Advertisement

Space-Time Hybridizable Discontinuous Galerkin Method for the Advection–Diffusion Equation on Moving and Deforming Meshes

  • Sander Rhebergen
  • Bernardo Cockburn

Abstract

We present the first space-time hybridizable discontinuous Galerkin finite element method for the advection–diffusion equation. Space-time discontinuous Galerkin methods have been proven to be very well suited for moving and deforming meshes which automatically satisfy the so-called Geometric Conservation law, for being able to provide higher-order accurate approximations in both time and space by simply increasing the degree of the polynomials used for the space-time finite elements, and for easily handling space-time adaptivity strategies. The hybridizable discontinuous Galerkin methods we introduce here add to these advantages their distinctive feature, namely, that the only globally-coupled degrees of freedom are those of the approximate trace of the scalar unknown. This results in a significant reduction of the size of the matrices to be numerically inverted, a more efficient implementation, and even better accuracy. We introduce the method, discuss its implementation and numerically explore its convergence properties.

Keywords

Discontinuous Galerkin methods Advection–diffusion equations Space-time methods 

Notes

Acknowledgements

All test cases were implemented using hpGEM [23] for which we thank V.R. Ambati for technical support. Sander Rhebergen gratefully acknowledges funding by a Rubicon Fellowship from the Netherlands Organisation for Scientific Research (NWO) and the Marie Curie Cofund Action. Bernardo Cockburn was supported in part by the National Science Foundation (Grant DMS-0712955) and by the University of Minnesota Supercomputing Institute.

