Int J Numer Methods Fluids 40(1–2):197–207
Blasius H (1908) Grenzschichten in fluessigkeiten mit kleiner reibung. Z Math Phys 56(1):1–37
Bui-Thanh T (2015) From Godunov to a unified hybridized discontinuous Galerkin framework for partial differential equations. J Comput Phys 295:114–146
Cangiani A, Dong Z, Georgoulis EH, Houston P (2017) \(hp\)-version discontinuous Galerkin methods on polygonal and polyhedral meshes. Springer, Berlin
Carter JE (1972) Numerical solutions of the Navier–Stokes equations for the supersonic laminar flow over a two-dimensional compression corner. Tech. rep., NASA-TR-R-385
Casoni E, Peraire J, Huerta A (2012) One-dimensional shock-capturing for high-order discontinuous Galerkin methods. Int J Numer Methods Fluids 71(6):737–755
Chalot F, Normand PE (2010) Higher-order stabilized finite elements in an industrial Navier–Stokes code. In: Kroll N, Bieler H, Deconinck H, Couaillier V, van der Ven H, Sørensen K (eds) ADIGMA—a European initiative on the development of adaptive higher-order variational methods for aerospace applications, pp 145–165. Springer
Chalot FL, Perrier P (2004) Industrial aerodynamics. Encyclopedia of computational mechanics
Chassaing JC, Khelladi S, Nogueira X (2013) Accuracy assessment of a high-order moving least squares finite volume method for compressible flows. Comput Fluids 71:41–53
Chiocchia G (1985) Exact solutions to transonic and supersonic flows. AGARD Technical Report AR–211
Cockburn B (2001) Devising discontinuous Galerkin methods for non-linear hyperbolic conservation laws. J Comput Appl Math 128(1–2):187–204
Cockburn B (2016) Static condensation, hybridization, and the devising of the HDG methods. In: Lecture notes in computational science and engineering, pp 129–177. Springer
Cockburn B, Di Pietro DA, Ern A (2016) Bridging the hybrid high-order and hybridizable discontinuous Galerkin methods. ESAIM Math Model Numer Anal 50(3):635–650
Cockburn B, Fu G (2016) Superconvergence by M-decompositions. Part II: construction of two-dimensional finite elements. ESAIM Math Model Numer Anal 51(1):165–186
Cockburn B, Fu G (2016) Superconvergence by M-decompositions. Part III: construction of three-dimensional finite elements. ESAIM Math Model Numer Anal 51(1):365–398
Cockburn B, Fu G (2017) Devising superconvergent HDG methods with symmetric approximate stresses for linear elasticity by M-decompositions. IMA J Numer Anal 38(2):566–604
Cockburn B, Fu G, Qiu W (2016) A note on the devising of superconvergent HDG methods for Stokes flow by M-decompositions. IMA J Numer Anal 37:730–749
Cockburn B, Fu G, Sayas FJ (2016) Superconvergence by M-decompositions. Part I: general theory for HDG methods for diffusion. Math Comput 86(306):1609–1641
Cockburn B, Gopalakrishnan J (2004) A characterization of hybridized mixed methods for second order elliptic problems. SIAM J Numer Anal 42(1):283–301
Cockburn B, Gopalakrishnan J (2005) Incompressible finite elements via hybridization. Part I: the Stokes system in two space dimensions. SIAM J Numer Anal 43(4):1627–1650
Cockburn B, Gopalakrishnan J (2005) Incompressible finite elements via hybridization. Part II: the Stokes system in three space dimensions. SIAM J Numer Anal 43(4):1651–1672
Cockburn B, Gopalakrishnan J (2009) The derivation of hybridizable discontinuous Galerkin methods for Stokes flow. SIAM J Numer Anal 47(2):1092–1125
Cockburn B, Gopalakrishnan J, Lazarov R (2009) Unified hybridization of discontinuous Galerkin, mixed, and continuous Galerkin methods for second order elliptic problems. SIAM J Numer Anal 47(2):1319–1365
Cockburn B, Karniadakis GE, Shu CW (2000) The development of discontinuous Galerkin methods. In: Discontinuous Galerkin methods, pp 3–50. Springer, Berlin
Cockburn B, Lin SY, Shu CW (1989) TVB Runge–Kutta local projection discontinuous Galerkin finite element method for conservation laws III: one-dimensional systems. J Comput Phys 84(1):90–113
Cockburn B, Shi K (2012) Superconvergent HDG methods for linear elasticity with weakly symmetric stresses. IMA J Numer Anal 33(3):747–770
Cockburn B, Shu CW (1998) The local discontinuous Galerkin method for time-dependent convection-diffusion systems. SIAM J Numer Anal 35(6):2440–2463
