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On the Connection Between the Spectral Difference Method and the Discontinuous Galerkin Method

Published online by Cambridge University Press:  20 August 2015

Georg May
Georg May*
Affiliation:
Aachen Institute for Advanced Study in Computational Engineering Sciences, RWTH Aachen University, Schinkelstr. 2, 52062 Aachen, Germany
*
*Corresponding author.Email:[email protected]
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Abstract

In this short note we present a derivation of the Spectral Difference Scheme from a Discontinuous Galerkin (DG) discretization of a nonlinear conservation law. This allows interpretation of the Spectral Difference Scheme as a particular discretization under the quadrature-free nodal DG paradigm. Moreover, it enables identification of the key differences between the Spectral Difference Scheme and standard nodal DG schemes.

Keywords

65N3065N9965M6065M99Spectral difference methoddiscontinuous Galerkin methodhyperbolic conservation lawsquadrature-free schemesnodal schemes
Type
Short Note
Information
Communications in Computational Physics , Volume 9 , Issue 4 , April 2011 , pp. 1071 - 1080
Copyright
Copyright © Global Science Press Limited 2011

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References

[1]Cockburn, B. and Shu, C. W., TVB Runge-Kutta local projection Discontinuous Galerkin finite element method for conservation laws II: general framework, Math. Comput., 52(186) (1988), 411–435.Google Scholar
[2]Karniadakis, G. E. and Sherwin, S., Spectral/hp Element Methods for Computational Fluid Dynamics, 2nd Edition, Oxford University Press,, 2005.Google Scholar
[3]Liu, Y., Vinokur, M. and Wang, Z. J., Discontinuous Spectral Difference method for conservation laws on unstructured grids, in: Proceedings of the 3rd International Conference on Computational Fluid Dynamics, July 12-16, 2004, Toronto, Canada,Springer, 2004.Google Scholar
[4]Liu, Y., Vinokur, M. and Wang, Z. J., Spectral Difference method for unstructured grids I: basic formulation, J. Comp. Phys., 216(2) (2006), 780–801.Google Scholar
[5]Wang, Z. J.Liu, Y., May, G. and Jameson, A., Spectral Difference method for unstructured grids II: extension to the Euler equations, J. Sci. Comput., 32(1) (2007), 54–71.Google Scholar
[6]Kopriva, D. A. and Kolias, J. H., A conservative staggered-grid Chebyshev multidomain method for compressible flows, J. Comp. Phys., 125 (1996), 244–261.Google Scholar
[7]Atkins, H. L. and Shu, C. W., Quadrature-free implementation of discontinuous Galerkin method for hyperbolic equations, AIAA, 36(5) (1998), 775–782.Google Scholar
[8]Hesthaven, J. S. and Gottlieb, D., Stable spectral methods for conservation laws on triangles with unstructured grids, Comput. Meth. Appl. Mech. Engrg., 175 (1999), 361–381.CrossRefGoogle Scholar
[9]Hesthaven, J. S. and Teng, C. H., Stable spectral methods on tethrahedral elements, SIAM J. Sci. Comput., 21(6) (2000), 2352–2380.Google Scholar
[10]Hesthaven, J. S. and Warburton, T., Nodal Discontinuous Galerkin Methods: Algorithms, Analysis, and Applications, No. 54 in Texts in Applied Mathematics, Springer Verlag, 2007.Google Scholar
[11]Wang, Z. J. and Gao, H., A unifying lifting collocation penalty formulation including the discontinuous galerkin, spectral volume/difference methods for conservation laws on mixed grids, J. Comput. Phys., 228(21) (2009), 8161–8186.Google Scholar
[12]Harten, A., High-resolution schemes for hyperbolic conservation laws, J. Comput. Phys., 49(3) (1983), 357–393.Google Scholar
[13]Carpenter, M. H. and Gottlieb, D., Spectral methods on arbitrary grids, J. Comput. Phys., 129 (1996), 74–86.CrossRefGoogle Scholar
[14]Blyth, M. G. and Pozrikidis, C., A Lobatto interpolation grid over the triangle, IMA J. Appl. Math., 71 (2006), 153–169.CrossRefGoogle Scholar
[15]Chen, Q. and Babuška, I., Approximate optimal points for polynomial interpolation of real functions in an interval and in a triangle, Comput. Meth. Appl. Mech. Eng., 128(3-4) (1995), 405–417.Google Scholar
[16]Hesthaven, J. S., From electrostatics to almost optimal nodal sets for polynomial interpolation in a simplex, SIAM J. Num. Anal., 35(2) (1998), 655–676.Google Scholar
[17]Taylor, M. A., Wingate, B. A. and Vincent, R. E., An algorithm for computing Fekete points in the triangle, SIAM J. Numer. Anal., 38(5) (2000), 1707–1720.Google Scholar
[18]Chen, Q. and Babuška, I., The optimal symmetrical points for polynomial interpolation of real functions in the tetrahedron, Comput. Meth. Appl. Mech. Eng., 137 (1996), 89–94.Google Scholar