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

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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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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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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Keywords

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