Advertisement

Summation-by-Parts and Correction Procedure via Reconstruction

  • Hendrik RanochaEmail author
  • Philipp Öffner
  • Thomas Sonar
Conference paper
Part of the Lecture Notes in Computational Science and Engineering book series (LNCSE, volume 119)

Abstract

The correction procedure via reconstruction (CPR, also known as flux reconstruction), is a framework of high order methods for conservation laws, unifying some discontinuous Galerkin, spectral difference and spectral volume methods. These methods are embedded in the framework of summation-by-parts (SBP) operators with simultaneous approximation terms (SATs), recovering the linearly stable methods of Vincent et al. (J Comput Phys 230(22): 8134–8154, 2011; J Sci Comput 47(1):50–72, 2011; Comput Methods Appl Mech Eng 296:248–272, 2015). The introduction of new correction terms enables stability for Burgers’ equation using nodal bases not including boundary nodes, i.e. Gauss nodes. Extended notions of SBP operators and split-forms are used to obtain stability.

References

  1. 1.
    Y. Allaneau, A. Jameson, Connections between the filtered discontinuous Galerkin method and the flux reconstruction approach to high order discretizations. Comput. Methods Appl. Mech. Eng. 200(49), 3628–3636 (2011)CrossRefzbMATHMathSciNetGoogle Scholar
  2. 2.
    D. De Grazia, G. Mengaldo, D. Moxey, P.E. Vincent, S.J. Sherwin, Connections between the discontinuous Galerkin method and high-order flux reconstruction schemes. Int. J. Numer. Methods Fluids 75(12), 860–877 (2014)CrossRefMathSciNetGoogle Scholar
  3. 3.
    D.C.D.R. Fernández, P.D. Boom, D.W. Zingg, A generalized framework for nodal first derivative summation-by-parts operators. J. Comput. Phys. 266, 214–239 (2014)CrossRefzbMATHMathSciNetGoogle Scholar
  4. 4.
    D.C.D.R. Fernández, J.E. Hicken, D.W. Zingg, Review of summation-by-parts operators with simultaneous approximation terms for the numerical solution of partial differential equations. Comput. Fluids 95, 171–196 (2014)CrossRefMathSciNetGoogle Scholar
  5. 5.
    T.C. Fisher, M.H. Carpenter, J. Nordström, N.K. Yamaleev, C. Swanson, Discretely conservative finite-difference formulations for nonlinear conservation laws in split form: theory and boundary conditions. J. Comput. Phys. 234, 353–375 (2013)CrossRefzbMATHMathSciNetGoogle Scholar
  6. 6.
    G.J. Gassner, 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)CrossRefzbMATHMathSciNetGoogle Scholar
  7. 7.
    J. Glaubitz, H. Ranocha, P. Öffner, T. Sonar, Enhancing stability of correction procedure via reconstruction using summation-by-parts operators II: modal filtering (2016). arXiv: 1606.01056 [math.NA] (Submitted)
  8. 8.
    J.E. Hicken, D.C. Del Rey Fernández, D.W. Zingg, Multidimensional summation-by-parts operators: general theory and application to simplex elements. SIAM J. Sci. Comput. 38(4), A1935–A1958 (2016)CrossRefzbMATHMathSciNetGoogle Scholar
  9. 9.
    H. Huynh, A flux reconstruction approach to high-order schemes including discontinuous Galerkin methods. AIAA Paper 4079, 2007 (2007)Google Scholar
  10. 10.
    A. Jameson, A proof of the stability of the spectral difference method for all orders of accuracy. J. Sci. Comput. 45(1–3), 348–358 (2010)CrossRefzbMATHMathSciNetGoogle Scholar
  11. 11.
    A. Jameson, P.E. Vincent, P. Castonguay, On the non-linear stability of flux reconstruction schemes. J. Sci. Comput. 50(2), 434–445 (2012)CrossRefzbMATHMathSciNetGoogle Scholar
  12. 12.
