iBet uBet web content aggregator. Adding the entire web to your favor.
iBet uBet web content aggregator. Adding the entire web to your favor.



Link to original content: https://unpaywall.org/10.1007/S10915-018-0811-X
Hybridized Discontinuous Galerkin Methods for Wave Propagation | Journal of Scientific Computing Skip to main content
Springer Nature Link
Log in

Hybridized Discontinuous Galerkin Methods for Wave Propagation

  • Published:
Journal of Scientific Computing Aims and scope Submit manuscript

Abstract

We present the recent development of hybridizable and embedded discontinuous Galerkin (DG) methods for wave propagation problems in fluids, solids, and electromagnetism. In each of these areas, we describe the methods, discuss their main features, display numerical results to illustrate their performance, and conclude with bibliography notes. The main ingredients in devising these DG methods are (1) a local Galerkin projection of the underlying partial differential equations at the element level onto spaces of polynomials of degree k to parametrize the numerical solution in terms of the numerical trace; (2) a judicious choice of the numerical flux to provide stability and consistency; and (3) a global jump condition that enforces the continuity of the numerical flux to obtain a global system in terms of the numerical trace. These DG methods are termed hybridized DG methods, because they are amenable to hybridization (static condensation) and hence to more efficient implementations. They share many common advantages of DG methods and possess some unique features that make them well-suited to wave propagation problems.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11

Similar content being viewed by others

Notes

  1. Strictly speaking, the finite element mesh can only partition the problem domain if \(\partial \Omega \) is piecewise p-th degree polynomial. For simplicity of exposition, and without loss of generality, we assume hereinafter that \(\mathcal {T}_h\) actually partitions \(\Omega \).

  2. Note the amplification factor \(N_1\) in the y-axis is a logarithmic quantity.

  3. Note the amplitude of the instabilities in Fig. 3 is non-dimensionalized with respect to the freestream velocity.

  4. The mismatch between the simulation and the experimental data near the leading edge is due to the missing vortex upwash induced by the finite extent of the computational domain, and not due to discretization errors [49, 50].

References

  1. Ahnert, T., Bärwolff, G.: Numerical comparison of hybridized discontinuous Galerkin and finite volume methods for incompressible flow. Int. J. Numer. Methods Fluids 76(5), 267–281 (2014)

    MathSciNet  Google Scholar 

  2. Alexander, R.: Diagonally implicit Runge–Kutta methods for stiff ODEs. SIAM J. Numer. Anal. 14, 1006–1021 (1977)

    MathSciNet  MATH  Google Scholar 

  3. Balan, A., Woopen, M., May, G.: Adjoint-based hp-adaptation for a class of high-order hybridized finite element schemes for compressible flows. In: 21st AIAA Computational Fluid Dynamics Conference (2013)

  4. Bonnasse-Gahot, M., Calandra, H., Diaz, J., Lanteri, S.: Hybridizable discontinuous galerkin method for the 2-d frequency-domain elastic wave equations. Geophys. J. Int. 213(1), 637–659 (2018)

    Google Scholar 

  5. Bui-Thanh, T.: From Godunov to a unified hybridized discontinuous Galerkin framework for partial differential equations. J. Comput. Phys. 295, 114–146 (2015)

    MathSciNet  MATH  Google Scholar 

  6. Cai, X.C., Sarkis, M.: A restricted additive Schwarz preconditioner for general sparse linear systems. SIAM J. Sci. Comput. 21, 792–797 (1999)

    MathSciNet  MATH  Google Scholar 

  7. Celiker, F., Cockburn, B., Shi, K.: Hybridizable discontinuous Galerkin methods for Timoshenko beams. J. Sci. Comput. 44(1), 1–37 (2010)

    MathSciNet  MATH  Google Scholar 

  8. Cesmelioglu, A., Cockburn, B., Nguyen, N.C., Peraire, J.: Analysis of HDG methods for Oseen equations. J. Sci. Comput. 55, 392–431 (2013)

    MathSciNet  MATH  Google Scholar 

  9. Cesmelioglu, A., Cockburn, B., Qiu, W.: Analysis of a hybridizable discontinuous Galerkin method for the steady-state incompressible Navier–Stokes equations. Math. Comput. 86(306), 1643–1670 (2017)

