Spectral Difference Method for Unstructured Grids II: Extension to the Euler Equations
- 377 Downloads
An efficient, high-order, conservative method named the spectral difference method has been developed recently for conservation laws on unstructured grids. It combines the best features of structured and unstructured grid methods to achieve high-computational efficiency and geometric flexibility; it utilizes the concept of discontinuous and high-order local representations to achieve conservation and high accuracy; and it is based on the finite-difference formulation for simplicity. The method is easy to implement since it does not involve surface or volume integrals. Universal reconstructions are obtained by distributing solution and flux points in a geometrically similar manner for simplex cells. In this paper, the method is further extended to nonlinear systems of conservation laws, the Euler equations. Accuracy studies are performed to numerically verify the order of accuracy. In order to capture both smooth feature and discontinuities, monotonicity limiters are implemented, and tested for several problems in one and two dimensions. The method is more efficient than the discontinuous Galerkin and spectral volume methods for unstructured grids.
KeywordsHigh-order conservation laws unstructured grids spectral difference spectral collocation method Euler equations
Unable to display preview. Download preview PDF.
- 1.Barth, J., and Frederickson, P. O. (1990). High-order solution of the Euler equations on unstructured grids using quadratic reconstruction. AIAA Paper No. 90-0013.Google Scholar
- 3.Canuto C., Hussaini M.Y., Quarteroni A., Zang T.A. (1987). Spectral Methods in Fluid Dynamics. Springer-Verlag, New YorkGoogle Scholar
- 7.Delanaye, M., and Liu, Y. (1999). Quadratic reconstruction finite volume schemes on 3D arbitrary unstructured polyhedral grids. AIAA Paper No. 99-3259-CPGoogle Scholar
- 11.Jameson A. (1995). Analysis and design of numerical schemes for gas dynamics 2: artificial diffusion and discrete shock structure. Int. J. Comput. Fluid Dyn. 5, 1–38Google Scholar
- 15.Liu, Y., Vinokur, M., and Wang, Z. J. (2004). Discontinuous spectral difference method for conservation laws on unstructured grids. Proceedings of the 3rd International Conference in CFD, Toronto, Canada.Google Scholar
- 22.Shu, C.-W. (1998). Essentially non-oscillatory and weighted essentially non-oscillatory schemes for hyperbolic conservation laws. In B. Cockburn, C. Johnson, C.-W. Shu and E. Tadmor (A. Quarteroni, ed.), Advanced Numerical Approximation of Nonlinear Hyperbolic Equations, Lecture Notes in Mathematics, Vol. 1697, Springer, Berlin, pp. 325–432.Google Scholar