A Fully Implicit Jacobian-Free High-Order Discontinuous Galerkin Mesoscale Flow Solver

  • Amik St-Cyr
  • David Neckels
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5545)


In this work it is shown how to discretize the compressible Euler equations around a vertically stratified base state using the discontinuous Galerkin approach on collocated Gauss type grids. A stiffly stable Rosenbrock W-method is combined with an approximate evaluation of the Jacobian to integrate in time the resulting system of ODEs. Simulations with fully compressible equations for a rising thermal bubble are performed. Also included are simulations of an inertia gravity wave in a periodic channel. The proposed time-stepping method accelerates the simulation times with respect to explicit Runge-Kutta time stepping procedures having the same number of stages.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Cockburn, B., Karniadakis, G.E., Shu, C.W.: Discontinuous Galerkin Methods: Theory, Computation, and Applications. Lecture Notes in Computational Science and Engineering, vol. 11. Springer, New York (2000)MATHGoogle Scholar
  2. 2.
    St-Cyr, A., Thomas, S.J.: Parallel atmospheric modeling with high-order continuous and discontinuous galerkin methods. In: Deane, A., Periaux, J., Ecer, A., Satofuka, N., McDonough, J. (eds.) Parallel Computational Fluid Dynamics 2005: Theory and Applications: Proceedings of the Parallel CFD 2005 Conference, pp. 485–492. Elsevier Science, Amsterdam (2006)Google Scholar
  3. 3.
    Bhanot, G., Dennis, J.M., Edwards, J., Grabowski, W., Gupta, M., Jordan, K., Loft, R.D., Sexton, J., St-Cyr, A., Thomas, S.J., Tufo, H.M., Voran, T., Walkup, R., Wyszogrodzki, A.A.: Early experiences with the 360tf ibm bluegene/l platform. Int. J. Comput. Meth. 5(2), 237–253 (2008)CrossRefMATHGoogle Scholar
  4. 4.
    Skamarock, W.C., Klemp, J.B., Dudhia, J., Gill, D.O., Barker, D.M., Wang, W., Powers, J.G.: A description of the advanced research WRF version 2. NCAR Tech. Note TN-468+STR, National Center for Atmospheric Research, 1850 Table Mesa Drive, Boulder, Colorado, 80305, USA (2005) (revised January 2007)Google Scholar
  5. 5.
    Bassi, F., Rebay, S.: High-order accurate discontinuous finite element solution of the 2d euler equations. Journal of Computational Physics 138(2), 251–285 (1997)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Giraldo, F.X., Hesthaven, J.S., Warburton, T.: Nodal high-order discontinuous galerkin methods for the spherical shallow water equations. Journal of Computational Physics 181(2), 499–525 (2002)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Ronquist, E.M.: Optimal Spectral Element Methods for the Unsteady Three-Dimensional Incom- pressible Navier-Stokes Equations. Ph.D thesis. MIT, Cambridge, MA, USA (1988)Google Scholar
  8. 8.
    Hairer, E., Wanner, G.: Solving Ordinary Differential Equations II, 2nd edn. Springer Series in Computational Mathematics, vol. 14. Springer, Heidelberg (1996)CrossRefMATHGoogle Scholar
  9. 9.
    Steihaug, T., Wolfbrandt, A.: An attempt to avoid exact jacobian and nonlinear equations in the numerical solution of stiff differential equations. Math. Comp. 33, 521–534 (1979)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Rang, J., Angermann, L.: New Rosenbrock W-methods of order 3 for partial differential algebraic equations of index 1. BIT Numerical Mathematics 45, 761–787 (2005)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Meister, A.: Asymptotic based preconditioning technique for low Mach number flows. ZAMM 83(1), 3–25 (2003)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Viozat, C.: Calcul d’écoulements stationnaires et instationnaires à petit nombre de Mach, et en maillages étirés. Ph.D thesis, Université de Nice-Sophia Antipolis, Nice, France (October 1998)Google Scholar
  13. 13.
    Guillard, H., Viozat, C.: On the behaviour of upwind schemes in the low mach number limit. Comput. Fluids 28, 63–86 (1999)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Mavriplis, D.J.: Multigrid strategies for viscous flow solvers on anisotropic unstructured meshes. Journal of Computational Physics 145, 141–165 (1998)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Wicker, L.J., Skamarock, W.C.: Time splitting methods for elastic models using forward time schemes. Mon. Wea. Rev. 130, 2088–2097 (2002)CrossRefGoogle Scholar
  16. 16.
    Skamarock, W.C., Klemp, J.B.: Efficiency and accuracy of the Klemp-Wilhelmson time-splitting technique. Mon. Wea. Rev. 122, 2623–2630 (1994)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2009

Authors and Affiliations

  • Amik St-Cyr
    • 1
  • David Neckels
    • 2
  1. 1.National Center for Atmospheric Research (NCAR)BoulderUSA
  2. 2.Previously NCAR now Beckman Coulter Inc.FullertonUSA

Personalised recommendations