Journal of Scientific Computing

, Volume 73, Issue 2–3, pp 1115–1144 | Cite as

A Direct Flux Reconstruction Scheme for Advection–Diffusion Problems on Triangular Grids

  • J. RomeroEmail author
  • F. D. Witherden
  • A. Jameson


The direct flux reconstruction (DFR) scheme is a high-order numerical method which is an alternative realization of the flux reconstruction (FR) approach. In 1D, the DFR scheme has been shown to be equivalent to the FR variant of the nodal discontinuous Galerkin scheme. In this article, the DFR approach is extended to triangular elements for advection and advection–diffusion problems. This was accomplished by combining aspects of the SD–RT variant of the spectral difference (SD) scheme for triangles, with modifications motivated by characteristics of the DFR scheme in one dimension. Von Neumann analysis is applied to the new scheme and linear stability is found to be dependent on the location of internal collocation points. This is in contrast to the standard FR scheme. This analysis indicates certain internal point sets can result in schemes which exhibit weak stability; however, stable and accurate solutions to a number of linear and nonlinear benchmark problems are readily obtained.


High order methods Flux reconstruction Spectral difference Triangular elements Compressible navier–stokes 



The authors would like to thank the Air Force Office of Scientific Research for their support via Grant FA9550-14-1-0186. The first author would like to acknowledge support from the Morgridge Family Stanford Graduate Fellowship.


