A Discontinuous Galerkin Method in Moving Domains

  • I. Lomtev
  • R. M. Kirby
  • G. E. Karniadakis
Part of the Lecture Notes in Computational Science and Engineering book series (LNCSE, volume 11)


We present a matrix-free discontinuous Galerkin method for simulating compressible viscous flows in two- and three-dimensional moving domains in an Arbitrary Lagrangian Eulerian (ALE) framework. Spatial discretization is based on standard structured and unstructured grids but using an orthogonal spectral hierarchical basis. The method is third-order accurate in time, and converges exponentially fast in space for smooth solutions. We also report on open issues related to quadrature crimes and over-integration.


Discontinuous Galerkin Method Vortex Tube Arbitrary Lagrangian Eulerian Spectral Element Method Grid Velocity 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    M. Dubiner. Spectral methods on triangles and other domains. J. Sci. Comp., 6: 345, 1991.MathSciNetzbMATHCrossRefGoogle Scholar
  2. 2.
    S.J. Sherwin. Hierarchical hp finite elements in hybrid domains. Finite Element in Analysis and Design, 27: 109–119, 1997.MathSciNetzbMATHGoogle Scholar
  3. 3.
    T.J.R. Hughes, W.K. Liu., and T.K. Zimmerman. Lagrangian-Eulerian finite element formulation for incompressible viscous flows. Comput. Methods Appl. Mech. Eng., 29: 329, 1981.zbMATHCrossRefGoogle Scholar
  4. 4.
    C.S. Venkatasubban. A new finite element formulation for ALE Arbitrary Lagrangian Eulerian compressible fluid mechanics. Int. J. Engng Sci., 33 (12): 1743–1762, 1995.MathSciNetzbMATHCrossRefGoogle Scholar
  5. 5.
    R. Lohner and C. Yang. Improved ale mesh velocities for moving bodies. Comm. Num. Meth. Eng. Phys., 12: 599–608, 1996.CrossRefGoogle Scholar
  6. 6.
    G. Di Battista, P. Eades, R. Tamassia, and I.G. Tollis. Graph Drawing. Prentice Hall, 1998.Google Scholar
  7. 7.
    G. Jiang and C.W. Shu. On a cell entropy inequality for discontinuous Galerkin methods. Math. Comp., 62: 531, 1994.MathSciNetzbMATHCrossRefGoogle Scholar
  8. 8.
    I. Lomtev, C. Quillen, and G.E. Karniadakis. Spectral/hp methods for viscous compressible flows on unstructured 2D meshes. J. Comp. Phys., 144: 325–357, 1998.MathSciNetzbMATHCrossRefGoogle Scholar
  9. 9.
    K.S. Bey, A. Patra, and J.T. Oden. hp version discontinuous Galerkin methods for hyperbolic conservation laws. Comp. Meth. Appl. Mech. Eng., 133: 259–286, 1996.zbMATHCrossRefGoogle Scholar
  10. 10.
    L.-W. Ho. A Legendre spectral element method for simulation of incompressible unsteady free-surface flows. PhD thesis, Massachustts Institute of Technology, 1989.Google Scholar
  11. 11.
    B. Koobus and C. Farhat. Second-order schemes that satisfy GCL for flow computations on dynamic grids. In AIAA 98–0113, 36th AIAA Aerospace Sciences Meeting and Exhibit, Reno, NV, January 12–15, 1998.Google Scholar
  12. 12.
    G.E. Karniadakis and S.J. Sherwin. Spectral/hp Element Methods for CFD. Oxford University Press, 1999.Google Scholar
  13. 13.
    I. Lomtev and G.E. Karniadakis. A discontinuous Galerkin method for the Navier-Stokes equations. Int. J. Num. Meth. Fluids, 29: 587–603, 1999.MathSciNetzbMATHCrossRefGoogle Scholar
  14. 14.
    I. Lomtev, R.M. Kirby, and G.E. Karniadakis. A discontinuous Galerkin ALE method for viscous compressible flows in moving domains. J. Comp. Phys., to appear, 1999.Google Scholar
  15. 15.
    B. Cockburn, S. Hou, and C.-W. Shu. The Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws IV: The multidimensional case. J. Comp. Phys., 54: 545, 1990.MathSciNetzbMATHGoogle Scholar
  16. 16.
    F. Bassi and S. Rebay. A high-order accurate discontinuous finite element method for the numerical solution of the compressible Navier-Stokes equations. J. Comp. Phys., 131: 267, 1997.MathSciNetzbMATHCrossRefGoogle Scholar
  17. 17.
    H. Liu and K. Kawachi. A numerical study of insect flight. J. Comp. Phys., 146: 124–156, 1998.zbMATHCrossRefGoogle Scholar
  18. 18.
    G. Karypis and V. Kumar. METIS: Unstructured graph partitioning and sparse matrix orderingsystem version 2.0. Technical report, Department of Computer Science, University of Minnesota, Minneapolis, MN 55455, 1995.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2000

Authors and Affiliations

  • I. Lomtev
    • 1
  • R. M. Kirby
    • 1
  • G. E. Karniadakis
    • 1
  1. 1.Division of Applied MathematicsBrown UniversityUSA

Personalised recommendations