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Journal of Scientific Computing

, Volume 34, Issue 3, pp 260–286 | Cite as

A Discontinuous Galerkin Scheme based on a Space-Time Expansion II. Viscous Flow Equations in Multi Dimensions

  • G. Gassner
  • F. Lörcher
  • C.-D. Munz
Article

Abstract

In part I of these two papers we introduced for inviscid flow in one space dimension a discontinuous Galerkin scheme of arbitrary order of accuracy in space and time. In the second part we extend the scheme to the compressible Navier-Stokes equations in multi dimensions. It is based on a space-time Taylor expansion at the old time level in which all time or mixed space-time derivatives are replaced by space derivatives using the Cauchy-Kovalevskaya procedure. The surface and volume integrals in the variational formulation are approximated by Gaussian quadrature with the values of the space-time approximate solution. The numerical fluxes at grid cell interfaces are based on the approximate solution of generalized Riemann problems for both, the inviscid and viscous part. The presented scheme has to satisfy a stability restriction similar to all other explicit DG schemes which becomes more restrictive for higher orders. The loss of efficiency, especially in the case of strongly varying sizes of grid cells is circumvented by use of different time steps in different grid cells. The presented time accurate numerical simulations run with local time steps adopted to the local stability restriction in each grid cell. In numerical simulations for the two-dimensional compressible Navier-Stokes equations we show the efficiency and the optimal order of convergence being p+1, when a polynomial approximation of degree p is used.

Keywords

Compressible Navier-Stokes equations Discontinuous Galerkin scheme Space-time expansion Local time stepping Diffusive generalized Riemann problem 

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References

  1. 1.
    Arnold, D.N., Brezzi, F., Cockburn, B., Marini, L.D.: Discontinuous Galerkin methods for elliptic problems. In: Cockburn, B., Karniadakis, G., Shu, C.-W. (eds.) Discontinuous Galerkin Methods. Lecture Notes in Computational Science and Engineering, pp. 89–101. Springer, New York (2000) Google Scholar
  2. 2.
    Arnold, D.N., Brezzi, F., Cockburn, B., Marini, L.D.: Unified analysis of discontinuous Galerkin methods for elliptic problems. SIAM J. Numer. Anal. 39(5), 1749–1779 (2002) MATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Babucke, A., Kloker, M.J., Rist, U.: DNS of a plane mixing layer for the investigation of sound generation mechanisms. Comput. Fluids (2007). doi: 10.1016/j.compfluid.2007.02.002 MATHGoogle Scholar
  4. 4.
    Bassi, F., Rebay, S.: 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) MATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Bassi, F., Rebay, S.: Numerical evaluation of two discontinuous Galerkin methods for the compressible Navier-Stokes equations. Int. J. Numer. Methods Fluids 40, 197–207 (2002) MATHCrossRefGoogle Scholar
  6. 6.
    Bassi, F., Rebay, S., Mariotti, G., Pedinotti, S., Savini, M.: A high-order accurate discontinuous finite element method fir inviscid an viscous turbomachinery flows. In: Decuypere, R., Dibelius, G. (eds.) Proceedings of 2nd European Conference on Turbomachinery, Fluid and Thermodynamics, Technologisch Instituut, Antwerpen, Belgium, 1997, pp. 99–108 Google Scholar
  7. 7.
    Cockburn, B., Shu, C.-W.: TVB Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws II: general framework. Math. Comput. 52, 411–435 (1989) MATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Cockburn, B., Karniadakis, G.E., Shu, C.-W.: Discontinuous Galerkin Methods. Lecture Notes in Computational Science and Engineering. Springer, New York (2000) MATHGoogle Scholar
  9. 9.
    Colonius, T., Lele, S.K., Moin, P.: Sound generation in a mixing layer. J. Fluid Mech. 330, 375–409 (1997) MATHCrossRefGoogle Scholar
  10. 10.
    Drela, M.: Two-dimensional transonic aerodynamic design and analysis using the Euler equations. Dissertation, Massachusetts Institute of Technology, Cambridge, MA (Feb. 1986). Gas Turbine Laboratory Report No. 187 Google Scholar
  11. 11.
    Dumbser, M., Munz, C.-D.: Arbitrary high order discontinuous Galerkin schemes. In: Cordier, S., Goudon, T., Gutnic, M. and Sonnendrucker, E. (eds.) Numerical Methods for Hyperbolic and Kinetic Problems. IRMA Series in Mathematics and Theoretical Physics, pp. 295–333. EMS Publishing House (2005) Google Scholar
  12. 12.
    Dumbser, M., Munz, C.-D.: Building blocks for arbitrary high order discontinuous Galerkin schemes. J. Sci. Comput. 27, 215–230 (2006) MATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Flaherty, J.E., Loy, R.M., Shephard, M.S., Szymanski, B.K., Teresco, J.D., Ziantz, L.H.: Adaptive local refinement with octree load balancing for the parallel solution of three-dimensional conservation laws. J. Parallel Distrib. Comput. 47(2), 139–152 (1997) CrossRefGoogle Scholar
  14. 14.
    Gassner, G., Lörcher, F., Munz, C.-D.: A contribution to the construction of diffusion fluxes for finite volume and discontinuous Galerkin schemes. J. Comput. Phys. (2006). doi: 10.1016/j.jcp.2006.11.004 MATHGoogle Scholar
  15. 15.
    Harten, A., Engquist, B., Osher, S., Chakravarthy, S.: Uniformly high order essentially non-oscillatory schemes, III. J. Comput. Phys. 71, 231–303 (1987) MATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    Hartmann, R., Houston, P.: Symmetric interior penalty DG methods for the compressible Navier–Stokes equations I: method formulation. Int. J. Numer. Anal. Model. 3(1), 1–20 (2006) MATHMathSciNetGoogle Scholar
  17. 17.
    Klaij, C., van der Vegt, J.J.W., van der Ven, H.: Spacetime discontinuous Galerkin method for the compressible Navier–Stokes equations. J. Comput. Phys. 217(2), 589–611 (2006) MATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    Lörcher, F., Gassner, G., Munz, C.-D.: A discontinuous Galerkin scheme based on a space-time expansion. I. Inviscid compressible flow in one space dimension. J. Sci. Comput. (2007). doi: 10.1007/s10915-007-9128-x Google Scholar
  19. 19.
    Qiu, J.: A numerical comparison of the Lax-Wendroff discontinuous Galerkin method based on different numerical fluxes. J. Sci. Comput. (2006). doi: 10.1007/s10915-006-9109-5 Google Scholar
  20. 20.
    Qiu, J., Khoo, B.C., Shu, C.-W.: A numerical study for the performance of the Runge-Kutta discontinuous Galerkin method based on different numerical fluxes. J. Comput. Phys. 212, 540–565 (2006) MATHCrossRefMathSciNetGoogle Scholar
  21. 21.
    Titarev, V., Toro, E.: ADER: Arbitrary high order Godunov approach. J. Sci. Comput. 17(14), 609–618 (2002) MATHCrossRefMathSciNetGoogle Scholar
  22. 22.
    Toro, E.: Riemann Solvers and Numerical Methods for Fluid Dynamics. Springer, New York (1999) MATHGoogle Scholar
  23. 23.
    Toro, E., Titarev, V.A.: Solution of the generalized Riemann problem for advection-reaction equations. Proc. R. Soc. Lond. 458, 271–281 (2002) MATHMathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2007

Authors and Affiliations

  1. 1.Institut für Aerodynamik und GasdynamikUniversity of StuttgartStuttgartGermany

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