  1. 1.
    Van den Abeele, K., Lacor, C., Wang, Z.: On the stability and accuracy of the spectral difference method. J. Sci. Comput. 37(2), 162–188 (2008)CrossRefzbMATHMathSciNetGoogle Scholar
  2. 2.
    Allaneau, Y., Jameson, A.: 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
  3. 3.
    Balan, A., May, G., Schöberl, J.: A stable high-order spectral difference method for hyperbolic conservation laws on triangular elements. J. Comput. Phy. 231(5), 2359–2375 (2012)CrossRefzbMATHMathSciNetGoogle Scholar
  4. 4.
    Castonguay, P., Vincent, P.E., Jameson, A.: A new class of high-order energy stable flux reconstruction schemes for triangular elements. J. Sci. Comput. 51(1), 224–256 (2012)CrossRefzbMATHMathSciNetGoogle Scholar
  5. 5.
    Castonguay, P., Williams, D., Vincent, P., Jameson, A.: Energy stable flux reconstruction schemes for advection-diffusion problems. Comput. Methods Appl. Mech. Eng. 267, 400–417 (2013)CrossRefzbMATHMathSciNetGoogle Scholar
  6. 6.
    Chan, A.S., Dewey, P.A., Jameson, A., Liang, C., Smits, A.J.: Vortex suppression and drag reduction in the wake of counter-rotating cylinders. J. Fluid Mech. 679, 343–382 (2011)CrossRefzbMATHGoogle Scholar
  7. 7.
    Cockburn, B., Hou, S., Shu, C.W.: The runge-kutta local projection discontinuous galerkin finite element method for conservation laws. iv. the multidimensional case. Math. Comput. 54(190), 545–581 (1990)zbMATHMathSciNetGoogle Scholar
  8. 8.
    Cockburn, B., Lin, S.Y., Shu, C.W.: Tvb runge-kutta local projection discontinuous galerkin finite element method for conservation laws iii: one-dimensional systems. J. Comput. Phys. 84(1), 90–113 (1989)CrossRefzbMATHMathSciNetGoogle Scholar
  9. 9.
    Cockburn, B., Shu, C.W.: Tvb runge-kutta local projection discontinuous galerkin finite element method for conservation laws. ii. general framework. Math. Comput. 52(186), 411–435 (1989)zbMATHMathSciNetGoogle Scholar
  10. 10.
    Cockburn, B., Shu, C.W.: The runge-kutta local projection \( p^1\)-discontinuous-galerkin finite element method for scalar conservation laws. RAIRO-Modélisation mathématique et analyse numérique 25(3), 337–361 (1991)zbMATHMathSciNetGoogle Scholar
  11. 11.
    Cockburn, B., Shu, C.W.: The local discontinuous galerkin method for time-dependent convection-diffusion systems. SIAM J. Numer. Anal. 35(6), 2440–2463 (1998)CrossRefzbMATHMathSciNetGoogle Scholar
  12. 12.
    Cockburn, B., Shu, C.W.: The runge-kutta discontinuous galerkin method for conservation laws v: multidimensional systems. J. Comput. Phys. 141(2), 199–224 (1998)CrossRefzbMATHMathSciNetGoogle Scholar
  13. 13.
    Cox, C., Liang, C., Plesniak, M.W.: A high-order solver for unsteady incompressible navier-stokes equations using the flux reconstruction method on unstructured grids with implicit dual time stepping. J. Comput. Phys. 314, 414–435 (2016)CrossRefzbMATHMathSciNetGoogle Scholar
  14. 14.
    De Grazia, D., Mengaldo, G., Moxey, D., Vincent, P., Sherwin, S.: Connections between the discontinuous galerkin method and high-order flux reconstruction schemes. Int. J. Numer. Methods Fluids 75(12), 860–877 (2014)CrossRefMathSciNetGoogle Scholar
  15. 15.
    Hesthaven, J.S., Warburton, T.: Nodal Discontinuous Galerkin Methods: algorithms, analysis, and applications, 54. Springer Verlag, New York (2008)Google Scholar
  16. 16.
    Huynh, H.: A flux reconstruction approach to high-order schemes including discontinuous Galerkin methods. AIAA Pap. 4079, 2007 (2007)Google Scholar
  17. 17.
    Huynh, H.T.: A reconstruction approach to high-order schemes including discontinuous Galerkin for diffusion. In: 47th AIAA Aerospace Sciences Meeting including the New Horizons Forum and Aerospace Exposition, p. 403 (2009)Google Scholar
  18. 18.
    Jameson, A.: A proof of the stability of the spectral difference method for all orders of accuracy. J. Sci. Comp. 45(1–3), 348–358 (2010)CrossRefzbMATHMathSciNetGoogle Scholar
  19. 19.
    Jameson, A., Baker, T.: Solution of the Euler equations for complex configurations. In: 6th Computational Fluid Dynamics Conference Danvers, p. 1929 (1983)Google Scholar
  20. 20.
    Kennedy, C.A., Carpenter, M.H., Lewis, R.M.: Low-storage, explicit runge-kutta schemes for the compressible navier–stokes equations. Appl. Numer. Math. 35(3), 177–219 (2000)CrossRefzbMATHMathSciNetGoogle Scholar
