Abstract
While second order methods for computational simulations of fluid flow provide the basis of widely used commercial software, there is a need for higher order methods for more accurate simulations of turbulent and vortex dominated flows. The discontinuous Galerkin (DG) method is the subject of much current research toward this goal. The spectral difference (SD) method has recently emerged as a promising alternative which can reduce the computational costs of higher order simulations. There remains some questions, however, about the stability of the SD method. This paper presents a proof that for the case of one dimensional linear advection the SD method is stable for all orders of accuracy in a norm of Sobolev type, provided that the interior fluxes collocation points are placed at the zeros of the corresponding Legendre polynomial.
Similar content being viewed by others
References
Cockburn, B., Shu, C.-W.: TVB Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws II: General framework. Math. Comput. 52, 411–435 (1989)
Cockburn, B., Lin, S.Y., Shu, C.-W.: TVB Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws III: One-dimensional systems. J. Comput. Phys. 84, 90–113 (1989)
Cockburn, B., Hou, S., Shu, C.-W.: TVB Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws IV: The multidimensional case. Math. Comput. 54, 545–581 (1990)
Cockburn, B., Shu, C.-W.: TVB Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws V: Multidimensional system. J. Comput. Phys. 141, 199–224 (1998)
Hesthaven, J.-S., Warburton, T.: Nodal Discontinuous Galerkin Methods—Algorithms, Analysis, and Applications. Springer, Berlin (2008)
Kopriva, D.A., Kolias, J.H.: A conservative staggered-grid Chebyshev multidomain method for compressible flows. J. Comput. Phys. 125(1), 244–261 (1996)
Liu, Y., Vinokur, M., Wang, Z.J.: Spectral difference method for unstructured grids I: Basic formulation. J. Comput. Phys. 216, 780–801 (2006)
May, G.: The spectral difference scheme as a quadrature-free discontinuous Galerkin method. AICES, 11 (2008)
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(1), 45–71 (2007)
Liang, C., Premasuthan, S., Jameson, A.: High-order accurate simulation of flow past two side-by-side cylinders with spectral difference method. J. Comput. Struct. 87, 812–817 (2009)
Liang, C., Premasuthan, S., Jameson, A., Wang, Z.J.: Large eddy simulation of compressible turbulent channel flow with spectral difference method. AIAA paper, AIAA-2009-402, Orlando, FL (2009)
Liang, C., Jameson, A., Wang, Z.J.: Spectral difference method for two-dimensional compressible flow on unstructured grids with mixed elements. J. Comput. Phys. 228, 2847–2858 (2009)
Premasuthan, S., Liang, C., Jameson, A., Wang, Z.J.: A P-multigrid spectral difference method for viscous flow. AIAA paper, AIAA-2009-950, Orlando, FL (2009)
Ou, K., Liang, C., Premasuthan, S., Jameson, A.: High-order spectral difference simulation of laminar compressible flow over two counter-rotating cylinders. AIAA-2009-3956, 27th AIAA AA Conference, San Antonio, June 2009
Mohammad, A.H., Wang, Z.J., Liang, C.: LES of turbulent flow past a cylinder using spectral difference method. AIAA applied aerodynamics meeting, Hawaii. AIAA-2008-7184 (2008)
Van den Abeele, K., Lacor, C., Wang, Z.J.: On the stability and accuracy of the spectral difference method. J. Sci. Comput. 37, 162–188 (2008)
Huynh, H.T.: A flux reconstruction approach to high-order schemes including discontinuous Galerkin methods. AIAA 2007-4079, 18th AIAA CFD Conference, Miami, June 2008
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Jameson, A. A Proof of the Stability of the Spectral Difference Method for All Orders of Accuracy. J Sci Comput 45, 348–358 (2010). https://doi.org/10.1007/s10915-009-9339-4
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10915-009-9339-4