Journal of Scientific Computing

, Volume 75, Issue 1, pp 83–110 | Cite as

Simultaneous Approximation Terms for Multi-dimensional Summation-by-Parts Operators

  • David C. Del Rey FernándezEmail author
  • Jason E. Hicken
  • David W. Zingg


This paper is concerned with the accurate, conservative, and stable imposition of boundary conditions and inter-element coupling for multi-dimensional summation-by-parts (SBP) finite-difference operators. More precisely, the focus is on diagonal-norm SBP operators that are not based on tensor products and are applicable to unstructured grids composed of arbitrary elements. We show how penalty terms—simultaneous approximation terms (SATs)—can be adapted to discretizations based on multi-dimensional SBP operators to enforce boundary and interface conditions. A general SAT framework is presented that leads to conservative and stable discretizations of the variable-coefficient advection equation. This framework includes the case where there are no nodes on the boundary of the SBP element at which to apply penalties directly. This is an important generalization, because elements analogous to Legendre–Gauss collocation, i.e. without boundary nodes, typically have higher accuracy for the same number of degrees of freedom. Symmetric and upwind examples of the general SAT framework are created using a decomposition of the symmetric part of an SBP operator; these particular SATs enable the pointwise imposition of boundary and inter-element conditions. We illustrate the proposed SATs using triangular-element SBP operators with and without nodes that lie on the boundary. The accuracy, conservation, and stability properties of the resulting SBP–SAT discretizations are verified using linear advection problems with spatially varying divergence-free velocity fields.


Summation by parts Finite-difference Unstructured mesh High-order methods Simultaneous approximation terms 

Supplementary material

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  1. 1.
    Parsani, M., Carpenter, M.H., Nielsen, E.J.: Entropy stable wall boundary conditions for the three-dimensional compressible Navier–Stokes equations. J. Comput. Phys. 292(C), 88–113 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Carpenter, M.H., Fisher, T.C., Nielsen, E.J., Frankel, S.H.: Entropy stable spectral collocation schemes for the Navier–Stokes equations: discontinuous interfaces. SIAM J. Sci. Comput. 36(5), B835–B867 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Fisher, T.C., Carpenter, M.H.: High-order entropy stable finite difference schemes for nonlinear conservation laws: finite domains. J. Comput. Phys. 252(1), 518–557 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Kreiss, H.-O., Scherer, G.: Finite element and finite difference methods for hyperbolic partial differential equations. In: de Boor, C. (ed.) Mathematical Aspects of Finite Elements in Partial Differential Equations, pp. 195–212. Academic Press, New York (1974)Google Scholar
  5. 5.
    Strand, B.: Summation by parts for finite difference approximations for d/dx. J. Comput. Phys. 110(1), 47–67 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Del Rey Fernández, D.C., 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(22), 171–196 (2014)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Svärd, M., Nordström, J.: Review of summation-by-parts schemes for initial-boundary-value-problems. J. Comput. Phys. 268(1), 17–38 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Del Rey Fernández, D.C., Zingg, D.W.: Generalized summation-by-parts operators for the second derivative with a variable coefficient. SIAM J. Sci. Comput. 37(6), A2840–A2864 (2015)CrossRefzbMATHGoogle Scholar
  9. 9.
    Del Rey Fernández, D.C., Boom, P.D., Zingg, D.W.: A generalized framework for nodal first derivative summation-by-parts operators. J. Comput. Phys. 266(1), 214–239 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Gassner, G.J.: A skew-symmetric discontinuous Galerkin spectral element discretization and its relation to SBP-SAT finite difference methods. SIAM J. Sci. Comput. 35(3), A1233–A1253 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Carpenter, M.H., Gottlieb, D.: Spectral methods on arbitrary grids. J. Comput. Phys. 129(1), 74–86 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Nordström, J., Forsberg, K., Adamsson, C., Eliasson, P.: Finite volume methods, unstructured meshes and strict stability for hyperbolic problems. Appl. Numer. Math. 45, 453–473 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Hicken, J.E., Del Rey Fernández, D.C., Zingg, D.W.: Multi-dimensional summation-by-parts operators: general theory and application to simplex elements. SIAM J. Sci. Comput. 38(4), A1935–A1958 (2016)CrossRefzbMATHGoogle Scholar
  14. 14.
    Hicken, J.E., Del Rey Fernández, D.C., Zingg, D.W.: Opportunities for efficient high-order methods based on the summation-by-parts property. In: 22nd AIAA Computational Fluid Dynamics Conference, No. AIAA–2015–3198, Dallas, Texas (2015)Google Scholar
  15. 15.
    Cools, R.: Monomial cubature rules since “Stroud”: a compilation—part 2. J. Comput. Appl. Math. 112(1–2), 21–27 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Carpenter, M.H., Gottlieb, D., Abarbanel, S.: Time-stable boundary conditions for finite-difference schemes solving hyperbolic systems: methodology and application to high-order compact schemes. J. Comput. Phys. 111(2), 220–236 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    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
  18. 18.
    Nordström, J., Carpenter, M.H.: Boundary and interface conditions for high-order finite-difference methods applied to the Euler and Navier–Stokes equations. J. Comput. Phys. 148(2), 621–645 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Nordström, J., Carpenter, M.H.: High-order finite-difference methods, multidimensional linear problems, and curvilinear coordinates. J. Comput. Phys. 173(1), 149–174 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Hesthaven, J.S., Warburton, T.: Nodal Discontinuous Galerkin Methods: Algorithms, Analysis, and Applications. Springer, New York (2008)CrossRefzbMATHGoogle Scholar
  21. 21.
    Harten, A.: On the symmetric form of systems of conservation laws with entropy. J. Comput. Phys. 49(1), 151–164 (1983)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Hughes, T.J.R., Franca, L.P., Mallet, M.: A new finite element formulation for computational fluid dynamics: I. Symmetric forms of the compressible Navier–Stokes equations and the second law of thermodymaics. Comput. Methods Appl. Mech. Eng. 54(2), 223–234 (1986)CrossRefzbMATHGoogle Scholar
  23. 23.
    Gassner, G.J., Winters, A.R., Kopriva, D.A.: Split form nodal discontinuous Galerkin schemes with summation-by-parts property for the compressible Euler equations, pp. 28–50. arXiv:1604.06618v2 [math.NA] (2016)
  24. 24.
    Kopriva, D.A., Gassner, G.J.: An energy stable discontinuous Galerkin spectral element discretization for variable coefficient advection problems. SIAM J. Sci. Comput. 4(36), A2076–A2099 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Gustafsson, B., Kreiss, H.-O., Oliger, J.: Time-Dependent Problems and Difference Methods. Pure and Applied Mathematics, 2nd edn. Wiley, New York (2013)CrossRefzbMATHGoogle Scholar
  26. 26.
    Kreiss, H.-O., Lorenz, J.: Initial-Boundary Value Problems and the Navier–Stokes Equations, Vol. 47 of Classics in Applied Mathematics. SIAM, New Delhi (2004)CrossRefGoogle Scholar
  27. 27.
    Thomas, P.D., Lombard, C.K.: Geometric conservation law and its application to flow computations on moving grids. AIAA J. 17(10), 1030–1037 (1979)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    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

Copyright information

© Springer Science+Business Media, LLC 2017

Authors and Affiliations

  1. 1.University of Toronto Institute for Aerospace StudiesNorth YorkCanada
  2. 2.Rensselaer Polytechnic InstituteTroyUSA

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