Journal of Scientific Computing

, Volume 57, Issue 2, pp 254–277 | Cite as

Stabilization of the Spectral Element Method in Convection Dominated Flows by Recovery of Skew-Symmetry

  • Johan Malm
  • Philipp Schlatter
  • Paul F. Fischer
  • Dan S. Henningson


We investigate stability properties of the spectral element method for advection dominated incompressible flows. In particular, properties of the widely used convective form of the nonlinear term are studied. We remark that problems which are usually associated with the nonlinearity of the governing Navier–Stokes equations also arise in linear scalar transport problems, which implicates advection rather than nonlinearity as a source of difficulty. Thus, errors arising from insufficient quadrature of the convective term, commonly referred to as ‘aliasing errors’, destroy the skew-symmetric properties of the convection operator. Recovery of skew-symmetry can be efficiently achieved by the use of over-integration. Moreover, we demonstrate that the stability problems are not simply connected to underresolution. We combine theory with analysis of the linear advection-diffusion equation in 2D and simulations of the incompressible Navier–Stokes equations in 2D of thin shear layers at a very high Reynolds number and in 3D of turbulent and transitional channel flow at moderate Reynolds number. For the Navier–Stokes equations, where the divergence-free constraint needs to be enforced iteratively to a certain accuracy, small divergence errors can be detrimental to the stability of the method and it is therefore advised to use additional stabilization (e.g. so-called filter-based stabilization, spectral vanishing viscosity or entropy viscosity) in order to assure a stable spectral element method.


Spectral element method (SEM) Stability Over-integration Skew-symmetry 



The authors gratefully acknowledge funding by VR (The Swedish Research Council) Computer time was provided by SNIC (Swedish National Infrastructure for Computing) with a generous grant by the Knut and Alice Wallenberg (KAW) Foundation. The simulations were run at the Centre for Parallel Computers (PDC) at the Royal Institute of Technology (KTH). The third author was supported by the U.S. Department of Energy under Contract DE-AC02-06CH11357.


