J Sci Comput (2008) 37: 162–188 DOI 10.1007/s10915-008-9201-0 On the Stability and Accuracy of the Spectral Difference Method Kris Van den Abeele · Chris Lacor · Z.J. Wang Received: 4 October 2007 / Revised: 18 January 2008 / Accepted: 27 February 2008 / Published online: 9 April 2008 © Springer Science+Business Media, LLC 2008 Abstract In this article, it is shown that under certain conditions, the spectral difference (SD) method is independent of the position of the solution points. This greatly simplifies the design of such schemes, and it also offers the possibility of a significant increase in the efficiency of the method. Furthermore, an accuracy and stability study, based on wave propagation analysis, is presented for several 1D and 2D SD schemes. It was found that higher than second-order 1D SD schemes using the Chebyshev–Gauss–Lobatto nodes as the flux points have a weak instability. New flux points were identified which produce accurate and stable SD schemes. In addition, a weak instability was also found in 2D third- and fourth-order SD schemes on triangular grids. Several numerical tests were performed to verify the analysis. Keywords Spectral difference method · Wave propagation properties · Stability 1 Introduction The spectral difference (SD) method for simplex cells was first presented in an article by Liu et al. [10], and later extended to the Euler equations by Wang et al. [22]. An implementation for the Navier–Stokes equations on unstructured hexahedral grids was reported in Sun et al. [15]. For 1D, 2D quadrilateral and 3D hexahedral grids, the SD method is identical to the multi-domain spectral method proposed in Kopriva and Kolias [9] and Kopriva [8]. Further contributions were reported in Huang et al. [7] and in May and Jameson [12]. The SD method is closely related to the discontinuous Galerkin (DG) method, see for instance the work of Cockburn, Shu et al. [1–4], the spectral volume (SV) method, see the work of Wang, K. Van den Abeele ( ) · C. Lacor Department of Mechanical Engineering, Fluid Dynamics and Thermodynamics Research Group, Vrije Universiteit Brussel, Pleinlaan 2, 1050 Brussels, Belgium e-mail: kvdabeel@vub.ac.be Z.J. Wang Department of Aerospace Engineering, Iowa State University, 2271 Howe Hall, Ames, IA 50011, USA