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Journal of Scientific Computing

, Volume 75, Issue 2, pp 906–940 | Cite as

A Dual Consistent Finite Difference Method with Narrow Stencil Second Derivative Operators

  • Sofia Eriksson
Article

Abstract

We study the numerical solutions of time-dependent systems of partial differential equations, focusing on the implementation of boundary conditions. The numerical method considered is a finite difference scheme constructed by high order summation by parts operators, combined with a boundary procedure using penalties (SBP–SAT). Recently it was shown that SBP–SAT finite difference methods can yield superconvergent functional output if the boundary conditions are imposed such that the discretization is dual consistent. We generalize these results so that they include a broader range of boundary conditions and penalty parameters. The results are also generalized to hold for narrow-stencil second derivative operators. The derivations are supported by numerical experiments.

Keywords

Finite differences Summation by parts Simultaneous approximation term Dual consistency Superconvergence Narrow stencil 

Mathematics Subject Classification

65M06 65M20 65M12 

Notes

Acknowledgements

The author would like to sincerely thank the anonymous referees for their valuable comments and suggestions.

References

  1. 1.
    Berg, J., Nordström, J.: Superconvergent functional output for time-dependent problems using finite differences on summation-by-parts form. J. Comput. Phys. 231(20), 6846–6860 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Berg, J., Nordström, J.: On the impact of boundary conditions on dual consistent finite difference discretizations. J. Comput. Phys. 236, 41–55 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Berg, J., Nordström, J.: Duality based boundary conditions and dual consistent finite difference discretizations of the Navier–Stokes and Euler equations. J. Comput. Phys. 259, 135–153 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Carpenter, M.H., Nordström, J., Gottlieb, D.: A stable and conservative interface treatment of arbitrary spatial accuracy. J. Comput. Phys. 148(2), 341–365 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Eriksson, S., Nordström, J.: Analysis of the order of accuracy for node-centered finite volume schemes. Appl. Numer. Math. 59(10), 2659–2676 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Fernández, D.C.D.R., Hicken, J.E., Zingg, D.W.: Review of summation-by-parts operators with simultaneous approximation terms for the numerical solution of partial differential equations. Comput. Fluids 95, 171–196 (2014)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Gustafsson, B., Kreiss, H.O., Oliger, J.: Time-Dependent Problems and Difference Methods. Wiley, New York (2013)CrossRefzbMATHGoogle Scholar
  8. 8.
    Hicken, J.E.: Output error estimation for summation-by-parts finite-difference schemes. J. Comput. Phys. 231(9), 3828–3848 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Hicken, J.E., Zingg, D.W.: Superconvergent functional estimates from summation-by-parts finite-difference discretizations. SIAM J. Sci. Comput. 33(2), 893–922 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Hicken, J.E., Zingg, D.W.: Summation-by-parts operators and high-order quadrature. J. Comput. Appl. Math. 237(1), 111–125 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Kreiss, H.O., Lorenz, J.: Initial-Boundary Value Problems and the Navier–Stokes Equations. Academic Press, New York (1989)zbMATHGoogle Scholar
  12. 12.
    Mattsson, K.: Summation by parts operators for finite difference approximations of second-derivatives with variable coefficients. J. Sci. Comput. 51(3), 650–682 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Mattsson, K., Nordström, J.: Summation by parts operators for finite difference approximations of second derivatives. J. Comput. Phys. 199(2), 503–540 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Nordström, J., Eriksson, S., Eliasson, P.: Weak and strong wall boundary procedures and convergence to steady-state of the Navier–Stokes equations. J. Comput. Phys. 231(14), 4867–4884 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Nordström, J., Svärd, M.: Well-posed boundary conditions for the Navier–Stokes equations. SIAM J. Numer. Anal. 43(3), 1231–1255 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Quarteroni, A., Sacco, R., Saleri, F.: Numerical Mathematics. Springer, Berlin (2000)zbMATHGoogle Scholar
  17. 17.
    Strand, B.: Summation by parts for finite difference approximation for d/dx. J. Comput. Phys. 110(1), 47–67 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Svärd, M., Nordström, J.: On the order of accuracy for difference approximations of initial-boundary value problems. J. Comput. Phys. 218(1), 333–352 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Svärd, M., Nordström, J.: Review of summation-by-parts schemes for initial-boundary-value problems. J. Comput. Phys. 268, 17–38 (2014)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2017

Authors and Affiliations

  1. 1.Department of MathematicsTechnische Universität DarmstadtDarmstadtGermany

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