  1. 1.
    Babuška, I.: The finite element method with Lagrangian multipliers. Numer. Math. 20(3), 179–192 (1973)CrossRefMATHGoogle Scholar
  2. 2.
    Bell, J.B., Colella, P., Glaz, H.M.: A second-order projection method for the incompressible navier-stokes equations. J. Comput. Phys. 85(2), 257–283 (1989)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Blackburn, H.M., Schmidt, S.: Spectral element filtering techniques for large-eddy simulation with dynamic estimation. J. Comput. Phys. 186(2), 610–629 (2003)CrossRefMATHGoogle Scholar
  4. 4.
    Blaisdell, G.A., Spyropoulos, E.T., Qin, J.H.: The effect of the formulation of nonlinear terms on aliasing errors in spectral methods. Appl. Numer. Math. 21(3), 207–219 (1996)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Boyd, J.P.: Two comments on filtering (artificial viscosity) for chebyshev and legendre spectral and spectral element methods: preserving boundary conditions and interpretation of the filter as a diffusion. J. Comput. Phys. 143(1), 283–288 (1998)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Brezzi, F.: On the existence, uniqueness and approximation of saddle-point problems arising from Lagrangian multipliers. RAIRO Anal. Numér. 8, 129–151 (1974)MathSciNetMATHGoogle Scholar
  7. 7.
    Canuto, C., Hussaini, M.Y., Quarteroni, A., Zang, T.A.: Spectral Methods in Fluid Dynamics. Springer, Berlin (1988)CrossRefMATHGoogle Scholar
  8. 8.
    Canuto, C., Russo, A., van Kemenade, V.: Stabilized spectral methods for the Navier-Stokes equations: residual-free bubbles and preconditioning. Comput. Meth. Appl. Mech. Eng. 166, 65–83 (1998)CrossRefMATHGoogle Scholar
  9. 9.
    Cottrell, J.A., Reali, A., Bazilevs, Y., Hughes, T.J.R.: Isogeometric analysis of structural vibrations. Comput. Methods Appl. Mech. Eng. 195, 5257–5296 (2006)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Deville, M., Fischer, P.F., Mund, E.: High-Order Methods for Incompressible Fluid Flow. Cambridge University Press, Cambridge (2002)CrossRefMATHGoogle Scholar
  11. 11.
    Ervin, V., Layton, W., Neda, M.: Numerical analysis of filter based stabilization for evolution equations. Technical Report, University of Pittsburgh (2010)Google Scholar
  12. 12.
    Fischer, P., Mullen, J.: Filter-based stabilization of spectral element methods. C.R. Acad. Sci. Paris t. 332(Serie I), 265–270 (2001) Google Scholar
  13. 13.
    Fischer, P.F.: An overlapping Schwarz method for spectral element solution of the incompressible Navier-Stokes equations. J. Comput. Phys. 133(1), 84–101 (1997)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Fischer, P.F., Kruse, G.W., Loth, F.: Spectral element methods for transitional flows. J. Sci. Comput. 17, 81–98 (2002)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Fischer, P.F., Lottes, J., Pointer, D., Siegel, A.: Petascale algorithms for reactor hydrodynamics. J. Phys. Conf. Ser. 125, 012076 (2008)Google Scholar
  16. 16.
    Fischer, P.F., Lottes, J.W., Kerkemeier, S.G.: nek5000 Web page (2008)
  17. 17.
    Gervasio, P., Saleri, F.: Stabilized spectral element approximation for the Navier-Stokes equations. Numer. Methods Partial Differ. Equ. 14, 115–141 (1998)MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Gilbert, N., Kleiser, L.: Near-wall phenomena in transition to turbulence. In Kline, S.J., Afgan, N.H. (eds.) Near-Wall Turbulence, pp. 7–27. New York, USA (1990). 1988 Zoran Zarić Memorial ConferenceGoogle Scholar
  19. 19.
    Gottlieb, D., Hesthaven, J.S.: Spectral methods for hyperbolic problems. J. Comput. Appl. Math. 128, 83–131 (2001)MathSciNetCrossRefMATHGoogle Scholar
  20. 20.
    Gottlieb, D., Orszag, S.A.: Numerical Analysis of Spectral Methods: Theory and Applications. SIAM-CBMS (1977)Google Scholar
  21. 21.
    Guermond, Jean-Luc, Pasquetti, Richard, Popov, Bojan: Entropy viscosity method for nonlinear conservation laws. J. Comput. Phys. 230(11), 4248–4267 (2011)MathSciNetCrossRefMATHGoogle Scholar
  22. 22.
    Hesthaven, J.S. Kirby, R.M.: Filtering in Legendre spectral methods. Math. Comput. 77, 1425–1452 (2008)Google Scholar
  23. 23.
    Hughes, T.J.R., Mazzei, L., Jansen, K.E.: Large eddy simulation and the variational multiscale method. Comput. Visual. Sci. 3, 47–59 (2000)CrossRefMATHGoogle Scholar
  24. 24.
    Jeong, J., Hussain, F.: On the identification of a vortex. J. Fluid Mech. 285, 69–94 (1995)MathSciNetCrossRefMATHGoogle Scholar
  25. 25.
    Karamanos, G.-S., Karniadakis, G.E.: A spectral vanishing viscosity method for large-eddy simulations. J. Comput. Phys. 163(1), 22–50 (2000)MathSciNetCrossRefMATHGoogle Scholar
  26. 26.
    Karniadakis, G.E., Sherwin, S.J.: Spectral/hp Element Methods for Computational Fluid Dynamics. Oxford University Press, New York (2005)CrossRefMATHGoogle Scholar
  27. 27.
    Kirby, R.M., Karniadakis, G.E.: De-alising on non-uniform grids: algorithms and applications. J. Comput. Phys. 191, 249–264 (2003)CrossRefMATHGoogle Scholar
  28. 28.
    Kirby, R.M., Sherwin, S.J.: Stabilisation of spectral/hp element methods through spectral vanishing viscosity: application to fluid mechanics modelling. Comput. Methods Appl. Mech. Eng. 195, 3128–3144 (2006)MathSciNetCrossRefMATHGoogle Scholar
  29. 29.
    Kline, S.J., Reynolds, W.C., Schraub, F.A., Runstadler, P.W.: The structure of turbulent boundary layers. J. Fluid Mech. 30, 741–773 (1967)CrossRefGoogle Scholar
  30. 30.
    Lee, S., Fischer, P.F., Bassiouny, H., Loth, F.: Direct numerical simulation of transitional flow in a stenosed carotid bifurcation. J. Biomech. 41, 2551–2561 (2008)CrossRefGoogle Scholar
