Journal of Scientific Computing

, Volume 47, Issue 1, pp 50–72 | Cite as

A New Class of High-Order Energy Stable Flux Reconstruction Schemes

Article

Abstract

The flux reconstruction approach to high-order methods is robust, efficient, simple to implement, and allows various high-order schemes, such as the nodal discontinuous Galerkin method and the spectral difference method, to be cast within a single unifying framework. Utilizing a flux reconstruction formulation, it has been proved (for one-dimensional linear advection) that the spectral difference method is stable for all orders of accuracy in a norm of Sobolev type, provided that the interior flux collocation points are located at zeros of the corresponding Legendre polynomials. In this article the aforementioned result is extended in order to develop a new class of one-dimensional energy stable flux reconstruction schemes. The energy stable schemes are parameterized by a single scalar quantity, which if chosen judiciously leads to the recovery of various well known high-order methods (including a particular nodal discontinuous Galerkin method and a particular spectral difference method). The analysis offers significant insight into why certain flux reconstruction schemes are stable, whereas others are not. Also, from a practical standpoint, the analysis provides a simple prescription for implementing an infinite range of energy stable high-order methods via the intuitive flux reconstruction approach.

Keywords

High-order methods Flux reconstruction Nodal discontinuous Galerkin method Spectral difference method Stability 

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References

  1. 1.
    Reed, W.H., Hill, T.R.: Triangular mesh methods for the neutron transport equation. Technical Report LA-UR-73-479, Los Alamos National Laboratory, Los Alamos, New Mexico, USA (1973) Google Scholar
  2. 2.
    Cockburn, B., Shu, C.: Runge–Kutta discontinuous Galerkin methods for convection-dominated problems. J. Sci. Comput. 16, 173 (2001) MathSciNetMATHCrossRefGoogle Scholar
  3. 3.
    Arnold, D.N., Brezzi, F., Cockburn, B., Marini, L.D.: Unified analysis of discontinuous Galerkin methods for elliptic problems. SIAM J. Numer. Anal. 39, 1749 (2001) MathSciNetCrossRefGoogle Scholar
  4. 4.
    Hesthaven, J.S., Warburton, T.: Nodal Discontinuous Galerkin Methods—Algorithms, Analysis, and Applications. Springer, Berlin (2008) MATHCrossRefGoogle Scholar
  5. 5.
    Hesthaven, J.S., Warburton, T.: Nodal high-order methods on unstructured grids. J. Comput. Phys. 181, 186 (2002) MathSciNetMATHCrossRefGoogle Scholar
  6. 6.
    Giraldo, F.X., Hesthaven, J.S., Warburton, T.: Nodal high-order discontinuous Galerkin methods for the spherical shallow water equations. J. Comput. Phys. 181, 499 (2002) MathSciNetMATHCrossRefGoogle Scholar
  7. 7.
    Kopriva, D.A., Kolias, J.H.: A conservative staggered-grid Chebyshev multidomain method for compressible flows. J. Comput. Phys. 125, 244 (1996) MathSciNetMATHCrossRefGoogle Scholar
  8. 8.
    Liu, Y., Vinokur, M., Wang, Z.J.: Spectral difference method for unstructured grids I: basic formulation. J. Comput. Phys. 216, 780 (2006) MathSciNetMATHCrossRefGoogle Scholar
  9. 9.
    Wang, Z.J., Liu, Y., May, G., Jameson, A.: Spectral difference method for unstructured grids II: extension to the Euler equations. J. Sci. Comput. 32, 45 (2007) MathSciNetMATHCrossRefGoogle Scholar
  10. 10.
    Liang, C., Premasuthan, S., Jameson, A.: High-order accurate simulation of low-Mach laminar flow past two side-by-side cylinders using spectral difference method. Comput. Struct. 87, 812 (2009) CrossRefGoogle Scholar
  11. 11.
    Liang, C., Jameson, A., Wang, Z.J.: Spectral difference method for compressible flow on unstructured grids with mixed elements. J. Comput. Phys. 228, 2847 (2009) MATHCrossRefGoogle Scholar
  12. 12.
    Huynh, H.T.: A flux reconstruction approach to high-order schemes including discontinuous Galerkin methods. In: AIAA Computational Fluid Dynamics Meeting (2007) Google Scholar
  13. 13.
    Jameson, A.: A proof of the stability of the spectral difference method for all orders of accuracy. J. Sci. Comput. 45(1–3), 348–358 (2010) MathSciNetMATHCrossRefGoogle Scholar
  14. 14.
    Roe, P.L.: Approximate Riemann solvers, parameter vectors, and difference schemes. J. Comput. Phys. 43, 357 (1981) MathSciNetMATHCrossRefGoogle Scholar
  15. 15.
    Carpenter, M.H., Kennedy, C.: Fourth-order 2N-storage Runge-Kutta schemes. Technical Report TM 109112, NASA, NASA Langley Research Center (1994) Google Scholar

Copyright information

© Springer Science+Business Media, LLC 2010

Authors and Affiliations

  1. 1.Department of Aeronautics and AstronauticsStanford UniversityStanfordUSA

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