Skip to main content
Springer Nature Link
Log in

Superconvergent Functional Estimates from Tensor-Product Generalized Summation-by-Parts Discretizations in Curvilinear Coordinates

  • Published:
Journal of Scientific Computing Aims and scope Submit manuscript

Abstract

We investigate superconvergent functional estimates in curvilinear coordinates for diagonal-norm tensor-product generalized summation-by-parts operators. We show that interpolation/extrapolation operators of degree greater than or equal to 2p are required to preserve at least 2p quadrature accuracy and functional superconvergence in curvilinear coordinates when: (1) the Jacobian of the coordinate transformation is approximated by the same generalized summation-by-parts operator that is used to approximate the flux terms and (2) the degree of the generalized summation-by-parts operator is lower than the degree of the polynomial used to represent the geometry of interest. Legendre–Gauss–Lobatto and Legendre–Gauss element-type operators are considered. When the aforementioned condition (2) is violated for the Legendre–Gauss operators, there is an even–odd quadrature convergence pattern that is explained by the cancellation of the leading truncation error terms for the interpolation/extrapolation operators that correspond to the odd-degree Legendre–Gauss operators. The theory developed is confirmed through numerical examples with a steady one-dimensional problem and the unsteady two-dimensional linear convection equation.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7

Similar content being viewed by others

References

  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)

    Article  MathSciNet  Google Scholar 

  2. Boom, P.D.: High-order implicit time-marching methods for unsteady fluid flow simulation. Ph.D. thesis, University of Toronto (2015)

  3. Boom, P.D., Zingg, D.W.: High-order implicit time-marching methods based on generalized summation-by-parts operators. SIAM J. Sci. Comput. 37(6), A2682–A2709 (2015)

    Article  MathSciNet  Google Scholar 

  4. Carpenter, M.H., Gottlieb, D.: Spectral methods on arbitrary grids. J. Comput. Phys. 129(1), 74–86 (1996)

    Article  MathSciNet  Google Scholar 

  5. 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, 220–236 (1994)

    Article  MathSciNet  Google Scholar 

  6. Cockburn, B., Wang, Z.: Adjoint-based, superconvergent Galerkin approximations of linear functionals. J. Sci. Comput. 73(2–3), 644–666 (2017)

    Article  MathSciNet  Google Scholar 

  7. Crean, J., Hicken, J.E., Del Rey Fernández, D.C., Zingg, D.W., Carpenter, M.H.: Entropy-stable summation-by-parts discretization of the Euler equations on general curved elements. J. Comput. Phys. 356, 410–438 (2018)

    Article  MathSciNet  Google Scholar 

  8. Del Rey Fernández, D.C.: Generalized summation-by-parts operators for first and second derivatives. Ph.D. thesis, University of Toronto (2015)

  9. Del Rey Fernández, D.C., Boom, P.D., Carpenter, M.H., Zingg, D.W.: Extension of tensor-product generalized and dense-norm summation-by-parts operators to curvilinear coordinates. J. Sci. Comput. 80(3), 1957–1996 (2019)

    Article  MathSciNet  Google Scholar 

  10. Del Rey Fernández, D.C., Boom, P.D., Shademan, M., Zingg, D.W.: Numerical investigation of tensor-product summation-by-parts discretization strategies and operators. 55th AIAA Aerospace Sciences Meeting (2017). AIAA paper 2017-0530

  11. 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, 214–239 (2014)

    Article  MathSciNet  Google Scholar 

  12. 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, 171–196 (2014)

    Article  MathSciNet  Google Scholar 

  13. Funaro, D., Gottlieb, D.: A new method of imposing boundary conditions in pseudospectral approximations of hyperbolic equations. Math. Comput. 51(184), 599–613 (1988)

    Article  MathSciNet  Google Scholar 

  14. 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)

    Article  MathSciNet  Google Scholar 

  15. 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. J. Comput. Phys. 327, 39–66 (2016)

    Article  MathSciNet  Google Scholar 

  16. Hartmann, R.: Adjoint consistency analysis of discontinuous Galerkin discretizations. SIAM J. Numer. Anal. 45(6), 2671–2696 (2007)

    Article  MathSciNet  Google Scholar 

  17. Hicken, J.E.: Output error estimation for summation-by-parts finite-difference schemes. J. Comput. Phys. 231(9), 3828–3848 (2012)

    Article  MathSciNet  Google Scholar 

  18. Hicken, J.E., Del Rey Fernández, D.C., Zingg, D.W.: Multidimensional summation-by-parts operators: general theory and application to simplex elements. SIAM J. Sci. Comput. 38(4), A1935–A1958 (2016)

    Article  MathSciNet  Google Scholar 

  19. Hicken, J.E., Zingg, D.W.: Parallel Newton–Krylov solver for the Euler equations discretized using simultaneous approximation terms. AIAA J. 46(11), 2773–2786 (2008)

