A Proof of the Stability of the Spectral Difference Method for All Orders of Accuracy

• Published in: Journal of scientific computing Vol. 45; no. 1-3; pp. 348 - 358
• Main Author: Jameson, Antony
• Format: Journal Article
• Language: English
• Published: Boston Springer US 01.10.2010; Springer Nature B.V
• Subjects: Accuracy; Algorithms; Computational fluid dynamics; Computational Mathematics and Numerical Analysis; Computer simulation; Computing costs; Fluid flow; Mathematical and Computational Engineering; Mathematical and Computational Physics; Mathematics; Mathematics and Statistics; Polynomials; Proving; Simulation; Spectra; Stability; Theoretical; Turbulence; Turbulent flow; Discontinuous Galerkin; Spectral difference; Stability proof; High order methods
• ISSN: 0885-7474; 1573-7691
• DOI: 10.1007/s10915-009-9339-4

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.