    K. Mattsson, M. Svärd, J. Nordström, Stable and accurate artificial dissipation. J. Sci. Comput. 21(1), 57–79 (2004)CrossRefzbMATHMathSciNetGoogle Scholar
  13. 13.
    J. Nordström, P. Eliasson, New developments for increased performance of the SBP-SAT finite difference technique, in IDIHOM: Industrialization of High-Order Methods-A Top-Down Approach (Springer, Berlin, 2015), pp. 467–488Google Scholar
  14. 14.
    P. Olsson, J. Oliger, Energy and maximum norm estimates for nonlinear conservation laws. Technical Report NASA-CR-195091, NASA, Research Institute for Advanced Computer Science; Moffett Field, CA, United States (1994)Google Scholar
  15. 15.
    H. Ranocha, SBP operators for CPR methods. Master’s thesis, TU Braunschweig (2016)Google Scholar
  16. 16.
    H. Ranocha, Shallow water equations: split-form, entropy stable, well-balanced, and positivity preserving numerical methods. GEM–Int. J. Geomath. (2016). DOI 10.1007/s13137-016-0089-9. arXiv: 1609.08029 [math.NA]
  17. 17.
    H. Ranocha, J. Glaubitz, P. Öffner, T. Sonar, Enhancing stability of correction procedure via reconstruction using summation-by-parts operators I: artificial dissipation. (2016). arXiv: 1606.00995 [math.NA] (Submitted)
  18. 18.
    H. Ranocha, P. Öffner, T. Sonar, Summation-by-parts operators for correction procedure via reconstruction. J. Comput. Phys. 311, 299–328 (2016). DOI 10.1016/j.jcp.2016.02.009. arXiv: 1511.02052 [math.NA]
  19. 19.
    H. Ranocha, P. Öffner, T. Sonar, Extended skew-symmetric form for summation-by-parts operators and varying Jacobians. J. Comput. Phys. 342, 13–28 (2017). DOI 10.1016/j.jcp.2017.04.044. arXiv: 1511.08408 [math.NA]
  20. 20.
    M. Svärd, J. Nordström, Review of summation-by-parts schemes for initial-boundary-value problems. J. Comput. Phys. 268, 17–38 (2014)CrossRefzbMATHMathSciNetGoogle Scholar
  21. 21.
    E. Tadmor, Skew-selfadjoint form for systems of conservation laws. J. Math. Anal. Appl. 103(2), 428–442 (1984)CrossRefzbMATHMathSciNetGoogle Scholar
  22. 22.
    E. Tadmor, The numerical viscosity of entropy stable schemes for systems of conservation laws. I. Math. Comput. 49(179), 91–103 (1987)CrossRefzbMATHMathSciNetGoogle Scholar
  23. 23.
    P.E. Vincent, P. Castonguay, A. Jameson, Insights from von Neumann analysis of high-order flux reconstruction schemes. J. Comput. Phys. 230(22), 8134–8154 (2011)CrossRefzbMATHMathSciNetGoogle Scholar
  24. 24.
    P.E. Vincent, P. Castonguay, A. Jameson, A new class of high-order energy stable flux reconstruction schemes. J. Sci. Comput. 47(1), 50–72 (2011)CrossRefzbMATHMathSciNetGoogle Scholar
  25. 25.
    P.E. Vincent, A.M. Farrington, F.D. Witherden, A. Jameson, An extended range of stable-symmetric-conservative flux reconstruction correction functions. Comput. Methods Appl. Mech. Eng. 296, 248–272 (2015)CrossRefMathSciNetGoogle Scholar
  26. 26.
    J. von Neumann, R.D. Richtmyer, A method for the numerical calculation of hydrodynamic shocks. J. Appl. Phys. 21(3), 232–237 (1950)CrossRefzbMATHMathSciNetGoogle Scholar
  27. 27.
    F.D. Witherden, A.M. Farrington, P.E. Vincent, PyFR: an open source framework for solving advection-diffusion type problems on streaming architectures using the flux reconstruction approach. Comput. Phys. Commun. 185(11), 3028–3040 (2014)CrossRefzbMATHGoogle Scholar

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  • Hendrik Ranocha
    • 1
    Email author
  • Philipp Öffner
    • 1
  • Thomas Sonar
    • 1
  1. 1.Institute Computational MathematicsTU BraunschweigBraunschweigGermany

Personalised recommendations