    MathSciNet  MATH  Google Scholar 

  10. Chabaud, B., Cockburn, B.: Uniform-in-time superconvergence of HDG methods for the heat equation. Math. Comput. 81, 107–129 (2012)

    MathSciNet  MATH  Google Scholar 

  11. Chaurasia, H.K.: A time-spectral hybridizable discontinuous Galerkin method for periodic flow problems. Ph.D. thesis, Massachusetts Institute of Technology (2014)

  12. Chaurasia, H.K., Nguyen, N.C., Peraire, J.: A Time-spectral hybridizable discontinuous Galerkin method for periodic flow problems. In: 21st AIAA Computational Fluid Dynamics Conference, Fluid Dynamics and Co-located Conferences, AIAA 2013-2861. American Institute of Aeronautics and Astronautics (2013)

  13. Chen, G., Xie, X.: A robust weak galerkin finite element method for linear elasticity with strong symmetric stresses. Comput. Methods Appl. Math. 16(3), 389–408 (2016)

    MathSciNet  MATH  Google Scholar 

  14. Chen, H., Qiu, W., Shi, K., Solano, M.: A superconvergent HDG method for the Maxwell equations. J. Sci. Comput. 70(3), 1010–1029 (2017)

    MathSciNet  MATH  Google Scholar 

  15. Christophe, A., Descombes, S., Lanteri, S.: An implicit hybridized discontinuous Galerkin method for the 3D time-domain Maxwell equations. Appl. Math. Comput. 319(Supplement C), 395–408 (2018)

    MathSciNet  Google Scholar 

  16. Ciuca, C.: Implicit hybridized discontinuous Galerkin methods for magnetohydrodynamics. Master’s thesis, Imperial College London (2018)

  17. Cockburn, B., Dong, B., Guzmán, J.: A superconvergent LDG-hybridizable Galerkin method for second-order elliptic problems. Math. Comput. 77, 1887–1916 (2008)

    MathSciNet  MATH  Google Scholar 

  18. Cockburn, B., Dong, B., Guzmán, J., Restelli, M., Sacco, R.: A hybridizable discontinuous Galerkin method for steady-state convection-diffusion-reaction problems. SIAM J. Sci. Comput. 31(5), 3827–3846 (2009)

    MathSciNet  MATH  Google Scholar 

  19. Cockburn, B., Fu, G.: Superconvergence by \(m\)-decompositions. Part ii: construction of two-dimensional finite elements. ESAIM Math. Model. Numer. Anal. 51(1), 165–186 (2017)

    MathSciNet  MATH  Google Scholar 

  20. Cockburn, B., Fu, G.: Superconvergence by \(m\)-decompositions. Part iii: construction of three-dimensional finite elements. ESAIM Math. Model. Numer. Anal. 51(1), 365–398 (2017)

    MathSciNet  MATH  Google Scholar 

  21. Cockburn, B., Fu, G.: Devising superconvergent hdg methods with symmetric approximate stresses for linear elasticity by M-decompositions. IMA J. Numer. Anal. 38(2), 566–604 (2018)

    MathSciNet  Google Scholar 

  22. Cockburn, B., Fu, G., Qiu, W.: A note on the devising of superconvergent hdg methods for stokes flow by m-decompositions. IMA J. Numer. Anal. 37(2), 730–749 (2017)

    MathSciNet  MATH  Google Scholar 

  23. Cockburn, B., Fu, G., Sayas, F.: Superconvergence by \(m\)-decompositions. Part i: general theory for hdg methods for diffusion. Math. Comput. 86(306), 1609–1641 (2017)

    MathSciNet  MATH  Google Scholar 

  24. Cockburn, B., Fu, Z., Hungria, A., Ji, L., Sánchez, M.A., Sayas, F.-J.: Stormer-Numerov HDG methods for acoustic waves. J. Sci. Comput. 75(2), 597–624 (2017)

    MathSciNet  MATH  Google Scholar 

  25. Cockburn, B., Gopalakrishnan, J., Lazarov, R.: Unified hybridization of discontinuous Galerkin, mixed, and continuous Galerkin methods for second order elliptic problems. SIAM J. Numer. Anal. 47(2), 1319–1365 (2009)