Cockburn B, Shu CW (1998) The Runge–Kutta discontinuous Galerkin method for conservation laws V. J Comput Phys 141(2):199–224
Cook AW, Cabot WH (2005) Hyperviscosity for shock–turbulence interactions. J Comput Phys 203(2):379–385
Degrez G, Boccadoro CH, Wendt JF (1987) The interaction of an oblique shock wave with a laminar boundary layer revisited. An experimental and numerical study. J Fluid Mech 177:247–263
Di Pietro D, Ern A (2015) A hybrid high-order locking-free method for linear elasticity on general meshes. Comput Methods Appl Mech Eng 283:1–21
Di Pietro DA, Ern A, Lemaire S (2014) An arbitrary-order and compact-stencil discretization of diffusion on general meshes based on local reconstruction operators. Comput Methods Appl Math 14(4):461–472
Donea J, Huerta A (2003) Finite element methods for flow problems. Wiley, Hoboken
Drikakis D (2003) Advances in turbulent flow computations using high-resolution methods. Prog Aerosp Sci 39(6–7):405–424
Dumbser M, Balsara DS (2016) A new efficient formulation of the HLLEM Riemann solver for general conservative and non-conservative hyperbolic systems. J Comput Phys 304:275–319
Dumbser M, Käser M, Titarev VA, Toro EF (2007) Quadrature-free non-oscillatory finite volume schemes on unstructured meshes for nonlinear hyperbolic systems. J Comput Phys 226(1):204–243
Egger H, Schöberl J (2009) A hybrid mixed discontinuous Galerkin finite-element method for convection-diffusion problems. IMA J Numer Anal 30(4):1206–1234
Egger H, Waluga C (2012) \(hp\)-analysis of a hybrid DG method for Stokes flow. IMA J Numer Anal 33(2):687–721
Egger H, Waluga C (2012) A hybrid mortar method for incompressible flow. Int J Numer Anal Model 9(4):793–812
Einfeldt B (1988) On Godunov-type methods for gas dynamics. SIAM J Numer Anal 25(2):294–318
Einfeldt B, Munz C, Roe P, Sjögreen B (1991) On Godunov-type methods near low densities. J Comput Phys 92(2):273–295
Ekaterinaris JA (2005) High-order accurate, low numerical diffusion methods for aerodynamics. Prog Aerosp Sci 41(3–4):192–300
Ekelschot D, Moxey D, Sherwin S, Peiró J (2017) A p-adaptation method for compressible flow problems using a goal-based error indicator. Comput Struct 181:55–69
Ern A, Di Pietro DA (2011) Mathematical aspects of discontinuous Galerkin methods. Springer, Berlin
Fernández P, Nguyen C, Peraire J (2018) A physics-based shock capturing method for unsteady laminar and turbulent flows. AIAA paper
Fernández P, Nguyen NC, Peraire J (2017) The hybridized discontinuous Galerkin method for implicit large-Eddy simulation of transitional turbulent flows. J Comput Phys 336:308–329
Fish J, Belytschko T (2007) A first course in finite elements. Wiley, Hoboken
Fleischmann N, Adami S, Hu XY, Adams NA (2019) A low dissipation method to cure the grid-aligned shock instability. J Comput Phys 401:109004
Gerhold T (2005) Overview of the hybrid RANS code TAU. In: MEGAFLOW-numerical flow simulation for aircraft design, pp 81–92. Springer
Giacomini M, Karkoulias A, Sevilla R, Huerta A (2018) A superconvergent HDG method for Stokes flow with strongly enforced symmetry of the stress tensor. J Sci Comput 77(3):1679–1702
Giacomini M, Sevilla R (2019) Discontinuous Galerkin approximations in computational mechanics: hybridization, exact geometry and degree adaptivity. SN Appl Sci 1:1047
Giacomini M, Sevilla R, Huerta A (2020) Tutorial on hybridizable discontinuous Galerkin (HDG) formulation for incompressible flow problems. Springer, Cham, pp 163–201
Giorgiani G, Fernández-Méndez S, Huerta A (2014) Hybridizable discontinuous Galerkin with degree adaptivity for the incompressible Navier–Stokes equations. Comput Fluids 98:196–208
Godunov SK, Bohachevsky I (1959) Finite difference method for numerical computation of discontinuous solutions of the equations of fluid dynamics. Mat Sb 47(3):271–306
Guyan RJ (1965) Reduction of stiffness and mass matrices. AIAA Paper 3(2):380–380
Hakkinen RJ, Greber I, Trilling L, Abarbanel SS (1959) The interaction of an oblique shock wave with a laminar boundary layer. Tech. rep., NASA Memo 2-18-59W
