Skip to main content
SpringerLink
Log in

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

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

Abstract

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.

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

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

Similar content being viewed by others

References

  1. Babuška, I.: The finite element method with Lagrangian multipliers. Numer. Math. 20(3), 179–192 (1973)

    Article  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  6. Brezzi, F.: On the existence, uniqueness and approximation of saddle-point problems arising from Lagrangian multipliers. RAIRO Anal. Numér. 8, 129–151 (1974)

    MathSciNet  MATH  Google Scholar 

  7. Canuto, C., Hussaini, M.Y., Quarteroni, A., Zang, T.A.: Spectral Methods in Fluid Dynamics. Springer, Berlin (1988)

    Book  MATH  Google Scholar 

  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)

    Article  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  10. Deville, M., Fischer, P.F., Mund, E.: High-Order Methods for Incompressible Fluid Flow. Cambridge University Press, Cambridge (2002)

    Book  MATH  Google Scholar 

  11. Ervin, V., Layton, W., Neda, M.: Numerical analysis of filter based stabilization for evolution equations. Technical Report, University of Pittsburgh (2010)

  12. Fischer, P., Mullen, J.: Filter-based stabilization of spectral element methods. C.R. Acad. Sci. Paris t. 332(Serie I), 265–270 (2001)

  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)

    Article  MathSciNet  MATH  Google Scholar 

  14. Fischer, P.F., Kruse, G.W., Loth, F.: Spectral element methods for transitional flows. J. Sci. Comput. 17, 81–98 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  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. Fischer, P.F., Lottes, J.W., Kerkemeier, S.G.: nek5000 Web page (2008) http://nek5000.mcs.anl.gov

  17. Gervasio, P., Saleri, F.: Stabilized spectral element approximation for the Navier-Stokes equations. Numer. Methods Partial Differ. Equ. 14, 115–141 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  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 Conference

  19. Gottlieb, D., Hesthaven, J.S.: Spectral methods for hyperbolic problems. J. Comput. Appl. Math. 128, 83–131 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  20. Gottlieb, D., Orszag, S.A.: Numerical Analysis of Spectral Methods: Theory and Applications. SIAM-CBMS (1977)

  21. Guermond, Jean-Luc, Pasquetti, Richard, Popov, Bojan: Entropy viscosity method for nonlinear conservation laws. J. Comput. Phys. 230(11), 4248–4267 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  22. Hesthaven, J.S. Kirby, R.M.: Filtering in Legendre spectral methods. Math. Comput. 77, 1425–1452 (2008)

    Google Scholar 

  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)

    Article  MATH  Google Scholar 

  24. Jeong, J., Hussain, F.: On the identification of a vortex. J. Fluid Mech. 285, 69–94 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  25. Karamanos, G.-S., Karniadakis, G.E.: A spectral vanishing viscosity method for large-eddy simulations. J. Comput. Phys. 163(1), 22–50 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  26. Karniadakis, G.E., Sherwin, S.J.: Spectral/hp Element Methods for Computational Fluid Dynamics. Oxford University Press, New York (2005)

    Book  MATH  Google Scholar 

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

    Article  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  Google Scholar 

  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)

    Article  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MATH  Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

  36. Moin, P., Mahesh, K.: Direct numerical simulation: A tool in turbulence research. Annu. Rev. Fluid Mech. 30(1), 539–578 (1998)

    Article  MathSciNet  Google Scholar 

  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)

    Article  MATH  Google Scholar 

  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)

    Article  Google Scholar 

  39. Orszag, S.A.: Comparison of pseudospectral and spectral approximations. Stud. Appl. Math. 51, 253–259 (1972)

    MATH  Google Scholar 

  40. Orszag, S.A.: Spectral methods for problems in complex geometries. J. Comput. Phys. 37, 70–92 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  41. Pasquetti, R.: Spectral vanishing viscosity method for large-eddy simulation of turbulent flows. J. Sci. Comput. 27(1–3), 365–375 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  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. 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. 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. Pope, S.: Turbulent Flows. Cambridge University Press, New York (2000)

    Book  MATH  Google Scholar 

  46. Rønquist, E.M.: Optimal Spectral Element Methods for the Unsteady Three-Dimensional Incompressible Navier–Stokes Equations. PhD thesis, M.I.T., Cambridge (1988)

  47. Rønquist, E.M.: Convection treatment using spectral elements of different order. Int. J. Numer. Methods Fluids. 22(4), 241–264 (1996)

    Article  Google Scholar 

  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)

    Article  Google Scholar 

  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. 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)

    Article  Google Scholar 

  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)

    Article  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  53. Wilhelm, D., Kleiser, L.: Stable and unstable formulations of the convection operator in spectral element simulations. Appl. Numer. Math. 33, 275–280 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  55. Xu, C.: Stabilization methods for spectral element computations of incompressible flows. J. Sci. Comput. 27, 495–505 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  56. Xu, C., Pasquetti, R.: Stabilized spectral element computations of high Reynolds number incompressible flows. J. Comput. Phys. 196(2), 680–704 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  57. Zang, T.A.: On the rotation and skew-symmetric forms for incompressible flow simulations. Appl. Numer. Math. 7(1), 27–40 (1991)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgments

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.

Author information

Authors and Affiliations

  1. Linné FLOW Centre, Swedish e-Science Research Centre (SeRC), KTH Mechanics, 100 44 , Stockholm, Sweden

    Johan Malm, Philipp Schlatter & Dan S. Henningson

  2. MCS, Argonne National Laboratory, Lemont, IL, USA

    Paul F. Fischer

Authors
  1. Johan Malm
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Philipp Schlatter
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Paul F. Fischer
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Dan S. Henningson
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Johan Malm.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Malm, J., Schlatter, P., Fischer, P.F. et al. Stabilization of the Spectral Element Method in Convection Dominated Flows by Recovery of Skew-Symmetry. J Sci Comput 57, 254–277 (2013). https://doi.org/10.1007/s10915-013-9704-1

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10915-013-9704-1

Keywords

Search

Navigation