  31. 31.
    Lomtev, I., Kirby, R.M., Karniadakis, G.E.: A discontinuous Galerkin ALE method for compressible viscous flows in moving domains. J. Comput. Phys. 155, 128–159 (1999)MathSciNetCrossRefMATHGoogle Scholar
  32. 32.
    Maday, Y., Patera, A.: Spectral element methods for the Navier-Stokes equations. In Noor, A.K. (ed.) State of the Art Surveys in Computational Mechanics, pp. 71–143. ASME (1989)Google Scholar
  33. 33.
    Maday, Y., Patera, A.T., Rønquist, E.M.: An operator-integration-factor splitting method for time-dependent problems: application to incompressible fluid flow. J. Sci. Comput. 5(4), 263–292 (1990)MathSciNetCrossRefMATHGoogle Scholar
  34. 34.
    Maday, Y., Rønquist, E.M.: Optimal error analysis of spectral methods with emphasis on non-constant coefficients and deformed geometries. Comput. Methods Appl. Mech. Eng. 80(1–3), 91–115 (1990)CrossRefMATHGoogle Scholar
  35. 35.
    Marras, S., Kelly, J.F., Giraldo, F.X., Vázquez, M.: Variational multiscale stabilization of high-order spectral elements for the advection-diffusion equation. J. Comput. Phys. 231(21), 7187–7213 (2012)MathSciNetCrossRefGoogle Scholar
  36. 36.
    Moin, P., Mahesh, K.: Direct numerical simulation: A tool in turbulence research. Annu. Rev. Fluid Mech. 30(1), 539–578 (1998)MathSciNetCrossRefGoogle Scholar
  37. 37.
    Moser, R.D., Kim, J., Mansour, N.: Direct numerical simulation of turbulent channel flow up to \(\text{ Re }_{\tau }=590\). Phys. Fluids. 11(4), 943–945 (1999)CrossRefMATHGoogle Scholar
  38. 38.
    Orszag, S.A.: Transform method for the calculation of vector-coupled sums: Application to the spectral form of the vorticity equation. J. Atmos. Sci. 27, 890–895 (1970)CrossRefGoogle Scholar
  39. 39.
    Orszag, S.A.: Comparison of pseudospectral and spectral approximations. Stud. Appl. Math. 51, 253–259 (1972)MATHGoogle Scholar
  40. 40.
    Orszag, S.A.: Spectral methods for problems in complex geometries. J. Comput. Phys. 37, 70–92 (1980)MathSciNetCrossRefMATHGoogle Scholar
  41. 41.
    Pasquetti, R.: Spectral vanishing viscosity method for large-eddy simulation of turbulent flows. J. Sci. Comput. 27(1–3), 365–375 (2006)MathSciNetCrossRefMATHGoogle Scholar
  42. 42.
    Pasquetti, R., Xu, C.J.: Comments on “Filter-based stabilization of spectral element methods”. Note J. Comput. Phys. 182, 646–650 (2002)Google Scholar
  43. 43.
    Pasquetti, R., Xu, C.J.: High-order algorithms for large-eddy simulation of incompressible flows. J. Sci. Comput. 17, 273–284 (2002)Google Scholar
  44. 44.
    Patera, A.T.: A spectral element method for fluid dynamics: Laminar flow in a channel expansion. J. Comput. Phys. 54, 468–488 (1984)Google Scholar
  45. 45.
    Pope, S.: Turbulent Flows. Cambridge University Press, New York (2000)CrossRefMATHGoogle Scholar
  46. 46.
    Rønquist, E.M.: Optimal Spectral Element Methods for the Unsteady Three-Dimensional Incompressible Navier–Stokes Equations. PhD thesis, M.I.T., Cambridge (1988)Google Scholar
  47. 47.
    Rønquist, E.M.: Convection treatment using spectral elements of different order. Int. J. Numer. Methods Fluids. 22(4), 241–264 (1996)CrossRefGoogle Scholar
  48. 48.
    Schlatter, P., Stolz, S., Kleiser, L.: LES of transitional flows using the approximate deconvolution model. Int. J. Heat Fluid Flow. 25(3), 549–558 (2004)CrossRefGoogle Scholar
  49. 49.
    Schlatter, P., Stolz, S., Kleiser, L.: Large-eddy simulation of spatial transition in plane channel flow. J. Turbulence. 7(33), 1–24 (2006)Google Scholar
  50. 50.
    Smith, C.R., Metzler, S.P.: The characteristics of low-speed streaks in the near-wall region of a turbulent boundary layer. J. Fluid Mech. 129, 27–54 (1983)CrossRefGoogle Scholar
  51. 51.
    Stolz, S., Schlatter, P., Kleiser, L.: High-pass filtered eddy-viscosity models for large-eddy simulations of transitional and turbulent flow. Phys. Fluids. 17(6), 065103 (2005)CrossRefGoogle Scholar
  52. 52.
    Wasberg, C.E., Gjesdal, T., Reif, B.A.P., Andreassen, Ø.: Variational multiscale turbulence modelling in a high order spectral element method. J. Comput. Phys. 228(19), 7333–7356 (2009)MathSciNetCrossRefMATHGoogle Scholar
  53. 53.
    Wilhelm, D., Kleiser, L.: Stable and unstable formulations of the convection operator in spectral element simulations. Appl. Numer. Math. 33, 275–280 (2000)MathSciNetCrossRefMATHGoogle Scholar
  54. 54.
    Wilhelm, D., Kleiser, L.: Stability analysis for different formulations of the nonlinear term in \({P}_{N}-{P}_{N-2}\), spectral element discretizations of the Navier-Stokes Equations. J. Comput. Phys. 174, 306–326 (2001)MathSciNetCrossRefMATHGoogle Scholar
  55. 55.
    Xu, C.: Stabilization methods for spectral element computations of incompressible flows. J. Sci. Comput. 27, 495–505 (2006)MathSciNetCrossRefMATHGoogle Scholar
  56. 56.
    Xu, C., Pasquetti, R.: Stabilized spectral element computations of high Reynolds number incompressible flows. J. Comput. Phys. 196(2), 680–704 (2004)MathSciNetCrossRefMATHGoogle Scholar
  57. 57.
    Zang, T.A.: On the rotation and skew-symmetric forms for incompressible flow simulations. Appl. Numer. Math. 7(1), 27–40 (1991)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  • Johan Malm
    • 1
  • Philipp Schlatter
    • 1
  • Paul F. Fischer
    • 2
  • Dan S. Henningson
    • 1
  1. 1.Linné FLOW Centre, Swedish e-Science Research Centre (SeRC)KTH MechanicsStockholmSweden
  2. 2.MCSArgonne National LaboratoryLemontUSA

Personalised recommendations