Harten A, Hyman JM (1983) Self adjusting grid methods for one-dimensional hyperbolic conservation laws. J Comput Phys 50(2):235–269
Harten A, Lax PD, van Leer B (1983) On upstream differencing and Godunov-type schemes for hyperbolic conservation laws. SIAM Rev 25(1):35–61
Hartmann R, Houston P (2003) Adaptive discontinuous Galerkin finite element methods for nonlinear hyperbolic conservation laws. SIAM J Sci Comput 24(3):979–1004
Hesthaven JS (2017) Numerical methods for conservation laws. Society for Industrial and Applied Mathematics
Huerta A, Angeloski A, Roca X, Peraire J (2013) Efficiency of high-order elements for continuous and discontinuous Galerkin methods. Int J Numer Methods Eng 96(9):529–560
Huerta A, Casoni E, Peraire J (2011) A simple shock-capturing technique for high-order discontinuous Galerkin methods. Int J Numer Methods Fluids 69(10):1614–1632
Hung C, MacCormack R (1976) Numerical solutions of supersonic and hypersonic laminar compression corner flows. AIAA J 14(4):475–481
Jasak H (2009) OpenFOAM: open source CFD in research and industry. Int J Nav Archit Ocean Eng 1(2):89–94
Jaust A, Schuetz J, Woopen M (2014) A hybridized discontinuous Galerkin method for unsteady flows with shock-capturing. In: 44th AIAA fluid dynamics conference. American Institute of Aeronautics and Astronautics
Jaust A, Schütz J (2014) A temporally adaptive hybridized discontinuous Galerkin method for time-dependent compressible flows. Comput Fluids 98:177–185
Jaust A, Schütz J, Woopen M (2015) An HDG method for unsteady compressible flows. In: Lecture notes in computational science and engineering, pp 267–274. Springer
Katzer E (1989) On the lengthscales of laminar shock/boundary-layer interaction. J Fluid Mech 206:477–496
Kawai S, Lele S (2008) Localized artificial diffusivity scheme for discontinuity capturing on curvilinear meshes. J Comput Phys 227(22):9498–9526
Kawai S, Shankar SK, Lele SK (2010) Assessment of localized artificial diffusivity scheme for large-eddy simulation of compressible turbulent flows. J Comput Phys 229(5):1739–1762
Komala-Sheshachala S, Sevilla R, Hassan O (2020) A coupled HDG-FV scheme for the simulation of transient inviscid compressible flows. Comput Fluids 202:104495
Kotteda VK, Mittal S (2014) Stabilized finite-element computation of compressible flow with linear and quadratic interpolation functions. Int J Numer Methods Fluids 75(4):273–294
Krivodonova L, Berger M (2006) High-order accurate implementation of solid wall boundary conditions in curved geometries. J Comput Phys 211(2):492–512
Kroll N (2009) ADIGMA: a European project on the development of adaptive higher order variational methods for aerospace applications. Aerospace Sciences meetings. American Institute of Aeronautics and Astronautics
Lax PD (1954) Weak solutions of nonlinear hyperbolic equations and their numerical computation. Commun Pure Appl Math 7(1):159–193
Lehrenfeld C, Schöberl J (2016) High order exactly divergence-free hybrid discontinuous Galerkin methods for unsteady incompressible flows. Comput Methods Appl Mech Eng 307:339–361
Mengaldo G, Grazia DD, Witherden F, Farrington A, Vincent P, Sherwin S, Peiro J (2014) A guide to the implementation of boundary conditions in compact high-order methods for compressible aerodynamics. AIAA Paper
Mittal S, Yadav S (2001) Computation of flows in supersonic wind-tunnels. Comput Methods Appl Mech Eng 191(6–7):611–634
Montlaur A, Fernández-Méndez S, Huerta A (2008) Discontinuous Galerkin methods for the Stokes equations using divergence-free approximations. Int J Numer Methods Fluids 57(9):1071–1092
Moro D, Nguyen N, Peraire J (2011) Navier–Stokes solution using hybridizable discontinuous Galerkin methods. AIAA Paper
Moro D, Nguyen NC, Peraire J (2016) Dilation-based shock capturing for high-order methods. Int J Numer Methods Fluids 82(7):398–416
Moro D, Nguyen NC, Peraire J, Drela M (2017) Mesh topology preserving boundary-layer adaptivity method for steady viscous flows. AIAA Paper 55(6):1970–1985
Moura RC, Mengaldo G, Peiró J, Sherwin SJ (2017) On the eddy-resolving capability of high-order discontinuous Galerkin approaches to implicit LES/under-resolved DNS of Euler turbulence. J Comput Phys 330:615–623