    MathSciNet  MATH  Google Scholar 

  26. Cockburn, B., Gopalakrishnan, J., Nguyen, N.C., Peraire, J., Sayas, F.-J.: Analysis of HDG methods for Stokes flow. Math. Comput. 80, 723–760 (2011)

    MathSciNet  MATH  Google Scholar 

  27. Cockburn, B., Gopalakrishnan, J., Sayas, F.-J.: A projection-based error analysis of HDG methods. Math. Comput. 79, 1351–1367 (2010)

    MathSciNet  MATH  Google Scholar 

  28. Cockburn, B., Guzmán, J., Wang, H.: Superconvergent discontinuous Galerkin methods for second-order elliptic problems. Math. Comput. 78, 1–24 (2009)

    MathSciNet  MATH  Google Scholar 

  29. Cockburn, B., Nguyen, N.C., Peraire, J.: A comparison of HDG methods for Stokes flow. J. Sci. Comput. 45(1–3), 215–237 (2010)

    MathSciNet  MATH  Google Scholar 

  30. Cockburn, B., Nguyen, N.C., Peraire, J.: A comparison of HDG methods for Stokes flow. J. Sci. Comput. 45(1), 215–237 (2010)

    MathSciNet  MATH  Google Scholar 

  31. Cockburn, B., Qiu, W., Shi, K.: Conditions for superconvergence of hdg methods for second-order elliptic problems. Math. Comput. 81(279), 1327–1353 (2012)

    MathSciNet  MATH  Google Scholar 

  32. Cockburn, B., Qiu, W., Shi, K.: Superconvergent hdg methods on isoparametric elements for second-order elliptic problems. SIAM J. Numer. Anal. 50(3), 1417–1432 (2012)

    MathSciNet  MATH  Google Scholar 

  33. Cockburn, B., Quenneville-Bélair, V.: Uniform-in-time superconvergence of HDG methods for the acoustic wave equation. Math. Comput. 83, 65–85 (2014)

    MathSciNet  MATH  Google Scholar 

  34. Cockburn, B., Sayas, F.J.: Divergence-conforming HDG methods for Stokes flow. Math. Comput. 83, 1571–1598 (2014)

    MathSciNet  MATH  Google Scholar 

  35. Cockburn, B., Shi, K.: Conditions for superconvergence of HDG methods for Stokes flow. Math. Comput. 82, 651–671 (2013)

    MathSciNet  MATH  Google Scholar 

  36. Cockburn, B., Shi, K.: Superconvergent HDG methods for linear elasticity with weakly symmetric stresses. IMA J. Numer. Anal. 33(3), 747–770 (2013)

    MathSciNet  MATH  Google Scholar 

  37. Cockburn, B., Di Pietro, D.A., Ern, A.: Bridging the hybrid high-order and hybridizable discontinuous galerkin methods. ESAIM Math. Model. Numer. Anal. 50(3), 635–650 (2016)

    MathSciNet  MATH  Google Scholar 

  38. Cui, J., Zhang, W.: An analysis of HDG methods for the Helmholtz equation. IMA J. Numer. Anal. 34(1), 279–295 (2014)

    MathSciNet  MATH  Google Scholar 

  39. Dahm, J.P.S., Fidkowski, K.J.: Error estimation and adaptation in hybridized discontinuous Galerkin methods. In: 52nd Aerospace Sciences Meeting, AIAA 2014-0078 (2014)

  40. Dedner, A., Kemm, F., Kröner, D., Munz, C.D., Schnitzer, T., Wesenberg, M.: Hyperbolic divergence cleaning for the MHD equations. J. Comput. Phys. 175(2), 645–673 (2002)

    MathSciNet  MATH  Google Scholar 

  41. Di Pietro, D.A., Ern, A.: A hybrid high-order locking-free method for linear elasticity on general meshes. Comput. Methods Appl. Mech. Eng. 283, 1–21 (2015)

    MathSciNet  MATH  Google Scholar 

  42. Dong, B.: Optimally convergent HDG method for third-order Korteweg-de Vries type equations. J. Sci. Comput. 73(2), 712–735 (2017)

    MathSciNet  MATH  Google Scholar 

  43. Evans, C.R., Hawley, J.F.: Simulation of magnetohydrodynamic flows: a constrained transport method. Astrophys. J. 332(2), 659–677 (1988)

