Journal of Scientific Computing

, Volume 32, Issue 1, pp 45–71 | Cite as

Spectral Difference Method for Unstructured Grids II: Extension to the Euler Equations

  • Z. J. Wang
  • Yen Liu
  • Georg May
  • Antony Jameson

An efficient, high-order, conservative method named the spectral difference method has been developed recently for conservation laws on unstructured grids. It combines the best features of structured and unstructured grid methods to achieve high-computational efficiency and geometric flexibility; it utilizes the concept of discontinuous and high-order local representations to achieve conservation and high accuracy; and it is based on the finite-difference formulation for simplicity. The method is easy to implement since it does not involve surface or volume integrals. Universal reconstructions are obtained by distributing solution and flux points in a geometrically similar manner for simplex cells. In this paper, the method is further extended to nonlinear systems of conservation laws, the Euler equations. Accuracy studies are performed to numerically verify the order of accuracy. In order to capture both smooth feature and discontinuities, monotonicity limiters are implemented, and tested for several problems in one and two dimensions. The method is more efficient than the discontinuous Galerkin and spectral volume methods for unstructured grids.


High-order conservation laws unstructured grids spectral difference spectral collocation method Euler equations 


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  1. 1.
    Barth, J., and Frederickson, P. O. (1990). High-order solution of the Euler equations on unstructured grids using quadratic reconstruction. AIAA Paper No. 90-0013.Google Scholar
  2. 2.
    Bassi F., Rebay S. (1997). High-order accurate discontinuous finite element solution of the 2D Euler equations. J. Comput. Phys. 138, 251–285MATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Canuto C., Hussaini M.Y., Quarteroni A., Zang T.A. (1987). Spectral Methods in Fluid Dynamics. Springer-Verlag, New YorkGoogle Scholar
  4. 4.
    Chen Q., Babuska I. (1995). Approximate optimal points for polynomial interpolation of real functions in an interval and in a triangle. Comput. Methods Appl. Mech. Engrg. 128, 405–417MATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Cockburn B., Shu C.-W. (1989). TVB Runge–Kutta local projection discontinuous Galerkin finite element method for conservation laws II: general framework. Math. Comput. 52, 411–435MATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Cockburn B., Shu C.-W. (1998). The Runge–Kutta discontinuous Garlerkin method for conservation laws V: multidimensional systems. J. Comput. Phys. 141, 199–224MATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Delanaye, M., and Liu, Y. (1999). Quadratic reconstruction finite volume schemes on 3D arbitrary unstructured polyhedral grids. AIAA Paper No. 99-3259-CPGoogle Scholar
  8. 8.
    Godunov S.K. (1959). A finite-difference method for the numerical computation of discontinuous solutions of the equations of fluid dynamics. Mater. Sb. 47, 271MathSciNetGoogle Scholar
  9. 9.
    Harten A. (1983). High resolution schemes for hyperbolic conservation laws. J. Comput. Phys. 49, 357–393MATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Hesthaven J.S. (1998). From electrostatics to almost optimal nodal sets for polynomial interpolation in a simplex. SIAM J. Numer. Anal. 35, 655–676MATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Jameson A. (1995). Analysis and design of numerical schemes for gas dynamics 2: artificial diffusion and discrete shock structure. Int. J. Comput. Fluid Dyn. 5, 1–38Google Scholar
  12. 12.
    Kopriva D.A. (1996). A conservative staggered-grid Chebyshev multidomain method for compressible flows. II semi-structured method. J. Comput. Phys. 128, 475MATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Kopriva D.A. (1998). A staggered-grid multidomain spectral method for the compressible Navier–Stokes equations. J. Comput. Phys. 143(1):125MATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Liu Y., Vinokur M. (1998). Exact integration of polynomials and symmetric quadrature formulas over arbitrary polyhedral grids. J. Comput. Phys. 140, 122–147MATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    Liu, Y., Vinokur, M., and Wang, Z. J. (2004). Discontinuous spectral difference method for conservation laws on unstructured grids. Proceedings of the 3rd International Conference in CFD, Toronto, Canada.Google Scholar
