Advertisement

An HDG Method for Unsteady Compressible Flows

  • Alexander Jaust
  • Jochen Schütz
  • Michael Woopen
Part of the Lecture Notes in Computational Science and Engineering book series (LNCSE, volume 106)

Abstract

Recent gain of interest in discontinuous Galerkin (DG) methods shows their success in computational fluid dynamics. One potential drawback is the high number of globally coupled unknowns. By means of hybridization, this number can be significantly reduced. The hybridized DG (HDG) method has proven to be beneficial especially for steady flows. In this work we apply it to a time-dependent flow problem with shocks. Due to its inherently implicit structure, time integration methods such as diagonally implicit Runge-Kutta (DIRK) methods present themselves as natural candidates. Furthermore, as the application of flux limiting to HDG is not straightforward, an artificial viscosity model is applied to stabilize the method.

References

  1. 1.
    A.H. Al-Rabeh, Embedded DIRK methods for the numerical integration of stiff systems of ODEs. Int. J. Comput. Math. 21(1), 65–84 (1987)CrossRefMATHGoogle Scholar
  2. 2.
    R. Alexander, Diagonally implicit Runge-Kutta methods for stiff O.D.E.’s. SIAM J. Numer. Anal. 14(14), 1006–1021 (1977)Google Scholar
  3. 3.
    D.N. Arnold, F. Brezzi, B. Cockburn, L.D. Marini, Unified analysis of discontinuous Galerkin methods for elliptic problems. SIAM J. Numer. Anal. 39(5), 1749–1779 (2002)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    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)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    J.R. Cash, Diagonally implicit Runge-Kutta formulae with error estimates. J. Inst. Math. Appl. 24, 293–301 (1979)CrossRefMATHGoogle Scholar
  6. 6.
    B. Cockburn, J. Gopalakrishnan, R. Lazarov, Unified hybridization of discontinuous Galerkin, mixed, and continuous Galerkin methods for second order elliptic problems. SIAM J. Numer. Anal. 47(2), 1319–1365 (2009)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    K. Fidkowski, T. Oliver, J. Lu, D. Darmofal, p-Multigrid solution of high-order discontinuous Galerkin discretizations of the compressible Navier-Stokes equations. J. Comput. Phys. 207, 92–113 (2005)Google Scholar
  8. 8.
    E. Hairer, G. Wanner, Solving Ordinary Differential Equations II, Stiff and Differential-Algebraic Problems. Springer Series in Computational Mathematics (Springer, Berlin, 1991)Google Scholar
  9. 9.
    R. Hartmann, P. Houston, Symmetric interior penalty DG methods for the compressible Navier-Stokes equations I: method formulation. Int. J. Numer. Anal. Model. 3(1), 1–20 (2006)MathSciNetMATHGoogle Scholar
  10. 10.
    R. Hartmann, P. Houston, Symmetric interior penalty DG methods for the compressible Navier-Stokes equations II: goal-oriented a posteriori error estimation. Int. J. Numer. Anal. Model. 3(2), 141–162 (2006)MathSciNetMATHGoogle Scholar
  11. 11.
    J. Hesthaven, T. Warburton, Nodal Discontinuous Galerkin Methods: Algorithms, Analysis, and Applications. Texts in Applied Mathematics, vol. 54 (Springer, Berlin, 2008)Google Scholar
  12. 12.
    P. Houston, E. Süli, hp-adaptive discontinuous Galerkin finite element methods for first order hyperbolic problems. SIAM J. Sci. Comput. 23(4), 1226–1252 (2001)Google Scholar
  13. 13.
    A. Jaust, J. Schütz, A temporally adaptive hybridized discontinuous Galerkin method for instationary compressible flows. Comput. Fluids 98, 177–185 (2014)MathSciNetCrossRefGoogle Scholar
  14. 14.
    A. Jaust, J. Schütz, M. Woopen, A hybridized discontinuous Galerkin method for unsteady flows with shock-capturing. AIAA Paper 2014-2781, in 44th AIAA Fluid Dynamics Conference, 2014Google Scholar
  15. 15.
    N.C. Nguyen, J. Peraire, Hybridizable discontinuous Galerkin methods for partial differential equations in continuum mechanics. J. Comput. Phys. 231, 5955–5988 (2012)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    N.C. Nguyen, J. Peraire, B. Cockburn, An implicit high-order hybridizable discontinuous Galerkin method for linear convection-diffusion equations. J. Comput. Phys. 228(9), 3232–3254 (2009)MathSciNetCrossRefGoogle Scholar
  17. 17.
    N.C. Nguyen, J. Peraire, B. Cockburn, An implicit high-order hybridizable discontinuous Galerkin method for nonlinear convection-diffusion equations. J. Comput. Phys. 228(23), 8841–8855 (2009)MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    N.C. Nguyen, J. Peraire, B. Cockburn, High-order implicit hybridizable discontinuous Galerkin methods for acoustics and elastodynamics. J. Comput. Phys. 230, 3695–3718 (2011)MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    J. Peraire, N.C. Nguyen, B. Cockburn, A hybridizable discontinuous Galerkin method for the compressible Euler and Navier-Stokes equations. AIAA Paper 2010-362, 48th AIAA Aerospace Sciences Meeting and Exhibit, 2010Google Scholar
  20. 20.
    P.-O. Persson, J. Peraire, Sub-cell shock capturing for discontinuous Galerkin methods. AIAA Paper 2006-0112, American Institute of Aeronautics and Astronautics, 2006Google Scholar
  21. 21.
    J. Schütz, G. May, A hybrid mixed method for the compressible Navier-Stokes equations. J. Comput. Phys. 240, 58–75 (2013)MathSciNetCrossRefMATHGoogle Scholar
  22. 22.
    J. Schütz, M. Woopen, G. May. A combined hybridized discontinuous Galerkin / hybrid mixed method for viscous conservation laws. in Hyperbolic Problems: Theory, Numerics, Applications, ed. by F. Ancona, A. Bressan, P. Marcati, A. Marson, pp. 915–922 (American Institute of Mathematical Sciences, Springfield 2012)Google Scholar
  23. 23.
    P. Woodward, P. Colella, The numerical simulation of two-dimensional fluid flow with strong shocks. J. Comput. Phys. 54, 115–173 (1984)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  • Alexander Jaust
    • 1
  • Jochen Schütz
    • 2
  • Michael Woopen
    • 3
  1. 1.MathCCESRWTH Aachen UniversityAachenGermany
  2. 2.IGPMRWTH Aachen UniversityAachenGermany
  3. 3.AICESRWTH Aachen UniversityAachenGermany

Personalised recommendations