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.
This is a preview of subscription content, log in to check access.
Buy single article
Instant access to the full article PDF.
Price includes VAT for USA
Subscribe to journal
Immediate online access to all issues from 2019. Subscription will auto renew annually.
This is the net price. Taxes to be calculated in checkout.
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)
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)
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)
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)
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)
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)
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)
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)
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)
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)
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)
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)
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)
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)
Hesthaven, J.S., Warburton, T.: Nodal Discontinuous Galerkin Methods: algorithms, analysis, and applications, 54. Springer Verlag, New York (2008)
Huynh, H.: A flux reconstruction approach to high-order schemes including discontinuous Galerkin methods. AIAA Pap. 4079, 2007 (2007)
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)
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)
Jameson, A., Baker, T.: Solution of the Euler equations for complex configurations. In: 6th Computational Fluid Dynamics Conference Danvers, p. 1929 (1983)
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)
Kopriva, D.A., Kolias, J.H.: A conservative staggered-grid Chebyshev multidomain method for compressible flows. J. Comput. Phys. 125(1), 244–261 (1996)
Liu, Y., Vinokur, M., Wang, Z.: Spectral difference method for unstructured grids i: basic formulation. J. Comput. Phys. 216(2), 780–801 (2006)
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)
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)
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)
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)
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)
Romero, J., Asthana, K., Jameson, A.: A simplified formulation of the flux reconstruction method. J. Sci. Comput. 67(1), 351–374 (2016)
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)
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)
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)
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
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)
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)
Williams, D., Jameson, A.: Energy stable flux reconstruction schemes for advection-diffusion problems on tetrahedra. J. Sci. Comput. 59(3), 721–759 (2014)
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)
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)
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)
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)
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.
Dedicated to Professor Chi-Wang Shu on the occasion of his 60th birthday.
See the Table 17.
About this article
Cite this article
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
- High order methods
- Flux reconstruction
- Spectral difference
- Triangular elements
- Compressible navier–stokes