    Google Scholar 

  44. Eyck, A.T., Celiker, F., Lew, A.: Adaptive stabilization of discontinuous Galerkin methods for nonlinear elasticity: motivation, formulation, and numerical examples. Comput. Methods Appl. Mech. Eng. 197, 1–21 (2007)

    MathSciNet  MATH  Google Scholar 

  45. Eyck, A.T., Celiker, F., Lew, A.: Adaptive stabilization of discontinuous Galerkin methods for nonlinear elasticity: analytical estimates. Comput. Methods Appl. Mech. Eng. 197, 2989–3000 (2008)

    MathSciNet  MATH  Google Scholar 

  46. Feng, X., Lu, P., Xu, X.: A hybridizable discontinuous Galerkin method for the time-harmonic Maxwell equations with high wave number. Comput. Methods Appl. Math. 16(3), 429–445 (2016)

    MathSciNet  MATH  Google Scholar 

  47. Feng, X., Xing, Y.: Absolutely stable local discontinuous Galerkin methods for the Helmholtz equation with large wave number. Math. Comput. 82(283), 1269–1296 (2012)

    MathSciNet  MATH  Google Scholar 

  48. Fernandez, P.: The hybridized discontinuous Galerkin methods for large-eddy simulation of transitional and turbulent flows. PhD thesis, Department of Aeronautics and Astronautics, Massachusetts Institute of Technology (2018)

  49. Fernandez, P., Nguyen, N.C., Peraire, J.: Subgrid-scale modeling and implicit numerical dissipation in DG-based Large-Eddy Simulation. In: 23rd AIAA Computational Fluid Dynamics Conference, AIAA 2017-3951, Denver, Colorado, USA (2017)

  50. Fernandez, P., Nguyen, N.C., Peraire, J.: The hybridized discontinuous Galerkin method for implicit large-eddy simulation of transitional turbulent flows. J. Comput. Phys. 336, 308–329 (2017)

    MathSciNet  MATH  Google Scholar 

  51. Fernandez, P., Nguyen, N.C., Peraire, J.: A physics-based shock capturing method for unsteady laminar and turbulent flows. In: 56th AIAA Aerospace Sciences Meeting, Orlando, Florida, Jan 2018. American Institute of Aeronautics and Astronautics (2018)

  52. Fernandez, P., Nguyen, N.C., Peraire, J.: Entropy-stable hybridized discontinuous Galerkin methods for the compressible Euler and Navier–Stokes equations. Comput. Methods Appl. Mech. Eng. (Under Review). arXiv preprint arXiv:1808.05066 (2018)

  53. Fernandez, P., Nguyen, N.C., Peraire, J.: A physics-based shock capturing method for large-eddy simulation. J. Comput. Phys. (Under Review). arXiv preprint arXiv:1806.06449 (2018)

  54. Fernandez, P., Nguyen, N.C., Roca, X., Peraire, J.: Implicit large-eddy simulation of compressible flows using the interior embedded discontinuous Galerkin method. In: 54th AIAA Aerospace Sciences Meeting, San Diego, California, USA, Jan 2016. American Institute of Aeronautics and Astronautics (2016)

  55. Fernandez, P., Moura, R., Mengaldo, G., Peraire, J.: Non-modal analysis of spectral element methods: towards accurate and robust large-eddy simulations. arXiv preprint arXiv:1804.09712 (2018)

  56. Fidkowski, K.J.: A hybridized discontinuous Galerkin method on mapped deforming domains. Comput. Fluids 139, 80–91 (2016)

    MathSciNet  MATH  Google Scholar 

  57. Fu, G., Cockburn, B., Stolarski, H.: Analysis of an HDG method for linear elasticity. Int. J. Numer. Methods Eng. 102(3–4), 551–575 (2015)

    MathSciNet  MATH  Google Scholar 

  58. Giorgiani, G., Fernández-Méndez, S., Huerta, A.: Hybridizable discontinuous Galerkin p-adaptivity for wave propagation problems. Int. J. Numer. Methods Fluids 72(12), 1244–1262 (2013)

    MathSciNet  Google Scholar 

  59. Giorgiani, G., Fernández-Méndez, S., Huerta, A.: Hybridizable discontinuous Galerkin with degree adaptivity for the incompressible Navier–Stokes equations. Comput. Fluids 98, 196–208 (2014)