Moura RC, Sherwin SJ, Peiró J (2015) Linear dispersion–diffusion analysis and its application to under-resolved turbulence simulations using discontinuous Galerkin spectral/hp methods. J Comput Phys 298:695–710
Moura RC, Sherwin SJ, Peiró J (2015) On DG-based iLES approaches at very high Reynolds numbers. Tech. rep., Imperial College London
Nguyen N, Peraire J, Cockburn B (2015) A class of embedded discontinuous Galerkin methods for computational fluid dynamics. J Comput Phys 302:674–692
Nguyen NC, Peraire J (2011) An adaptive shock-capturing HDG method for compressible flows. AIAA Paper 3060
Nguyen NC, Peraire J (2012) Hybridizable discontinuous Galerkin methods for partial differential equations in continuum mechanics. J Comput Phys 231(18):5955–5988
Nguyen NC, Peraire J, Cockburn B (2009) An implicit high-order hybridizable discontinuous Galerkin method for linear convection–diffusion equations. J Comput Phys 228(9):3232–3254
Nguyen NC, Peraire J, Cockburn B (2009) An implicit high-order hybridizable discontinuous Galerkin method for nonlinear convection–diffusion equations. J Comput Phys 228(23):8841–8855
Nguyen NC, Peraire J, Cockburn B (2011) An implicit high-order hybridizable discontinuous Galerkin method for the incompressible Navier–Stokes equations. J Comput Phys 230(4):1147–1170
Oikawa I (2015) Analysis of a reduced-order HDG method for the Stokes equations. J Sci Comput 67(2):475–492
Peery K, Imlay S (1988) Blunt-body flow simulations. AIAA Paper 2904
Peraire J, Nguyen C, Cockburn B (2011) an embedded discontinuous Galerkin method for the compressible Euler and Navier–Stokes equations. AIAA Paper 3228
Peraire J, Nguyen NC, Cockburn B (2010) A hybridizable discontinuous Galerkin method for the compressible Euler and Navier–Stokes Equations. AIAA Paper 363
Persson PO (2013) Shock capturing for high-order discontinuous Galerkin simulation of transient flow problems. AIAA Paper 3061
Persson PO, Peraire J (2006) Sub-cell shock capturing for discontinuous Galerkin methods. AIAA Paper 0112
Perthame B, Shu CW (1996) On positivity preserving finite volume schemes for Euler equations. Numer Math 73(1):119–130
Qamar A, Hasan N, Sanghi S (2006) New scheme for the computation of compressible flows. AIAA J 44(5):1025–1039
Qiu J, Khoo BC, Shu CW (2006) A numerical study for the performance of the Runge–Kutta discontinuous Galerkin method based on different numerical fluxes. J Comput Phys 212(2):540–565
Qiu W, Shen J, Shi K (2017) An HDG method for linear elasticity with strong symmetric stresses. Math Comput 87(309):69–93
Qiu W, Shi K (2016) A superconvergent HDG method for the incompressible Navier–Stokes equations on general polyhedral meshes. IMA J Numer Anal 36(4):1943–1967
Quirk JJ (1994) A contribution to the great Riemann solver debate. Int J Numer Methods Fluids 18(6):555–574
Randall J, Leveque LRJ (2013) Finite volume methods for hyperbolic problems. Cambridge University Press, Cambridge
Roca X, Nguyen C, Peraire J (2013) Scalable parallelization of the hybridized discontinuous Galerkin method for compressible flow. AIAA Paper
Roe PL (1981) Approximate Riemann solvers, parameter vectors, and difference schemes. J Comput Phys 43(2):357–372
Rohde A (2001) Eigenvalues and eigenvectors of the Euler equations in general geometries. AIAA Paper
Samii A, Dawson C (2018) An explicit hybridized discontinuous Galerkin method for Serre–Green–Naghdi wave model. Comput Methods Appl Mech Eng 330:447–470. https://doi.org/10.1016/j.cma.2017.11.001
Samii A, Kazhyken K, Michoski C, Dawson C (2019) A comparison of the explicit and implicit hybridizable discontinuous Galerkin methods for nonlinear shallow water equations. J Sci Comput 80(3):1936–1956. https://doi.org/10.1007/s10915-019-01007-z
Schlichting H, Gersten K (2016) Boundary-layer theory. Springer, Berlin
Schütz J, May G (2013) An adjoint consistency analysis for a class of hybrid mixed methods. IMA J Numer Anal 34(3):1222–1239
Schütz J, May G (2013) A hybrid mixed method for the compressible Navier–Stokes equations. J Comput Phys 240:58–75
