Skip to main content
SpringerLink
Log in

On the Quadrature and Weak Form Choices in Collocation Type Discontinuous Galerkin Spectral Element Methods

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

Abstract

We examine four nodal versions of tensor product discontinuous Galerkin spectral element approximations to systems of conservation laws for quadrilateral or hexahedral meshes. They arise from the two choices of Gauss or Gauss-Lobatto quadrature and integrate by parts once (I) or twice (II) formulations of the discontinuous Galerkin method. We show that the two formulations are in fact algebraically equivalent with either Gauss or Gauss-Lobatto quadratures when global polynomial interpolations are used to approximate the solutions and fluxes within the elements. Numerical experiments confirm the equivalence of the approximations and indicate that using Gauss quadrature with integration by parts once is the most efficient of the four approximations.

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

Access this article

Log in via an institution

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

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Black, K.: A conservative spectral element method for the approximation of compressible fluid flow. Kybernetika 35(1), 133–146 (1999)

    MathSciNet  Google Scholar 

  2. Black, K.: Spectral element approximation of convection-diffusion type problems. Appl. Numer. Math. 33(1–4), 373–379 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  3. Canuto, C., Hussaini, M.Y., Quarteroni, A., Zang, T.A.: Spectral Methods: Fundamentals in Single Domains. Springer, Berlin (2006)

    MATH  Google Scholar 

  4. Canuto, C., Hussaini, M.Y., Quarteroni, A., Zang, T.A.: Spectral Methods: Evolution to Complex Geometries and Applications to Fluid Dynamics. Springer, Berlin (2007)

    MATH  Google Scholar 

  5. Castel, N., Cohen, G., Durufle, M.: Application of discontinuous Galerkin spectral method on hexahedral elements for aeroacoustic. J. Comput. Acoust. 17(2), 175–196 (2009)

    Article  MathSciNet  Google Scholar 

  6. Courant, R., Friedrichs, K.O.: Supersonic Flow and Shock Waves. Springer, Berlin (1976)

    MATH  Google Scholar 

  7. Deng, S.: Numerical simulation of optical coupling and light propagation in coupled optical resonators with size disorder. Appl. Numer. Math. 57(5–7), 475–485 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  8. Deng, S.Z., Cai, W., Astratov, V.N.: Numerical study of light propagation via whispering gallery modes in microcylinder coupled resonator optical waveguides. Opt. Express 12(26), 6468–6480 (2004)

    Article  Google Scholar 

  9. Fagherazzi, S., Furbish, D.J., Rasetarinera, P., Hussaini, M.Y.: Application of the discontinuous spectral Galerkin method to groundwater flow. Adv. Water Resour. 27, 129–140 (2004)

    Article  Google Scholar 

  10. Fagherazzi, S., Rasetarinera, P., Hussaini, M.Y., Furbish, D.J.: Numerical solution of the dam-break problem with a discontinuous Galerkin method. J. Hydraul. Eng. 130(6), 532–539 (2004)

    Article  Google Scholar 

  11. Giraldo, F.X., Hesthaven, J.S., Warburton, T.: Nodal high-order discontinuous Galerkin methods for the spherical shallow water equations. J. Comput. Phys. 181(2), 499–525 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  12. Giraldo, F.X., Restelli, M.: A study of spectral element and discontinuous Galerkin methods for the Navier–Stokes equations in nonhydrostatic mesoscale atmospheric modeling: Equation sets and test cases. J. Comput. Phys. 227, 3849–3877 (2008)

    MATH  MathSciNet  Google Scholar 

  13. Gordon, W.J., Hall, C.A.: Construction of curvilinear co-ordinate systems and their applications to mesh generation. Int. J. Numer. Methods Eng. Eng. 7, 461–477 (1973)

    Article  MATH  MathSciNet  Google Scholar 

  14. Hesthaven, J.S., Warburton, T.: Nodal Discontinuous Galerkin Methods: Algorithms, Analysis, and Applications. Springer, Berlin (2008)

    Book  MATH  Google Scholar 

  15. Kirby, R.M., Karniadakis, G.E.: De-aliasing on non-uniform grids: algorithms and applications. J. Comput. Phys. 191, 249–264 (2003)

    Article  MATH  Google Scholar 

  16. Kopriva, D.A.: Metric identities and the discontinuous spectral element method on curvilinear meshes. J. Sci. Comput. 26(3), 301–327 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  17. Kopriva, D.A., Woodruff, S.L., Hussaini, M.Y.: Discontinuous spectral element approximation of Maxwell’s Equations. In: B. Cockburn, G. Karniadakis, C.-W. Shu (eds.) Proceedings of the International Symposium on Discontinuous Galerkin Methods. Springer, New York (2000)

    Google Scholar 

  18. Kopriva, D.A., Woodruff, S.L., Hussaini, M.Y.: Computation of electromagnetic scattering with a non-conforming discontinuous spectral element method. Int. J. Numer. Methods Eng. 53, 105–122 (2002)

    Article  MATH  Google Scholar 

  19. Kopriva, D.A.: Implementing Spectral Methods for Partial Differential Equations. Scientific Computation. Springer, Berlin (2009)

    Book  MATH  Google Scholar 

  20. Lomtev, I., M Kirby, R., Karniadakis, G.E.: A discontinuous Galerkin ALE method for compressible viscous flows in moving domains. J. Comput. Phys. 155, 128–159 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  21. Rasetarinera, P., Hussaini, M.Y.: An efficient implicit discontinuous spectral Galerkin method. J. Comput. Phys. 172, 718–738 (2001)

    Article  MATH  Google Scholar 

  22. Rasetarinera, P., Kopriva, D.A., Hussaini, M.Y.: Discontinuous spectral element solution of acoustic radiation from thin airfoils. AIAA J. 39(11), 2070–2075 (2001)

    Article  Google Scholar 

  23. Restelli, M., Giraldo, F.X.: A conservative discontinuous Galerkin semi-implicit formulation for the Navier–Stokes equations in nonhydrostatic mesoscale modeling. SIAM J. Sci. Comput. 31(3), 2231–2257 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  24. Roe, P.L.: Approximate Riemann solvers, parameter vectors, and difference schemes. J. Comput. Phys. 135(2), 250–258 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  25. Stanescu, D., Farassat, F., Y Hussaini, M.: Aircraft engine noise scattering—parallel discontinuous Galerkin spectral element method. Paper 2002-0800, AIAA (2002)

  26. Stanescu, D., Hussaini, M.Y., Farassat, F.: Aircraft engine noise scattering by fuselage and wings: a computational approach. J. Sound Vib. 263(2), 319–333 (2003)

    Article  Google Scholar 

  27. Stanescu, D., Xu, J., Farassat, F., Hussaini, M.Y.: Computation of engine noise propagation and scattering off an aircraft. Aeroacoustics 1(4), 403–420 (2002)

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, The Florida State University, Tallahassee, FL, 32312, USA

    David A. Kopriva

  2. Institute for Aerodynamics and Gasdynamics, University of Stuttgart, Pfaffenwaldring 21, 70550, Stuttgart, Germany

    Gregor Gassner

Authors
  1. David A. Kopriva
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Gregor Gassner
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to David A. Kopriva.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Kopriva, D.A., Gassner, G. On the Quadrature and Weak Form Choices in Collocation Type Discontinuous Galerkin Spectral Element Methods. J Sci Comput 44, 136–155 (2010). https://doi.org/10.1007/s10915-010-9372-3

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10915-010-9372-3

Keywords

Search

Navigation