    Article  Google Scholar 

  20. 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)

    Article  MathSciNet  Google Scholar 

  21. Hicken, J.E., Zingg, D.W.: Summation-by-parts operators and high-order quadrature. J. Comput. Appl. Math. 237(1), 111–125 (2013)

    Article  MathSciNet  Google Scholar 

  22. Hicken, J.E., Zingg, D.W.: Dual consistency and functional accuracy: a finite-difference perspective. J. Comput. Phys. 256, 161–182 (2014)

    Article  MathSciNet  Google Scholar 

  23. Hunter, J.D.: Matplotlib: a 2d graphics environment. Comput. Sci. Eng. 9(3), 90–95 (2007)

    Article  Google Scholar 

  24. Koepf, W.: Hypergeometric Summation. Universitext. Springer, London (2014)

    Book  Google Scholar 

  25. Kreiss, H.O., Scherer, G.: Finite element and finite difference methods for hyperbolic partial differential equations. In: Mathematical Aspects of Finite Elements in Partial Differential Equations, pp. 195–212. Academic Press, New York/London (1974)

    Chapter  Google Scholar 

  26. Krivodonova, L., Xin, J., Remacle, J.F., Chevaugeon, N., Flaherty, J.E.: Shock detection and limiting with discontinuous Galerkin methods for hyperbolic conservation laws. Appl. Numer. Math. 48(3–4), 323–338 (2004)

    Article  MathSciNet  Google Scholar 

  27. Loken, C., Gruner, D., Groer, L., Peltier, R., Bunn, N., Craig, M., Henriques, T., Dempsey, J., Yu, C.H., Chen, J., Dursi, L.J., Chong, J., Northrup, S., Pinto, J., Knecht, N., Zon, R.V.: SciNet: lessons learned from building a power-efficient top-20 system and data centre. J. Phys. Conf. Ser. 256, 012026 (2010)

    Article  Google Scholar 

  28. Lu, J.C.C.: An a posteriori error control framework for adaptive precision optimization using discontinuous Galerkin finite element method. Ph.D. thesis, Massachusetts Institute of Technology (2005)

  29. Nordström, J.: Conservative finite difference formulations, variable coefficients, energy estimates and artificial dissipation. J. Sci. Comput. 29(3), 375–404 (2006)

    Article  MathSciNet  Google Scholar 

  30. Osusky, M., Zingg, D.W.: Parallel Newton–Krylov–Schur flow solver for the Navier–Stokes equations. AIAA J. 51(12), 2833–2851 (2013)

    Article  Google Scholar 

  31. Ranocha, H.: Generalised summation-by-parts operators and variable coefficients. J. Comput. Phys. 362, 20–48 (2018)

    Article  MathSciNet  Google Scholar 

  32. Ranocha, H., Öffner, P., Sonar, T.: Summation-by-parts operators for correction procedure via reconstruction. J. Comput. Phys. 311, 299–328 (2016)

    Article  MathSciNet  Google Scholar 

  33. Svärd, M., Carpenter, M.H., 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)

    Article  MathSciNet  Google Scholar 

  34. Svärd, M., Nordström, J.: Review of summation-by-parts schemes for initial-boundary-value problems. J. Comput. Phys. 268, 17–38 (2014)

    Article  MathSciNet  Google Scholar 

  35. 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)

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

The first author wishes to thank Dr. Pieter D. Boom for helpful discussions regarding some of the proofs in his Ph.D. thesis [2]. All figures were created using Matplotlib [23] with the convergence plots styled after those in [18].

Author information

Authors and Affiliations

  1. University of Toronto Institute for Aerospace Studies, Toronto, Canada

    David A. Craig Penner & David W. Zingg

Authors
  1. David A. Craig Penner
    View author publications

    You can also search for this author inPubMed Google Scholar

  2. David W. Zingg
    View author publications

    You can also search for this author inPubMed Google Scholar

Corresponding author

Correspondence to David A. Craig Penner.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

A preliminary version of this paper appeared in Generalized summation-by-parts methods: coordinate transformations, quadrature accuracy, and functional superconvergence, AIAA Aviation 2019 Forum, (2019). AIAA paper 2019-2952.

This work was supported by the Natural Sciences and Engineering Research Council of Canada and the University of Toronto. A portion of the computations were performed on the Niagara supercomputer at the SciNet HPC Consortium [27]. SciNet is funded by: the Canada Foundation for Innovation; the Government of Ontario; Ontario Research Fund - Research Excellence; and the University of Toronto.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Craig Penner, D.A., Zingg, D.W. Superconvergent Functional Estimates from Tensor-Product Generalized Summation-by-Parts Discretizations in Curvilinear Coordinates. J Sci Comput 82, 41 (2020). https://doi.org/10.1007/s10915-020-01147-7

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s10915-020-01147-7

Keywords

Mathematics Subject Classification