BIT Numerical Mathematics

, Volume 56, Issue 3, pp 951–966 | Cite as

Efficient fully discrete summation-by-parts schemes for unsteady flow problems

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Abstract

We make an initial investigation into the temporal efficiency of a fully discrete summation-by-parts approach for unsteady flows. As a model problem for the Navier–Stokes equations we consider a two-dimensional advection–diffusion problem with a boundary layer. The problem is discretized in space using finite difference approximations on summation-by-parts form together with weak boundary conditions, leading to optimal stability estimates. For the time integration part we consider various forms of high order summation-by-parts operators and compare with an existing popular fourth order diagonally implicit Runge–Kutta method. To solve the resulting fully discrete equation system, we employ a multi-grid scheme with dual time stepping.

Keywords

Summation-by-parts in time Unsteady flow calculations   Temporal efficiency 

Mathematics Subject Classification

65M06 65M55 

References

  1. 1.
    Berg, J., Nordström, J.: Stable Robin solid wall boundary conditions for the Navier–Stokes equations. J. Comput. Phys. 230(19), 7519–7532 (2011)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Bijl, H., Carpenter, M.: Iterative solution techniques for unsteady flow computations using higher order time integration schemes. Int. J. Numer. Methods Fluids 47, 857–862 (2005)CrossRefMATHGoogle Scholar
  3. 3.
    Birken, P., Jameson, A.: On nonlinear preconditioners in Newton-Krylov methods for unsteady flows. Int. J. Numer. Methods Fluids 62, 565–573 (2010)MathSciNetMATHGoogle Scholar
  4. 4.
    Boom, P., Zingg, D.: High-order implicit time-marching methods based on generalized summation-by-parts operators. Technical report (2014). arXiv:1410.0201
  5. 5.
    Boom, P., Zingg, D.: Runge–Kutta characterization of the generalized summation-by-parts approach in time. Technical report (2014). arXiv:1410.0202
  6. 6.
    Carpenter, M., Gottlieb, D.: Spectral methods on arbitrary grids. J. Comput. Phys. 129, 74–86 (1996)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Carpenter, M., Nordström, J., Gottlieb, D.: A stable and conservative interface treatment of arbitrary spatial accuracy. J. Comput. Phys. 148, 341–365 (1999)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Del Rey Fernandez, D., Boom, P., Zingg, D.: A generalized framework for nodal first derivative summation-by-parts operators. J. Comput. Phys. 266, 214–239 (2014)Google Scholar
  9. 9.
    Kennedy, C.A., Carpenter, M.H.: Additive Runge–Kutta schemes for convection–diffusion–reaction equations. Appl. Numer. Math. 44(1–2), 139–181 (2003)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Kleb, W., Wood, W., van Leer, B.: Efficient multi-stage time marching for viscous flows via local preconditioning. AIAA Paper 99-3267 (1999)Google Scholar
  11. 11.
    Knoll, D., Keyes, D.: Jacobian-free Newton–Krylov methods: a survey of approaches and applications. J. Comput. Phys. 193, 357–397 (2004)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Lundquist, T., Nordström, J.: The SBP–SAT technique for initial value problems. J. Comput. Phys. 270, 86–104 (2014)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Nordström, J., Carpenter, M.H.: High-order finite difference methods, multidimensional linear problems and curvilinear coordinates. J. Comput. Phys. 173, 149–174 (2001)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Nordström, J., Lundquist, T.: Summation-by-parts in time. J. Comput. Phys. 251, 487–499 (2013)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Svärd, M., Carpenter, M., Nordström, J.: A stable high-order finite difference scheme for the compressible Navier–Stokes equations: far-field boundary conditions. J. Comput. Phys. 225(1), 1020–1038 (2007)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Svärd, M., Nordström, J.: A stable high-order finite difference scheme for the compressible Navier–Stokes equations: no-slip wall boundary conditions. J. Comput. Phys. 227(10), 4805–4824 (2008)MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Svärd, M., Nordström, J.: Review of summation-by-parts schemes for initial-boundary-value problems. J. Comput. Phys. 268, 17–38 (2014)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2015

Authors and Affiliations

  1. 1.Department of Mathematics, Computational MathematicsLinköping UniversityLinköpingSweden

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