Schütz J, Woopen M, May G (2012) A hybridized DG/mixed scheme for nonlinear advection-diffusion systems, including the compressible Navier–Stokes equations. AIAA Paper
Sevilla R, Fernández-Méndez S, Huerta A (2008) NURBS-enhanced finite element method for Euler equations. Int J Numer Methods Fluids 57(9):1051–1069
Sevilla R, Fernández-Méndez S, Huerta A (2008) NURBS-enhanced finite element method (NEFEM). Int J Numer Methods Eng 76(1):56–83
Sevilla R, Giacomini M, Karkoulias A, Huerta A (2018) A superconvergent hybridisable discontinuous Galerkin method for linear elasticity. Int J Numer Methods Eng 116(2):91–116
Sevilla R, Gil AJ, Weberstadt M (2017) A high-order stabilised ale finite element formulation for the euler equations on deformable domains. Comput Struct 181:89–102
Sevilla R, Hassan O, Morgan K (2013) An analysis of the performance of a high-order stabilised finite element method for simulating compressible flows. Comput Methods Appl Mech Eng 253:15–27
Sevilla R, Huerta A (2016) Tutorial on hybridizable discontinuous Galerkin (HDG) for second-order elliptic problems. In: Schröder J, Wriggers P (eds) Advanced l. CISM International Centre for Mechanical Sciences, vol 566, pp 105–129. Springer
Shakib F, Hughes TJ, Johan Z (1991) A new finite element formulation for computational fluid dynamics: X. The compressible Euler and Navier–Stokes equations. Comput Methods Appl Mech Eng 89(1–3):141–219
Slotnick JP, Khodadoust A, Alonso JJ, Darmofal DL, Gropp W, Lurie E, Mavriplis DJ (2014) Cfd vision 2030 study: a path to revolutionary computational aerosciences. Tech. rep., NASA
Sørensen K, Hassan O, Morgan K, Weatherill N (2003) A multigrid accelerated hybrid unstructured mesh method for 3D compressible turbulent flow. Comput Mech 31(1–2):101–114
Thibert JJ, Granjacques M, Ohman LH (1979) NACA 0012 airfoil. AGARD advisory report AR–138 A1
Toro EF (2009) Riemann solvers and numerical methods for fluid dynamics. Springer, Berlin
Tuck EO (1991) A criterion for leading-edge separation. J Fluid Mech 222(–1):33
Von Neumann J, Richtmyer RD (1950) A method for the numerical calculation of hydrodynamic shocks. J Appl Phys 21(3):232–237
Vymazal M, Koloszar L, D’Angelo S, Villedieu N, Ricchiuto M, Deconinck H (2015) High-order residual distribution and error estimation for steady and unsteady compressible flow. In: Notes on numerical fluid mechanics and multidisciplinary design, pp 381–395. Springer
Wang Z, Liu Y (2006) Extension of the spectral volume method to high-order boundary representation. J Comput Phys 211(1):154–178
Wang ZJ, Fidkowski K, Abgrall R, Bassi F, Caraeni D, Cary A, Deconinck H, Hartmann R, Hillewaert K, Huynh HT, Kroll N, May G, Persson PO, van Leer B, Visbal M (2013) High-order CFD methods: current status and perspective. Int J Numer Methods Fluids 72(8):811–845
Williams DM (2018) An entropy stable, hybridizable discontinuous Galerkin method for the compressible Navier–Stokes equations. Math Comput 87(309):95–121
Woopen M, Balan A, May G, Schütz J (2014) A comparison of hybridized and standard DG methods for target-based hp-adaptive simulation of compressible flow. Comput Fluids 98:3–16
Woopen M, Ludescher T, May G (2014) A hybridized discontinuous Galerkin method for turbulent compressible flow. AIAA Paper
Yano M, Darmofal DL (2012) Case C1. 3: flow over the NACA 0012 airfoil: subsonic inviscid, transonic inviscid, and subsonic laminar flows. In: First international workshop on high-order CFD methods
Yoshihara H, Sacher P (1985) Test cases for inviscid flow field methods. AGARD advisory report AR-211
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Conflict of interest
The authors declare that they have no conflict of interest.
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
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).
Rights and permissions
About this article
Cite this article
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
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11831-020-09508-z
Keywords
- hybridisable discontinuous Galerkin
- compressible flows
- Riemann solvers
- HLL-type numerical fluxes
- high-order
- numerical benchmarks