    MathSciNet  MATH  Google Scholar 

  60. Gopalakrishnan, J., Li, F., Nguyen, N.C., Peraire, J.: Spectral approximations by the HDG method. Math. Comput. 84(293), 1037–1059 (2015)

    MathSciNet  MATH  Google Scholar 

  61. Griesmaier, R., Monk, P.: Error analysis for a hybridizable discontinuous Galerkin method for the Helmholtz equation. J. Sci. Comput. 49(3), 291–310 (2011)

    MathSciNet  MATH  Google Scholar 

  62. Gürkan, C., Kronbichler, M., Fernández-Méndez, S.: Extended hybridizable discontinuous Galerkin with heaviside enrichment for heat bimaterial problems. J. Sci. Comput. 72(2), 542–567 (2017)

    MathSciNet  MATH  Google Scholar 

  63. Güzey, S., Cockburn, B., Stolarski, H.K.: The embedded discontinuous Galerkin methods: application to linear shells problems. Int. J. Numer. Methods Eng. 70, 757–790 (2007)

    MathSciNet  MATH  Google Scholar 

  64. Hungria, A., Prada, D., Sayas, F.-J.: HDG methods for elastodynamics. Comput. Math. Appl. 74(11), 2671–2690 (2017)

    MathSciNet  MATH  Google Scholar 

  65. Huynh, L.N.T., Nguyen, N.C., Peraire, J., Khoo, B.C.: A high-order hybridizable discontinuous Galerkin method for elliptic interface problems. Int. J. Numer. Methods Fluids 93(2), 183–200 (2013)

    MathSciNet  MATH  Google Scholar 

  66. Jaust, A., Schütz, J.: A temporally adaptive hybridized discontinuous Galerkin method for time-dependent compressible flows. Comput. Fluids 98, 177–185 (2014)

    MathSciNet  MATH  Google Scholar 

  67. Kabaria, H., Lew, A., Cockburn, B.: A hybridizable discontinuous Galerkin formulation for non-linear elasticity. Comput. Methods Appl. Mech. Eng. 283, 303–329 (2015)

    MathSciNet  MATH  Google Scholar 

  68. Kolkman, L.N.: Implementation of an implicit-explicit scheme for hybridizable discontinuous Galerkin methods. Master’s thesis, Massachusetts Institute of Technology (2018)

  69. Lehrenfeld, C., Schöberl, J.: High order exactly divergence-free hybrid discontinuous Galerkin methods for unsteady incompressible flows. Comput. Methods Appl. Mech. Eng. 307, 339–361 (2016)

    MathSciNet  Google Scholar 

  70. Li, F., Xu, L., Yakovlev, S.: Central discontinuous Galerkin methods for ideal MHD equations with the exactly divergence-free magnetic field. J. Comput. Phys. 230(12), 4828–4847 (2011)

    MathSciNet  MATH  Google Scholar 

  71. Li, L., Lanteri, S., Mortensen, N.A., Wubs, M.: A hybridizable discontinuous Galerkin method for solving nonlocal optical response models. Comput. Phys. Commun. 219, 99–107 (2017)

    MathSciNet  Google Scholar 

  72. Li, L., Lanteri, S., Perrussel, R.: A hybridizable discontinuous Galerkin method combined to a Schwarz algorithm for the solution of 3d time-harmonic Maxwell’s equations. J. Comput. Phys. 256, 563–581 (2014)

    MathSciNet  MATH  Google Scholar 

  73. Li, L., Lanteri, S., Perrussel, R.: A class of locally well-posed hybridizable discontinuous Galerkin methods for the solution of time-harmonic Maxwell’s equations. Comput. Phys. Commun. 192, 23–31 (2015)

    MathSciNet  MATH  Google Scholar 

  74. Lu, P., Chen, H., Qiu, W.: An absolutely stable hp-HDG method for the time-harmonic Maxwell equations with high wave number. Math. Comput. 86(306), 1553–1577 (2017)

    MathSciNet  MATH  Google Scholar 

  75. McGhee, R., Walker, B., Millard, B.: Experimental Results for the Eppler 387 airfoil at Low Reynolds Number in the Langley Low-Turbulence Pressure Tunnel. Technical Report, NASA Langley Research Center, Langley (1988)

