Advertisement

On the Scaling of Entropy Viscosity in High Order Methods

  • Adeline KornelusEmail author
  • Daniel Appelö
Conference paper
Part of the Lecture Notes in Computational Science and Engineering book series (LNCSE, volume 119)

Abstract

In this work, we outline the entropy viscosity method and discuss how the choice of scaling influences the size of viscosity for a simple shock problem. We present examples to illustrate the performance of the entropy viscosity method under two distinct scalings.

Notes

Acknowledgements

Adeline Kornelus and Daniel Appelö are supported in part by NSF Grant DMS-1319054. Any conclusions or recommendations expressed in this paper are those of the author and do not necessarily reflect the views NSF.

References

  1. 1.
    F. Bassi, S. Rebay, 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)CrossRefzbMATHMathSciNetGoogle Scholar
  2. 2.
    J.-L. Guermond, R. Pasquetti, Entropy-based nonlinear viscosity for Fourier approximations of conservation laws. C. R. Math. 346, 801–806 (2008)CrossRefzbMATHMathSciNetGoogle Scholar
  3. 3.
    J.-L. Guermond, R. Pasquetti, Entropy viscosity method for high-order approximations of conservation laws, in Spectral and High Order Methods for Partial Differential Equations (Springer, Berlin, 2011), pp. 411–418CrossRefzbMATHGoogle Scholar
  4. 4.
    J.-L. Guermond, R. Pasquetti, Entropy viscosity method for higher-order approximations of conservation laws, in Spectral and High Order Methods for Partial Differential Equations. Lecture Notes in Computational Science and Engineering, vol. 76 (Springer, Berlin, 2011), pp. 411–418Google Scholar
  5. 5.
    J.-L. Guermond, R. Pasquetti, B. Popov, Entropy viscosity for conservation equations, in V European Conference on Computational Fluid Dynamics (Eccomas CFD 2010) (2010)Google Scholar
  6. 6.
    J.-L. Guermond, R. Pasquetti, B. Popov, Entropy viscosity method for nonlinear conservation laws. J. Comput. Phys. 230, 4248–4267 (2011)CrossRefzbMATHMathSciNetGoogle Scholar
  7. 7.
    J.-L. Guermond, R. Pasquetti, B. Popov, From suitable weak solutions to entropy viscosity. J. Sci. Comput. 49, 35–50 (2011)CrossRefzbMATHMathSciNetGoogle Scholar
  8. 8.
    J.S. Hesthaven, T. Warburton, Nodal Discontinuous Galerkin Methods: Algorithms, Analysis, and Applications, vol. 54 (Springer, New York, 2008)zbMATHGoogle Scholar
  9. 9.
    C. Johnson, A. Szepessy, P. Hansbo, On the convergence of shock-capturing streamline diffusion finite element methods for hyperbolic conservation laws. Math. Comput. 54, 107–129 (1990)CrossRefzbMATHMathSciNetGoogle Scholar
  10. 10.
    R.M. Kirby, G.E. Karniadakis, De-aliasing on non-uniform grids: algorithms and applications. J. Comput. Phys. 191, 249–264 (2003)CrossRefzbMATHGoogle Scholar
  11. 11.
    A. Kornelus, D. Appelö, Flux-conservative Hermite methods for nonlinear conservation laws (2017). Preprints submitted to J. Sci. Comp. arXiv1703.06848Google Scholar
  12. 12.
    R. LeVeque, Finite Volume Methods for Hyperbolic Problems, vol. 31 (Cambridge University Press, Cambridge, 2002)CrossRefzbMATHGoogle Scholar
  13. 13.
    P. Persson, J. Peraire, Sub-cell shock capturing for discontinuous Galerkin methods, in 44-th AIAA Aerospace Sciences Meeting and Exhibit (2006), pp. 1–13Google Scholar
  14. 14.
    C.-W. Shu, Essentially Non-oscillatory and Weighted Essentially Non-oscillatory Schemes for Hyperbolic Conservation Laws (Springer, Berlin, 1998)CrossRefzbMATHGoogle Scholar
  15. 15.
    H. Yee, B. Sjögreen, Development of low dissipative high order filter schemes for multiscale Navier–Stokes/MHD systems. J. Comput. Phys. 225, 910–934 (2007)CrossRefzbMATHMathSciNetGoogle Scholar
  16. 16.
    V. Zingan, J.-L. Guermond, J. Morel, B. Popov, Implementation of the entropy viscosity method with the discontinuous Galerkin method. Comput. Methods Appl. Mech. Eng. 253, 479–490 (2013)CrossRefzbMATHMathSciNetGoogle Scholar

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  1. 1.The School of Mathematical and Statistical SciencesArizona State UniversityTempeUSA
  2. 2.The Department of Applied MathematicsUniversity of ColoradoBoulderUSA

Personalised recommendations