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On the Spectral Difference Method with Modal Filtering Applied to the Euler Equations

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Recent Developments in the Numerics of Nonlinear Hyperbolic Conservation Laws

Part of the book series: Notes on Numerical Fluid Mechanics and Multidisciplinary Design ((NNFM,volume 120))

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Abstract

We extend the Spectral Difference method to Proriol-Koornwinder-Dubi-ner-polynomials (PKD) on triangular grids using a two dimensional Lobatto points extension as the set of fluxpoints. These polynomials form an orthogonal basis on triangles and fulfill a singular Sturm-Liouville-problem which can be used to construct modal filters in order to stabilize the scheme for nonlinear conservation laws. To avoid global filtering, we give an outlook of possible edge detection techniques in two dimensions based on the conjugated Fourier partial sum. Finally, we show numerical results for the Spectral Difference method using the proposed filter technique applied to the Euler equations and the nonlinear shock vortex interaction.

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References

  1. Van den Abeele, K., Lacor, C., Wang, Z.J.: On the Stability and Accuracy of the Spectral Difference Method. Journ. Sc. Comp. 37, 62–188 (2008)

    Google Scholar 

  2. Barter, G.E., Darmofal, D.L.: Shock Capturing with Higher-Order PDE-Based Artificial Viscosity. In: Proc. 18th AIAA CFD Conf. AIAA-2007-3823 (2007)

    Google Scholar 

  3. Blyth, M.G., Pozrikidis, C.: A Lobatto interpolation grid over the triangle. IMA Journ. Appl. Math. 71, 153–169 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  4. Carpenter, M.H., Kennedy, C.A.: Fourth-order 2N-storage Runge-Kutta schemes. NASA Report TM 109112 (1994)

    Google Scholar 

  5. Dubiner, M.: Spectral Methods on Triangles and Other Domains. Journ. Sc. Comp. 6, 345–390 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  6. Gelb, A., Tadmor, E.: Spectral reconstruction of piecewise smooth functions from their discrete data. Math. Model. and Num. Analysis 36, 155–175 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  7. Gelb, A., Tadmor, E.: Detection of Edges in Spectral Data. Appl. and Comp. Harm. Analysis 7, 101–135 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  8. Gottlieb, D., Hesthaven, J.S.: Spectral methods for hyperbolic problems. Journ. Comp. Appl. Math. 128, 83–131 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  9. Koornwinder, T.: Two-variable analogues of the classical orthogonal polynomials. In: Askey, R. (ed.) Theory and Applications of Special Functions. Academic Press, San Diego (1975)

    Google Scholar 

  10. Kopriva, D.A., Kolias, J.H.: A Conservative Staggered-Grid Chebyshev Multidomain Method for Compressible Flows. Journal of Computational Physics 125, 244–261 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  11. Liu, Y., Vinokur, M., Wang, Z.J.: Spectral difference method for unstructured grids I: Basic formulation. Journ. Comp. Physics 216, 780–801 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  12. Lukács, F.: Ueber die Bestimmung des Sprunges einer Funktion aus ihrer Fourierreihe. Journ. Reine u. Angew. Mathematik 150, 107–112 (1920)

    Google Scholar 

  13. Meister, A., Ortleb, S., Sonar, T.: Application of Spectral Filtering to Discontinuous Galerkin Methods on Triangulations. To be publ. in Num. Meth. for PDEs (2012)

    Google Scholar 

  14. Móricz, F.: Extension of a Theorem of Ferenc Lukács from Single to Double Conjugate Series. Journ. Math. Analysis and Appl. 259, 582–592 (2001)

    Article  MATH  Google Scholar 

  15. Tadmor, E.: Super Viscosity and Spectral Approximations of Nonlinear Conservation Laws. Numerical Methods for Fluid Dynamics 4, 69–82 (1993)

    MathSciNet  Google Scholar 

  16. Taylor, M.A., Wingate, B.A.: The natural function space for triangular and tetrahedral spectral elements. Los Alamos National Laboratory Report LA-UR-98-1711 (1998)

    Google Scholar 

  17. Wang, Z.J., Liu, Y., May, G., Jameson, A.: Spectral difference method for unstructured grids II: Extension to the Euler Equations. Journ. Sc. Comp. 32 (2007)

    Google Scholar 

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Authors and Affiliations

  1. Institut Computational Mathematics, Technische Universität Braunschweig, Pockelsstr. 14, 38106, Braunschweig, Germany

    Thomas Sonar & Martina Wirz

Authors
  1. Thomas Sonar
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  2. Martina Wirz
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Correspondence to Thomas Sonar .

Editor information

Editors and Affiliations

  1. , Department of Mathematics, University of Hamburg, Bundesstr. 55, Hamburg, 20146, Germany

    Rainer Ansorge

  2. , Department of Aerospace Engineering, Delft University of Technology, Delft, 2600, Netherlands

    Hester Bijl

  3. , Department of Mathematics and Natural Sc, University of Kassel, Heinrich-Plett-Str. 40, Kassel, 34132, Germany

    Andreas Meister

  4. Institute for Computational Mathematics, Technical University of Braunschweig, Pockelstr. 14, Braunschweig, 38106, Germany

    Thomas Sonar

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Sonar, T., Wirz, M. (2013). On the Spectral Difference Method with Modal Filtering Applied to the Euler Equations. In: Ansorge, R., Bijl, H., Meister, A., Sonar, T. (eds) Recent Developments in the Numerics of Nonlinear Hyperbolic Conservation Laws. Notes on Numerical Fluid Mechanics and Multidisciplinary Design, vol 120. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-33221-0_19

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  • DOI: https://doi.org/10.1007/978-3-642-33221-0_19

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-33220-3

  • Online ISBN: 978-3-642-33221-0

  • eBook Packages: EngineeringEngineering (R0)

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