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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
Article

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.

Keywords

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

Notes

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.

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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

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