  16. 16.
    Liu Y., Vinokur M., Wang Z.J. (2006). Spectral difference method for unstructured grids I: basic formulation. J. Comput. Phys. 216(2):780–801MATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    Liu Y., Vinokur M., Wang Z.J. (2006). Spectral (finite) volume method for conservation laws on unstructured grids V: extension to three-dimensional systems. J. Comput. Phys. 212, 454–472MATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    Roe P.L. (1981). Approximate Riemann solvers, parameter vectors, and difference schemes. J. Comput. Phys. 43, 357–372MATHCrossRefMathSciNetGoogle Scholar
  19. 19.
    Rusanov V.V. (1961). Calculation of interaction of non-steady shock waves with obstacles. J. Comput. Math. Phys. USSR 1, 267–279MathSciNetGoogle Scholar
  20. 20.
    Shu C.-W. (1987). TVB uniformly high-order schemes for conservation laws. Math. Comput. 49, 105–121MATHCrossRefGoogle Scholar
  21. 21.
    Shu C.-W. (1988). Total-variation-diminishing time discretizations. SIAM J. Sci. Stat. Comput. 9:1073MATHCrossRefGoogle Scholar
  22. 22.
    Shu, C.-W. (1998). Essentially non-oscillatory and weighted essentially non-oscillatory schemes for hyperbolic conservation laws. In B. Cockburn, C. Johnson, C.-W. Shu and E. Tadmor (A. Quarteroni, ed.), Advanced Numerical Approximation of Nonlinear Hyperbolic Equations, Lecture Notes in Mathematics, Vol. 1697, Springer, Berlin, pp. 325–432.Google Scholar
  23. 23.
    Spiteri R.J., Ruuth S.J. (2002). A new class of optimal high-order strong-stability-preserving time discretization methods. SIAM J. Numer. Anal. 40, 469–491MATHCrossRefMathSciNetGoogle Scholar
  24. 24.
    Sridar D., Balakrishnan N. (2003). An upwind finite difference scheme for meshless solvers. J. Comput. Phys. 189, 1–29MATHCrossRefMathSciNetGoogle Scholar
  25. 25.
    van Leer B. (1979). Towards the ultimate conservative difference scheme V. a second order sequel to Godunov’s method. J. Comput. Phys. 32, 101–136CrossRefGoogle Scholar
  26. 26.
    Wang Z.J. (2000). A fast nested multi-grid viscous flow solver for adaptive Cartesian/quad grids. Int. J. Numer. Methods Fluids 33, 657–680MATHCrossRefGoogle Scholar
  27. 27.
    Wang Z.J. (2002). Spectral (finite) volume method for conservation laws on unstructured grids: I. Basic formulation. J. Comput. Phys. 178, 210MATHCrossRefMathSciNetGoogle Scholar
  28. 28.
    Wang Z.J., Liu Y. (2002). Spectral (finite) volume method for conservation laws on unstructured grids II: extension to two-dimensional scalar equation. J. Comput. Phys. 179, 665–697MATHCrossRefMathSciNetGoogle Scholar
  29. 29.
    Wang Z.J., Liu Y. (2004).Spectral (finite) volume method for conservation laws on unstructured grids III: one-dimensional systems and partition optimization. J. Scientific Comput. 20, 137–157MATHCrossRefGoogle Scholar
  30. 30.
    Wang Z.J., Zhang L., Liu Y. (2004). Spectral (finite) volume method for conservation laws on unstructured grids IV: extension to two-dimensional systems. J. Comput. Phys. 194, 716–741MATHCrossRefMathSciNetGoogle Scholar
  31. 31.
    Woodward P., Colella P. (1984). The numerical simulation of two-dimensional fluid flow with strong shocks. J. Comput. Phys. 54, 115–173MATHCrossRefMathSciNetGoogle Scholar
  32. 32.
    Wolfram S. (1999). The Mathematica Book, 4th ed. Wolfram Media and Cambridge University Press, New YorkMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2006

Authors and Affiliations

  • Z. J. Wang
    • 1
  • Yen Liu
    • 2
  • Georg May
    • 3
  • Antony Jameson
    • 3
  1. 1.Department of Aerospace EngineeringIowa State UniversityAmesUSA
  2. 2.NASA Ames Research CenterMoffett FieldUSA
  3. 3.Department of Aeronautics and AstronauticsStanford UniversityStanfordUSA

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