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

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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.

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References

  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.

  2. Bassi F., Rebay S. (1997). High-order accurate discontinuous finite element solution of the 2D Euler equations. J. Comput. Phys. 138, 251–285

    Article  MATH  MathSciNet  Google Scholar 

  3. Canuto C., Hussaini M.Y., Quarteroni A., Zang T.A. (1987). Spectral Methods in Fluid Dynamics. Springer-Verlag, New York

    Google Scholar 

  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, 405–417

    Article  MATH  MathSciNet  Google Scholar 

  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, 411–435

    Article  MATH  MathSciNet  Google Scholar 

  6. Cockburn B., Shu C.-W. (1998). The Runge–Kutta discontinuous Garlerkin method for conservation laws V: multidimensional systems. J. Comput. Phys. 141, 199–224

    Article  MATH  MathSciNet  Google Scholar 

  7. Delanaye, M., and Liu, Y. (1999). Quadratic reconstruction finite volume schemes on 3D arbitrary unstructured polyhedral grids. AIAA Paper No. 99-3259-CP

  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, 271

    MathSciNet  Google Scholar 

  9. Harten A. (1983). High resolution schemes for hyperbolic conservation laws. J. Comput. Phys. 49, 357–393

    Article  MATH  MathSciNet  Google Scholar 

  10. Hesthaven J.S. (1998). From electrostatics to almost optimal nodal sets for polynomial interpolation in a simplex. SIAM J. Numer. Anal. 35, 655–676

    Article  MATH  MathSciNet  Google Scholar 

  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, 1–38

    Google Scholar 

  12. Kopriva D.A. (1996). A conservative staggered-grid Chebyshev multidomain method for compressible flows. II semi-structured method. J. Comput. Phys. 128, 475

    Article  MATH  MathSciNet  Google Scholar 

  13. Kopriva D.A. (1998). A staggered-grid multidomain spectral method for the compressible Navier–Stokes equations. J. Comput. Phys. 143(1):125

    Article  MATH  MathSciNet  Google Scholar 

  14. Liu Y., Vinokur M. (1998). Exact integration of polynomials and symmetric quadrature formulas over arbitrary polyhedral grids. J. Comput. Phys. 140, 122–147

    Article  MATH  MathSciNet  Google Scholar 

  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.

  16. Liu Y., Vinokur M., Wang Z.J. (2006). Spectral difference method for unstructured grids I: basic formulation. J. Comput. Phys. 216(2):780–801

    Article  MATH  MathSciNet  Google Scholar 

  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, 454–472

    Article  MATH  MathSciNet  Google Scholar 

  18. Roe P.L. (1981). Approximate Riemann solvers, parameter vectors, and difference schemes. J. Comput. Phys. 43, 357–372

    Article  MATH  MathSciNet  Google Scholar 

  19. Rusanov V.V. (1961). Calculation of interaction of non-steady shock waves with obstacles. J. Comput. Math. Phys. USSR 1, 267–279

    MathSciNet  Google Scholar 

  20. Shu C.-W. (1987). TVB uniformly high-order schemes for conservation laws. Math. Comput. 49, 105–121

    Article  MATH  Google Scholar 

  21. Shu C.-W. (1988). Total-variation-diminishing time discretizations. SIAM J. Sci. Stat. Comput. 9:1073

    Article  MATH  Google Scholar 

  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.

  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, 469–491

    Article  MATH  MathSciNet  Google Scholar 

  24. Sridar D., Balakrishnan N. (2003). An upwind finite difference scheme for meshless solvers. J. Comput. Phys. 189, 1–29

    Article  MATH  MathSciNet  Google Scholar 

  25. van Leer B. (1979). Towards the ultimate conservative difference scheme V. a second order sequel to Godunov’s method. J. Comput. Phys. 32, 101–136

    Article  Google Scholar 

  26. Wang Z.J. (2000). A fast nested multi-grid viscous flow solver for adaptive Cartesian/quad grids. Int. J. Numer. Methods Fluids 33, 657–680

    Article  MATH  Google Scholar 

  27. Wang Z.J. (2002). Spectral (finite) volume method for conservation laws on unstructured grids: I. Basic formulation. J. Comput. Phys. 178, 210

    Article  MATH  MathSciNet  Google Scholar 

  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, 665–697

    Article  MATH  MathSciNet  Google Scholar 

  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, 137–157

    Article  MATH  Google Scholar 

  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, 716–741

    Article  MATH  MathSciNet  Google Scholar 

  31. Woodward P., Colella P. (1984). The numerical simulation of two-dimensional fluid flow with strong shocks. J. Comput. Phys. 54, 115–173

    Article  MATH  MathSciNet  Google Scholar 

  32. Wolfram S. (1999). The Mathematica Book, 4th ed. Wolfram Media and Cambridge University Press, New York

    MATH  Google Scholar 

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Authors and Affiliations

  1. Department of Aerospace Engineering, Iowa State University, 2271 Howe Hall, Ames, IA, 50011, USA

    Z. J. Wang

  2. NASA Ames Research Center, Mail Stop T27B-1, Moffett Field, CA, 94035, USA

    Yen Liu

  3. Department of Aeronautics and Astronautics, Stanford University, Durand Building, 496 Lomita Mall, Stanford, CA, 94305-4035, USA

    Georg May & Antony Jameson

Authors
  1. Z. J. Wang
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  2. Yen Liu
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  3. Georg May
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  4. Antony Jameson
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Correspondence to Z. J. Wang.

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Wang, Z.J., Liu, Y., May, G. et al. Spectral Difference Method for Unstructured Grids II: Extension to the Euler Equations. J Sci Comput 32, 45–71 (2007). https://doi.org/10.1007/s10915-006-9113-9

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