A High-Order Local Discontinuous Galerkin Scheme for Viscoelastic Fluid Flow

  • Anne KikkerEmail author
  • Florian Kummer
Conference paper
Part of the Lecture Notes in Computational Science and Engineering book series (LNCSE, volume 124)


Coping with the so called high Weissenberg number problem (HWNP) is a key focus of research in computational rheology. By numerically simulating viscoelastic flow a breakdown in convergence often occurs for different computational approaches at critically high values of the Weissenberg number. This is due to two major problems concerning stability in the discretization. First, we have a mixed hyperbolic-elliptic problem weighted by a ratio parameter between retardation and relaxation time of viscoelastic fluid. Second, we have a convection-dominated convection-diffusion problem in the constitutive equations. We introduce a solver for viscoelastic Oldroyd B flow with an exclusively high-order Discontinuous Galerkin (DG) scheme for all equations using a local DG formulation in order to solve the hyperbolic constitutive equations and using a streamline upwinding formulation for the convective fluxes of the constitutive equations. The successful implementation of the local DG formulation for Newtonian fluid with appropriate fluxes containing stabilizing penalty parameters is shown in two results. First, a hk-convergence study is presented for a non-polynomial manufactured solution for the Stokes system. Second, numerical results are shown for the confined cylinder benchmark problem for Navier-Stokes flow and compared to the same flow using a symmetric interior penalty method without additional constitutive equations.


Local discontinuous Galerkin method Oldroyd B fluid 



This work is partially supported by the ‘Excellence Initiative’ of the German Federal and State Governments within the Graduate School of Computational Engineering at Technische Universität Darmstadt.


  1. 1.
    Owens, R.G., Phillips, T.N.: Computational Rheology. Imperial College Press, London (2002)CrossRefGoogle Scholar
  2. 2.
    Fortin, M., Fortin, A.: A new approach for the FEM simulation of viscoelastic flows. J. Non. Newton. Fluid. Mech. 32(3), 295–310 (1989). CrossRefGoogle Scholar
  3. 3.
    Cockburn, B., Karniadakis, G.E., Shu, C.-W.: Discontinuous Galerkin Methods. Lecture Notes in Computational Science and Engineering (11). Springer, Berlin (2000).
  4. 4.
    Phillips, T.N., Williams, A.J.: Viscoelastic flow through a planar contraction using a semi-Lagrangian finite volume method. J. Non. Newton. Fluid. Mech. 87(2–3), 215–246 (1999). CrossRefGoogle Scholar
  5. 5.
    Di Pietro, D.A., Ern, A.: Mathematical Aspects of Discontinuous Galerkin Methods. Mathématiques et Applications, vol. 69. Springer, Berlin (2011).
  6. 6.
    Li, B.Q.: Discontinuous Finite Elements in Fluid Dynamics and Heat Transfer. Computational Fluid and Solid Mechanics. Springer, London (2006).
  7. 7.
    Cockburn, B., Kanschat, G., Schötzau, D., Schwab, C.: Local discontinuous galerkin methods for the stokes system. SIAM J. Numer. Anal. 40(1), 319–343 (2002). MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Technische Universität DarmstadtDarmstadtGermany
  2. 2.Graduate School CE, Technische Universität DarmstadtDarmstadtGermany

Personalised recommendations