  76. Moro, D., Nguyen, N.C., Peraire, J.: Navier–Stokes Solution Using Hybridizable Discontinuous Galerkin methods. Hawaii, Technical Report, Honolulu (2011)

  77. Moro, D., Nguyen, N.C., Peraire, J.: Dilation-based shock capturing for high-order methods. Int. J. Numer. Methods Fluids 82(7), 398–416 (2016)

    MathSciNet  Google Scholar 

  78. Moro, D., Nguyen, N.C., Peraire, J., Drela, M.: Advances in the development of a high order, viscous-inviscid interaction solver. In: 21st AIAA Computational Fluid Dynamics Conference, AIAA 2013-2943, San Diego (2013)

  79. Moro, D., Nguyen, N.C., Peraire, J., Drela, M.: Mesh topology preserving boundary-layer adaptivity method for steady viscous flows. AIAA J. 55(6), 1970–1985 (2017)

    Google Scholar 

  80. Munz, C.D., Omnes, P., Schneider, R., Sonnendrücker, E., Voß, U.: Divergence correction techniques for Maxwell solvers based on a hyperbolic model. J. Comput. Phys. 161(2), 484–511 (2000)

    MathSciNet  MATH  Google Scholar 

  81. Nguyen, N.C., Peraire, J., Cockburn, B.: Hybridizable discontinuous Galerkin methods. In: Proceedings of the International Conference on Spectral and High Order Methods, Trondheim (2009)

  82. Nguyen, N.C., Peraire, J., Cockburn, B.: A hybridizable discontinuous Galerkin method for the incompressible Navier–Stokes equations. In: Proceedings of the 48th AIAA Aerospace Sciences Meeting and Exhibit, AIAA 2010-362, Orlando (2010)

  83. Nguyen, N.C., Peraire, J.: An adaptive shock-capturing HDG method for compressible flows. In: 20th AIAA Computational Fluid Dynamics Conference, AIAA 2011–3060, Reston, Virigina, 2011. American Institute of Aeronautics and Astronautics (2011)

  84. Nguyen, N.C., Peraire, J.: Hybridizable discontinuous Galerkin methods for partial differential equations in continuum mechanics. J. Comput. Phys. 231(18), 5955–5988 (2012)

    MathSciNet  MATH  Google Scholar 

  85. Nguyen, N.C., Peraire, J., Cockburn, B.: An implicit high-order hybridizable discontinuous Galerkin method for linear convection diffusion equations. J. Comput. Phys. 228(9), 3232–3254 (2009)

    MathSciNet  MATH  Google Scholar 

  86. Nguyen, N.C., Peraire, J., Cockburn, B.: An implicit high-order hybridizable discontinuous Galerkin method for nonlinear convection diffusion equations. J. Comput. Phys. 228(23), 8841–8855 (2009)

    MathSciNet  MATH  Google Scholar 

  87. Nguyen, N.C., Peraire, J., Cockburn, B.: A hybridizable discontinuous Galerkin method for Stokes flow. Comput. Methods Appl. Mech. Eng. 199(9–12), 582–597 (2010)

    MathSciNet  MATH  Google Scholar 

  88. Nguyen, N.C., Peraire, J., Cockburn, B.: An implicit high-order hybridizable discontinuous Galerkin method for the incompressible Navier–Stokes equations. J. Comput. Phys. 230(4), 1147–1170 (2011)

    MathSciNet  MATH  Google Scholar 

  89. Nguyen, N.C., Peraire, J., Cockburn, B.: High-order implicit hybridizable discontinuous Galerkin methods for acoustics and elastodynamics. J. Comput. Phys. 230(10), 3695–3718 (2011)

    MathSciNet  MATH  Google Scholar 

  90. Nguyen, N.C., Peraire, J., Cockburn, B.: Hybridizable discontinuous Galerkin methods for the time-harmonic Maxwell’s equations. J. Comput. Phys. 230(19), 7151–7175 (2011)

    MathSciNet  MATH  Google Scholar 

  91. Nguyen, N.C., Peraire, J., Cockburn, B.: A class of embedded discontinuous Galerkin methods for computational fluid dynamics. J. Comput. Phys. 302, 674–692 (2015)