References

  1. 1.
    Ambati, V.R., Bokhove, O.: Space-time discontinuous Galerkin finite element method for shallow water flows. J. Comput. Appl. Math. 204, 452–462 (2007) MathSciNetzbMATHCrossRefGoogle Scholar
  2. 2.
    Ambati, V.R., Bokhove, O.: Space-time discontinuous Galerkin discretization of rotating shallow water equations. J. Comput. Phys. 225, 1233–1261 (2007) MathSciNetzbMATHCrossRefGoogle Scholar
  3. 3.
    Cockburn, B., Dong, B., Guzmán, J.: A superconvergent LDG-hybridizable Galerkin method for second-order elliptic problems. Math. Comput. 77, 1887–1916 (2008) zbMATHCrossRefGoogle Scholar
  4. 4.
    Cockburn, B., Dong, B., Guzmán, J., Restelli, M., Sacco, R.: Superconvergent and optimally convergent LDG-hybridizable discontinuous Galerkin methods for convection-diffusion-reaction problems. SIAM J. Sci. Comput. 31, 3827–3846 (2009) MathSciNetzbMATHCrossRefGoogle Scholar
  5. 5.
    Cockburn, B., Gopalakrishnan, J.: A characterization of hybridized mixed methods for the Dirichlet problem. SIAM J. Numer. Anal. 42, 283–301 (2004) MathSciNetzbMATHCrossRefGoogle Scholar
  6. 6.
    Cockburn, B., Gopalakrishnan, J., Lazarov, R.: Unified hybridization of discontinuous Galerkin, mixed, and continuous Galerkin methods for second order elliptic problems. SIAM J. Numer. Anal. 47, 1319–1365 (2009) MathSciNetzbMATHCrossRefGoogle Scholar
  7. 7.
    Cockburn, B., Gopalakrishnan, J., Nguyen, N.C., Peraire, J., Sayas, F.J.: Analysis of an HDG method for Stokes flow. Math. Comput. 80, 723–760 (2011) MathSciNetzbMATHCrossRefGoogle Scholar
  8. 8.
    Cockburn, B., Gopalakrishnan, J., Sayas, F.-J.: A projection-based error analysis of HDG methods. Math. Comput. 79, 1351–1367 (2010) MathSciNetzbMATHCrossRefGoogle Scholar
  9. 9.
    Gopalakrishnan, J., Tan, S.: A convergent multigrid cycle for the hybridized mixed method. Numer. Linear Algebra Appl. 16, 689–714 (2009) MathSciNetzbMATHCrossRefGoogle Scholar
  10. 10.
    Hueber, B., Walhorn, E., Dinkler, D.: A monolithic approach to fluid-structure interaction using space-time finite elements. Comput. Methods Appl. Mech. Eng. 193, 2087–2104 (2004) CrossRefGoogle Scholar
  11. 11.
    Klaij, C.M., van der Vegt, J.J.W., van der Ven, H.: Space-time discontinuous Galerkin method for the compressible Navier–Stokes equations. J. Comput. Phys. 217, 589 (2006) MathSciNetzbMATHCrossRefGoogle Scholar
  12. 12.
    Klaij, C.M., van Raalte, M.H., van der Ven, H., van der Vegt, J.J.W.: Vegt, h-multigrid for space-time discontinuous Galerkin discretizations of the compressible Navier–Stokes equations. J. Comput. Phys. 227, 1024–1045 (2007) MathSciNetzbMATHCrossRefGoogle Scholar
  13. 13.
    Lesoinne, M., Farhat, C.: Geometric conservation laws for flow problems with moving boundaries and deformable meshes, and their impact on aeroelastic computations. Comput. Methods Appl. Mech. Eng. 134, 71–90 (1996) zbMATHCrossRefGoogle Scholar
  14. 14.
    Nguyen, N.C., Peraire, J., Cockburn, B.: An implicit high-order hybridizable discontinuous Galerkin method for linear convection–diffusion equations. J. Comput. Phys. 228, 3232–3254 (2009) MathSciNetCrossRefGoogle Scholar
  15. 15.
    Nguyen, N.C., Peraire, J., Cockburn, B.: An implicit high-order hybridizable discontinuous Galerkin method for nonlinear convection–diffusion equations. J. Comput. Phys. 228, 8841–8855 (2009) MathSciNetzbMATHCrossRefGoogle Scholar
  16. 16.
    Nguyen, N.C., Peraire, J., Cockburn, B.: Hybridizable discontinuous Galerkin methods. In: Proceedings of the International Conference on Spectral and High Order Methods, June 2009, Trondheim, Norway. LNCSE. Springer, Berlin (2009) Google Scholar
  17. 17.
    Nguyen, N.C., Peraire, J., Cockburn, B.: A hybridizable discontinuous Galerkin method for Stokes flow. Comput. Methods Appl. Mech. Eng. 193, 2087–2104 (2010). I have: 199, 582–597 (2010) MathSciNetGoogle Scholar
  18. 18.
    Nguyen, N.C., Peraire, J., Cockburn, B.: An implicit high-order hybridizable discontinuous Galerkin method for the incompressible Navier–Stokes equations. J. Comput. Phys. 230, 1147–1170 (2011) MathSciNetzbMATHCrossRefGoogle Scholar
  19. 19.
    Peraire, J., Nguyen, N.C., Cockburn, B.: A hybridizable discontinuous Galerkin method for the incompressible Navier–Stokes equations. In: AIAA, Orlando, Florida, p. 362 (2010) Google Scholar
