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

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Abstract

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.

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References

  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)

    Article  MATH  MathSciNet  Google Scholar 

  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)

    Article  MATH  MathSciNet  Google Scholar 

  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)

    Article  MATH  MathSciNet  Google Scholar 

  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)

    Article  MATH  MathSciNet  Google Scholar 

  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)

    Article  MATH  MathSciNet  Google Scholar 

  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)

    Article  MATH  Google Scholar 

  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)

    MATH  MathSciNet  Google Scholar 

  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)

    Article  MATH  MathSciNet  Google Scholar 

  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)

    MATH  MathSciNet  Google Scholar 

  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)

    MATH  MathSciNet  Google Scholar 

  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)

    Article  MATH  MathSciNet  Google Scholar 

  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)

    Article  MATH  MathSciNet  Google Scholar 

  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)

    Article  MATH  MathSciNet  Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

  15. Hesthaven, J.S., Warburton, T.: Nodal Discontinuous Galerkin Methods: algorithms, analysis, and applications, 54. Springer Verlag, New York (2008)

  16. Huynh, H.: A flux reconstruction approach to high-order schemes including discontinuous Galerkin methods. AIAA Pap. 4079, 2007 (2007)

    Google Scholar 

  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)

  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)

    Article  MATH  MathSciNet  Google Scholar 

  19. Jameson, A., Baker, T.: Solution of the Euler equations for complex configurations. In: 6th Computational Fluid Dynamics Conference Danvers, p. 1929 (1983)

  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)

    Article  MATH  MathSciNet  Google Scholar 

  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)

    Article  MATH  MathSciNet  Google Scholar 

  22. Liu, Y., Vinokur, M., Wang, Z.: Spectral difference method for unstructured grids i: basic formulation. J. Comput. Phys. 216(2), 780–801 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  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. 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)

    Article  MATH  MathSciNet  Google Scholar 

  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)

    Article  MATH  MathSciNet  Google Scholar 

  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)

    Article  Google Scholar 

  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)

  28. Romero, J., Asthana, K., Jameson, A.: A simplified formulation of the flux reconstruction method. J. Sci. Comput. 67(1), 351–374 (2016)

    Article  MATH  MathSciNet  Google Scholar 

  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)

    Article  Google Scholar 

  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)

    Article  MATH  Google Scholar 

  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)

  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). http://dl.acm.org/citation.cfm?id=3014904.3014906

  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)

    Article  MATH  MathSciNet  Google Scholar 

  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)

    Article  MATH  MathSciNet  Google Scholar 

  35. Williams, D., Jameson, A.: Energy stable flux reconstruction schemes for advection-diffusion problems on tetrahedra. J. Sci. Comput. 59(3), 721–759 (2014)

    Article  MATH  MathSciNet  Google Scholar 

  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)

    Article  MATH  MathSciNet  Google Scholar 

  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)

    Article  MATH  MathSciNet  Google Scholar 

  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)

    Article  MATH  Google Scholar 

  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)

    Article  MATH  MathSciNet  Google Scholar 

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Acknowledgements

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.

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Authors and Affiliations

  1. Department of Aeronautics and Astronautics, Stanford University, Stanford, CA, 94305, USA

    J. Romero, F. D. Witherden & A. Jameson

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  1. J. Romero
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  2. F. D. Witherden
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  3. A. Jameson
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Correspondence to J. Romero.

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Dedicated to Professor Chi-Wang Shu on the occasion of his 60th birthday.

Appendix

Appendix

See the Table 17.

Table 17 \(\epsilon \) point locations in barycentric coordinates, \(\varvec{\xi } = (\xi _1, \xi _2, \xi _3)^T\)
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Romero, J., Witherden, F.D. & Jameson, A. A Direct Flux Reconstruction Scheme for Advection–Diffusion Problems on Triangular Grids. J Sci Comput 73, 1115–1144 (2017). https://doi.org/10.1007/s10915-017-0472-1

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