    MathSciNet  MATH  Google Scholar 

  92. Nguyen, N.C., Peraire, J., Reitich, F., Cockburn, B.: A phase-based hybridizable discontinuous Galerkin method for the numerical solution of the Helmholtz equation. J. Comput. Phys. 290, 318–335 (2015)

    MathSciNet  MATH  Google Scholar 

  93. Nguyen, N.C., Roca, X., Moro, D., Peraire, J.: A hybridized multiscale discontinuous Galerkin method for compressible flows. In: 51st AIAA Aerospace Sciences Meeting iIncluding the New Horizons Forum and Aerospace Exposition, AIAA-2013-689 (2013)

  94. Park, H.-R., Chen, X., Nguyen, N.C., Oh, S.-H., Peraire, J.: Nanogap-enhanced Terahertz sensing of 1-nm-thick dielectric films. ACS Photonics 2(3), 417–424 (2015)

    Google Scholar 

  95. Peraire, J., Nguyen, N.C., Cockburn, B.: A hybridizable discontinuous Galerkin method for the compressible Euler and Navier–Stokes equations. In: 48th AIAA Aerospace Sciences Meeting Including the New Horizons Forum and Aerospace Exposition, AIAA 2010-363 (2010)

  96. Peraire, J., Nguyen, N.C., Cockburn, B.: An embedded discontinuous Galerkin method for the compressible Euler and Navier–Stokes equations. In 20th AIAA Computational Fluid Dynamics Conference, AIAA 2011-3228, Reston, Virigina, Jun 2011. American Institute of Aeronautics and Astronautics (2011)

  97. Qiu, W., Shen, J., Shi, K.: An HDG method for linear elasticity with strong symmetric stresses. Math. Comput. 87(309), 69–93 (2018)

    MathSciNet  MATH  Google Scholar 

  98. Qiu, W., Shi, K.: A superconvergent HDG method for the incompressible Navier–Stokes equations on general polyhedral meshes. IMA J. Numer. Anal. 36(4), 1943–1967 (2016)

    MathSciNet  MATH  Google Scholar 

  99. Rhebergen, S., Cockburn, B.: A space-time hybridizable discontinuous Galerkin method for incompressible flows on deforming domains. J. Comput. Phys. 231(11), 4185–4204 (2012)

    MathSciNet  MATH  Google Scholar 

  100. Roca, X., Nguyen, N.C., Peraire, J.: Scalable parallelization of the hybridized discontinuous Galerkin method for compressible flow. In: 21st AIAA Computational Fluid Dynamics Conference, AIAA-2013-2939 (2013)

  101. Saad, Y., Schultz, M.H.: GMRES: a generalized minimal residual algorithm for solving nonsymmetric linear systems. SIAM J. Sci. Stat. Comput. 7, 856–869 (1986)

    MathSciNet  MATH  Google Scholar 

  102. Samii, A., Dawson, C.: An explicit hybridized discontinuous Galerkin method for Serre–Green–Naghdi wave model. Comput. Methods Appl. Mech. Eng. 330(Supplement C), 447–470 (2018)

    MathSciNet  Google Scholar 

  103. Sánchez, M.A., Ciuca, C., Nguyen, N.C., Peraire, J., Cockburn, B.: Symplectic Hamiltonian HDG methods for wave propagation phenomena. J. Comput. Phys. 350(Supplement C), 951–973 (2017)

    MathSciNet  MATH  Google Scholar 

  104. Schütz, J., May, G.: A hybrid mixed method for the compressible Navier–Stokes equations. J. Comput. Phys. 240, 58–75 (2013)

    MathSciNet  MATH  Google Scholar 

  105. Schutz, J., May, G.: An adjoint consistency analysis for a class of hybrid mixed methods. IMA J. Numer. Anal. 34, 1222–1239 (2013)

    MathSciNet  MATH  Google Scholar 

  106. Sheldon, J.P., Miller, S.T., Pitt, J.S.: A hybridizable discontinuous Galerkin method for modeling fluid-structure interaction. J. Comput. Phys. 326(Supplement C), 91–114 (2016)

    MathSciNet  MATH  Google Scholar 

  107. Soon, S.-C.: Hybridizable discontinuosu Galerkin methods for solid mechanics. Ph.D thesis, University of Minnesota (2008)