  20. 20.
    Peraire, J., Nguyen, N.C., Cockburn, B.: A hybridizable discontinuous Galerkin finite method for the compressible Euler and Navier–Stokes equations. In: AIAA, Orlando, Florida, p. 363 (2010) Google Scholar
  21. 21.
    Persson, P.-O., Bonet, J., Peraire, J.: Discontinuous Galerkin solution of the Navier–Stokes equations on deformable domains. Comput. Methods Appl. Mech. Eng. 198, 1585–1595 (2009) zbMATHCrossRefGoogle Scholar
  22. 22.
    Persson, P.-O., Peraire, J.: Newton-GMRES preconditionning for discontinuous Galerkin discretizations of the Navier–Stokes equations. SIAM J. Sci. Comput. 30, 2709–2733 (2008) MathSciNetzbMATHCrossRefGoogle Scholar
  23. 23.
    Pesch, L., Bell, A., Sollie, W.E.H., Ambati, V.R., Bokhove, O., van der Vegt, J.J.W.: hpGEM—a software framework for discontinuous Galerkin finite element methods. ACM Trans. Math. Softw. 33 (2007) Google Scholar
  24. 24.
    Pesch, L., van der Vegt, J.J.W.: A discontinuous Galerkin finite element discretization of the Euler equations for compressible and incompressible fluids. J. Comput. Phys. 227, 5426–5446 (2008) MathSciNetzbMATHCrossRefGoogle Scholar
  25. 25.
    Reed, W.H., Hill, T.R.: Triangular mesh methods for the neutron transport equation. Los Alamos Scientific Laboratory, Report LA-UR-73-479 (1973) Google Scholar
  26. 26.
    Rhebergen, S., Bokhove, O., van der Vegt, J.J.W.: Discontinuous Galerkin finite element methods for hyperbolic nonconservative partial differential equations. J. Comput. Phys. 227, 1887 (2008) MathSciNetzbMATHCrossRefGoogle Scholar
  27. 27.
    Rhebergen, S., Bokhove, O., van der Vegt, J.J.W.: Discontinuous Galerkin finite element method for shallow two-phase flows. Comput. Methods Appl. Mech. Eng. 198, 819–830 (2009) zbMATHCrossRefGoogle Scholar
  28. 28.
    Rhebergen, S., van der Vegt, J.J.W., van der Ven, H.: Multigrid optimization for space-time discontinuous Galerkin discretizations of advection dominated flows. In: Kroll, N., Bieler, H., Deconinck, H., Couallier, V., Van der Ven, H., Sorensen, K. (eds.) ADIGMA—A European Initiative on the Development of Adaptive Higer-Order Variational Methods for Aerospace Applications. Notes on Numerical Fluid Mechanics and Multidisciplinary Design, vol. 113, pp. 257–269. Springer, Berlin (2010) CrossRefGoogle Scholar
  29. 29.
    Sollie, W.E.H., Bokhove, O., van der Vegt, J.J.W.: Space-time discontinuous Galerkin finite element method for two-fluid flows. J. Comput. Phys. 230, 789–817 (2011) MathSciNetzbMATHCrossRefGoogle Scholar
  30. 30.
    Sudirham, J.J., van der Vegt, J.J.W., van Damme, R.M.J.: Space-time discontinuous Galerkin method for advection-diffusion problems on time-dependent domains. Appl. Numer. Math. 56, 1491–1518 (2006) MathSciNetzbMATHCrossRefGoogle Scholar
  31. 31.
    Tassi, P.A., Rhebergen, S., Vionnet, C.A., Bokhove, O.: A discontinuous Galerkin finite element model for river bed evolution under shallow flows. Comput. Methods Appl. Mech. Eng. 197, 2930–2947 (2008) MathSciNetzbMATHCrossRefGoogle Scholar
  32. 32.
    van der Vegt, J.J.W., Sudirham, J.J.: A space-time discontinuous Galerkin method for the time-dependent Oseen equations. Appl. Numer. Math. 58, 1892–1917 (2008) MathSciNetzbMATHCrossRefGoogle Scholar
  33. 33.
    van der Vegt, J.J.W., van der Ven, H.: Space-time discontinuous Galerkin finite element method with dynamic grid motion for inviscid compressible flows I. General formulation. J. Comput. Phys. 182, 546–585 (2002) MathSciNetzbMATHCrossRefGoogle Scholar
  34. 34.
    van der Vegt, J.J.W., Rhebergen, S.: hp-Multigrid as smoother algorithm for higher order discontinuous Galerkin discretizations of advection dominated flows. Part I. Multilevel analysis. J. Comput. Phys. 231(22), 7537–7563 (2012) CrossRefGoogle Scholar
  35. 35.
    van der Vegt, J.J.W., Rhebergen, S.: hp-Multigrid as smoother algorithm for higher order discontinuous Galerkin discretizations of advection dominated flows. Part II. Optimization of the Runge-Kutta smoother. J. Comput. Phys. 231(22), 7564–7583 (2012) CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  1. 1.School of MathematicsUniversity of MinnesotaMinneapolisUSA

Personalised recommendations