  21. 21.
    Kopriva, D.A., Kolias, J.H.: A conservative staggered-grid Chebyshev multidomain method for compressible flows. J. Comput. Phys. 125(1), 244–261 (1996)CrossRefzbMATHMathSciNetGoogle Scholar
  22. 22.
    Liu, Y., Vinokur, M., Wang, Z.: Spectral difference method for unstructured grids i: basic formulation. J. Comput. Phys. 216(2), 780–801 (2006)CrossRefzbMATHMathSciNetGoogle Scholar
  23. 23.
    May, G., Schöberl, J.: Analysis of a Spectral Difference Scheme with Flux Interpolation on Raviart-Thomas Elements. Aachen Institute for Advanced Study in Computational Engineering Science, Aachen (2010)Google Scholar
  24. 24.
    Mengaldo, G., De Grazia, D., Moxey, D., Vincent, P.E., Sherwin, S.: Dealiasing techniques for high-order spectral element methods on regular and irregular grids. J. Comput. Phys. 299, 56–81 (2015)CrossRefzbMATHMathSciNetGoogle Scholar
  25. 25.
    Mengaldo, G., Grazia, D., Vincent, P.E., Sherwin, S.J.: On the connections between discontinuous galerkin and flux reconstruction schemes: extension to curvilinear meshes. J. Sci. Comput. 67(3), 1272–1292 (2016)CrossRefzbMATHMathSciNetGoogle Scholar
  26. 26.
    Park, J., Kwon, K., Choi, H.: Numerical solutions of flow past a circular cylinder at reynolds numbers up to 160. KSME Int. J. 12(6), 1200–1205 (1998)CrossRefGoogle Scholar
  27. 27.
    Raviart, P.A., Thomas, J.M.: A mixed finite element method for 2-nd order elliptic problems. In: Galligani, I., Magenes, E. (eds.) Mathematical Aspects of Finite Element Methods, pp. 292–315. Springer (1977)Google Scholar
  28. 28.
    Romero, J., Asthana, K., Jameson, A.: A simplified formulation of the flux reconstruction method. J. Sci. Comput. 67(1), 351–374 (2016)CrossRefzbMATHMathSciNetGoogle Scholar
  29. 29.
    Rusanov, V.V.: The calculation of the interaction of non-stationary shock waves and obstacles. USSR Comput. Math. Math. Phys. 1(2), 304–320 (1962)CrossRefGoogle Scholar
  30. 30.
    Sharman, B., Lien, F.S., Davidson, L., Norberg, C.: Numerical predictions of low reynolds number flows over two tandem circular cylinders. Int. J. Numer. Methods in Fluids 47(5), 423–447 (2005)CrossRefzbMATHGoogle Scholar
  31. 31.
    Vermeire, B., Witherden, F., Vincent, P.: On the utility of gpu accelerated high-order methods for unsteady flow simulations: a comparison with industry-standard tools. J. Comput. Phys. 334, 497–521 (2017)Google Scholar
  32. 32.
    Vincent, P., Witherden, F., Vermeire, B., Park, J.S., Iyer, A.: Towards green aviation with python at petascale. In: Proceedings of the International Conference for High Performance Computing, Networking, Storage and Analysis, SC ’16, pp. 1:1–1:11. IEEE Press, Piscataway (2016).
  33. 33.
    Vincent, P.E., Castonguay, P., Jameson, A.: Insights from von neumann analysis of high-order flux reconstruction schemes. J. Comput. Phys. 230(22), 8134–8154 (2011)CrossRefzbMATHMathSciNetGoogle Scholar
  34. 34.
    Vincent, P.E., Castonguay, P., Jameson, A.: A new class of high-order energy stable flux reconstruction schemes. J. Sci. Comput. 47(1), 50–72 (2011)CrossRefzbMATHMathSciNetGoogle Scholar
  35. 35.
    Williams, D., Jameson, A.: Energy stable flux reconstruction schemes for advection-diffusion problems on tetrahedra. J. Sci. Comput. 59(3), 721–759 (2014)CrossRefzbMATHMathSciNetGoogle Scholar
  36. 36.
    Williams, D., Shunn, L., Jameson, A.: Symmetric quadrature rules for simplexes based on sphere close packed lattice arrangements. J. Comput. Appl. Math. 266, 18–38 (2014)CrossRefzbMATHMathSciNetGoogle Scholar
  37. 37.
    Williams, D.M., Castonguay, P., Vincent, P.E., Jameson, A.: Energy stable flux reconstruction schemes for advection-diffusion problems on triangles. J. Comput. Phys. 250, 53–76 (2013)CrossRefzbMATHMathSciNetGoogle Scholar
  38. 38.
    Witherden, F.D., Farrington, A.M., Vincent, P.E.: 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
  39. 39.
    Witherden, F.D., Vincent, P.E.: An analysis of solution point coordinates for flux reconstruction schemes on triangular elements. J. Sci. Comput. 61(2), 398–423 (2014)CrossRefzbMATHMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2017

Authors and Affiliations

  1. 1.Department of Aeronautics and AstronauticsStanford UniversityStanfordUSA

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