High Order and Underresolution

  • Andrea BeckEmail author
  • Gregor Gassner
  • Claus-Dieter Munz
Part of the Notes on Numerical Fluid Mechanics and Multidisciplinary Design book series (NNFM, volume 120)


In this work, the accuracy of high order discontinuous Galerkin discretizations for underresolved problems is investigated. Whereas the superior behavior of high order methods for the well resolved case is undisputed, in case of underresolution, the answer is not as clear. The controversy originates from the fact that order of convergence is a concept for discretization parameters tending to zero, whereas underresolution is synonym for large discretization parameters. However, this work shows that even in the case of underresolution, high order discontinuous Galerkin approximations yield superior efficiency compared to their lower order variants due to the better dispersion and dissipation behavior. It is furthermore shown that a very high order accurate discretization (theoretically 16th order in this case) yields even better accuracy than state-of-the-art large eddy simulation methods for the same number of degrees of freedom for the considered example. This result is particularly surprising since those large eddy simulation methods are tuned specifically to capture coarsely resolved turbulence, whereas the considered high order method can be applied directly to a wide range of other multi-scale problems without additional parameter tuning.


Integration Point Discontinuous Galerkin Method Gauss Point Spectral Element Method Conservative Variable 
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  1. 1.
    Gassner, G.J., Kopriva, D.A.: A comparison of the dispersion and dissipation errors of Gauss and Gauss-Lobatto discontinuous Galerkin spectral element methods. SIAM J. Sci. Comput. 33, 2560–2579 (2011)MathSciNetzbMATHCrossRefGoogle Scholar
  2. 2.
    Gassner, G.J., Beck, A.D.: On the accuracy of high-order discretizations for underresolved turbulence simulations. Theoretical and Computational Fluid Dynamics (2012), doi:10.1007/s00162-011-0253-7Google Scholar
  3. 3.
    Brachet, M.E., Meiron, D.I., Orszag, S.A., Nickel, B.G., Morf, R.H., Frisch, U.: Small-scale structure of the Taylor-Green vortex. Journal of Fluid Mechanics 130, 411–452 (1983)zbMATHCrossRefGoogle Scholar
  4. 4.
    Toro, E.F.: Riemann Solvers and Numerical Methods for Fluid Dynamics. Springer (1999)Google Scholar
  5. 5.
    Bassi, F., Rebay, S.: A high-order accurate discontinuous finite element method for the numerical solution of the compressible Navier-Stokes equations. J. Comput. Phys. 131, 267–279 (1997)MathSciNetzbMATHCrossRefGoogle Scholar
  6. 6.
    Arnold, D.N., Brezzi, F., Cockburn, B., Marini, L.D.: Unified analysis of discontinuous Galerkin methods for elliptic problems. SIAM J. Numer. Anal. 39, 1749–1779 (2002)MathSciNetzbMATHCrossRefGoogle Scholar
  7. 7.
    Hesthaven, J.S., Warburton, T.: Nodal Discontinuous Galerkin Methods: Algorithms, Analysis, and Applications. Springer, New York (2008)zbMATHCrossRefGoogle Scholar
  8. 8.
    Kopriva, D.A.: Implementing Spectral Methods for Partial Differential Equations: Algorithms for Scientists and Engineers. Springer (2009)Google Scholar
  9. 9.
    Kirby, R.M., Karniadakis, G.E.: De-aliasing on non-uniform grids: algorithms and application. J. Comput. Phys. 191, 249–264 (2003)zbMATHCrossRefGoogle Scholar
  10. 10.
    Ohlsson, J., Schlatter, P., Fischer, P.F., Henningson, D.S.: Stabilization of the spectral-element method in turbulent flow simulations. Lecture Notes in Computational Science and Engineering, vol. 76, pp. 449–458. Springer (2011)Google Scholar
  11. 11.
    Hickel, S.: Implicit Turbulence Modeling for Large-Eddy Simulation. Dissertation, Technische Universität München, Munich, Germany (2008)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • Andrea Beck
    • 1
    Email author
  • Gregor Gassner
    • 1
  • Claus-Dieter Munz
    • 1
  1. 1.Institute of Aerodynamics and GasdynamicsUniversität StuttgartStuttgartGermany

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