  108. Soon, S.-C., Cockburn, B., Stolarski, H.K.: A hybridizable discontinuous Galerkin method for linear elasticity. Int. J. Numer. Methods Eng. 80(8), 1058–1092 (2009)

    MathSciNet  MATH  Google Scholar 

  109. Stanglmeier, M., Nguyen, N.C., Peraire, J., Cockburn, B.: An explicit hybridizable discontinuous Galerkin method for the acoustic wave equation. Comput. Methods Appl. Mech. Eng. 300, 748–769 (2016)

    MathSciNet  Google Scholar 

  110. Terrana, S., Vilotte, J.-P., Guillot, L.: A spectral hybridizable discontinuous galerkin method for elastic–acoustic wave propagation. Geophys. J. Int. 213(1), 574–602 (2018)

    Google Scholar 

  111. Ueckermann, M.P., Lermusiaux, P.F.J.: High-order schemes for 2D unsteady biogeochemical ocean models. Ocean Dyn. 60(6), 1415–1445 (2010)

    Google Scholar 

  112. Ueckermann, M.P., Lermusiaux, P.F.J.: Hybridizable discontinuous Galerkin projection methods for Navier–Stokes and Boussinesq equations. J. Comput. Phys. 306, 390–421 (2016)

    MathSciNet  MATH  Google Scholar 

  113. Vidal-Codina, F., Nguyen, N.C., Oh, S.-H., Peraire, J.: A hybridizable discontinuous Galerkin method for computing nonlocal electromagnetic effects in three-dimensional metallic nanostructures. J. Comput. Phys. 355, 548–565 (2018)

    MathSciNet  MATH  Google Scholar 

  114. Williams, D.: An entropy stable, hybridizable discontinuous Galerkin method for the compressible Navier–Stokes equations. Math. Comput. 87(309), 95–121 (2018)

    MathSciNet  MATH  Google Scholar 

  115. Woopen, M., Balan, A., May, G.: A hybridized hiscontinuous Galerkin method for three-dimensional compressible flow problems. In: 52nd Aerospace Sciences Meeting, AIAA 2014-0938 (2014)

  116. Woopen, M., Balan, A., May, G., Schütz, J.: A comparison of hybridized and standard DG methods for target-based hp-adaptive simulation of compressible flow. Comput. Fluids 98, 3–16 (2014)

    MathSciNet  MATH  Google Scholar 

  117. Woopen, M., May, G., Schütz, J.: Adjoint-based error estimation and mesh adaptation for hybridized discontinuous Galerkin methods. Int. J. Numer. Methods Fluids 76(11), 811–834 (2014)

    MathSciNet  Google Scholar 

  118. Yoo, D., Nguyen, N.C., Martin-Moreno, L., Mohr, D.A., Carretero-Palacios, S., Shaver, J., Peraire, J., Ebbesen, T.W., Oh, S.H.: High-throughput fabrication of resonant metamaterials with ultrasmall coaxial apertures via atomic layer lithography. Nano Lett. 16(3), 2040–2046 (2016)

    Google Scholar 

  119. Zhu, L., Huang, T.Z., Li, L.: A hybrid-mesh hybridizable discontinuous Galerkin method for solving the time-harmonic Maxwell’s equations. Appl. Math. Lett. 68, 109–116 (2017)

    MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

The authors acknowledge the Air Force Office of Scientific Research (FA9550-15-1-0276 and FA9550-16-1-0214), the NASA (NNX16AP15A), and Pratt & Whitney for supporting this work. P. Fernandez also acknowledges the financial support from the Zakhartchenko and “la Caixa” Fellowships.

Author information

Authors and Affiliations

  1. MIT Department of Aeronautics and Astronautics, 77 Massachusetts Ave., Cambridge, MA, 02139, USA

    P. Fernandez, A. Christophe, S. Terrana, N. C. Nguyen & J. Peraire

Authors
  1. P. Fernandez
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. A. Christophe
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. S. Terrana
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. N. C. Nguyen
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. J. Peraire
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to N. C. Nguyen.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Fernandez, P., Christophe, A., Terrana, S. et al. Hybridized Discontinuous Galerkin Methods for Wave Propagation. J Sci Comput 77, 1566–1604 (2018). https://doi.org/10.1007/s10915-018-0811-x

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10915-018-0811-x

Keywords

Search

Navigation