A Direct Flux Reconstruction Scheme for Advection–Diffusion Problems on Triangular Grids
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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.
KeywordsHigh 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.
- 15.Hesthaven, J.S., Warburton, T.: Nodal Discontinuous Galerkin Methods: algorithms, analysis, and applications, 54. Springer Verlag, New York (2008)Google Scholar
- 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)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)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
- 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
- 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.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