Spectral difference method for unstructured grids II: Extension to the Euler equations

Journal of Scientific Computing

Volumn 32, Issue 1, 2007, Pages 45-71

  Wang, Z J ^a   Liu, Yen ^b   May, Georg ^c   Jameson, Antony ^c  

^a Iowa State University of Science and Technology   (United States)

^b NASA AMES RESEARCH CENTER   (United States)

^c 496 Lomita Mall   (United States)

Author keywords

Conservation laws; Euler equations; High order; Spectral collocation method; Spectral difference; Unstructured grids

Indexed keywords

FLUX POINTS; GEOMETRIC FLEXIBILITY; SPECTRAL COLLOCATION METHOD; SPECTRAL VOLUME METHODS;

COMPUTATIONAL METHODS; EULER EQUATIONS; FEATURE EXTRACTION; GALERKIN METHODS; NONLINEAR SYSTEMS;

GRID COMPUTING;

EID: 34250893230     PISSN: 08857474     EISSN: None     Source Type: Journal    
DOI: 10.1007/s10915-006-9113-9     Document Type: Article

References (32)

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.
2. Bassi, F., and Rebay, S. (1997). High-order accurate discontinuous finite element solution of the 2D Euler equations. J. Comput. Phys. 138, 251-285.
3. Canuto, C., Hussaini, M. Y., Quarteroni, A., and Zang, T. A. (1987). Spectral Methods in Fluid Dynamics, Springer-Verlag, New York.
4. Chen, Q., and Babuska, I. (1995). Approximate optimal points for polynomial interpolation of real functions in an interval and in a triangle. Comput. Methods Appl. Mech. Engrg. 128, 405-417.
5. Cockburn, B., and Shu, C.-W. (1989). TVS Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws II: general framework. Math. Comput. 52, 411-435.
6. Cockburn, B., and Shu, C.-W. (1998). The Runge-Kutta discontinuous Garlerkin method for conservation laws V: multidimensional systems. J. Comput. Phys. 141, 199-224.
7. Delanaye, M., and Liu, Y. (1999). Quadratic reconstruction finite volume schemes on 3D arbitrary unstructured polyhedral grids. AIAA Paper No. 99-3259-CP
8. Godunov, S. K. (1959). A finite-difference method for the numerical computation of discontinuous solutions of the equations of fluid dynamics. Mater. Sb. 47, 271.
9. Harten, A. (1983). High resolution schemes for hyperbolic conservation laws. J. Comput. Phys. 49, 357-393.
10. Hesthaven, J. S. (1998). From electrostatics to almost optimal nodal sets for polynomial interpolation in a simplex. SIAM J. Numer. Anal. 35, 655-676.
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, 1-38.
12. Kopriva, D. A. (1996). A conservative staggered-grid Chebyshev multidomain method for compressible flows. II semi-structured method. J. Comput. Phys. 128, 475.
13. Kopriva, D. A. (1998). A staggered-grid multidomain spectral method for the compressible Navier-Stokes equations. J. Comput. Phys. 143(1), 125.
14. Liu, Y., and Vinokur, M. (1998). Exact integration of polynomials and symmetric quadrature formulas over arbitrary polyhedral grids. J. Comput. Phys. 140, 122-147.
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.
16. Liu, Y., Vinokur, M., and Wang, Z. J. (2006). Spectral difference method for unstructured grids I: basic formulation. J. Comput. Phys. 216(2), 780-801.
17. Liu, Y., Vinokur, M., and Wang, Z. J. (2006). Spectral (finite) volume method for conservation laws on unstructured grids V: extension to three-dimensional systems. J. Comput. Phys. 212, 454-472.
18. Roe, P. L. (1981). Approximate Riemann solvers, parameter vectors, and difference schemes. J. Comput. Phys. 43, 357-372.
19. Rusanov, V. V. (1961). Calculation of interaction of non-steady shock waves with obstacles. J. Comput. Math. Phys. USSR 1, 267-279.
20. Shu, C.-W. (1987). TVB uniformly high-order schemes for conservation laws. Math. Comput. 49, 105-121.
21. Shu, C.-W. (1988). Total-variation-diminishing time discretizations. SIAM J. Sci. Stat. Comput. 9, 1073.
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.
23. Spiteri, R. J., and Ruuth, S. J. (2002). A new class of optimal high-order strong-stability-preserving time discretization methods. SIAM J. Numer. Anal. 40, 469-491.
24. Sridar, D., and Balakrishnan, N. (2003). An upwind finite difference scheme for meshless solvers. J. Comput. Phys. 189, 1-29.
25. van Leer, B. (1979). Towards the ultimate conservative difference scheme V. a second order sequel to Godunov's method. J. Comput. Phys. 32, 101-136.
26. Wang, Z. J. (2000). A fast nested multi-grid viscous flow solver for adaptive Cartesian/quad grids. Int. J. Numer. Methods Fluids 33, 657-680.
27. Wang, Z. J. (2002). Spectral (finite) volume method for conservation laws on unstructured grids: I. Basic formulation. J. Comput. Phys. 178, 210.
28. Wang, Z. J., and Liu, Y. (2002). Spectral (finite) volume method for conservation laws on unstructured grids II: extension to two-dimensional scalar equation. J. Comput. Phys. 179, 665-697.
29. Wang, Z. J., and Liu, Y. (2004).Spectral (finite) volume method for conservation laws on unstructured grids III: one-dimensional systems and partition optimization. J. Scientific Comput. 20, 137-157.
30. Wang, Z. J., Zhang, L., and Liu, Y. (2004). Spectral (finite) volume method for conservation laws on unstructured grids IV: extension to two-dimensional systems. J. Comput. Phys. 194, 716-741.
31. Woodward, P., and Colella, P. (1984). The numerical simulation of two-dimensional fluid flow with strong shocks. J. Comput. Phys. 54, 115-173.
32. Wolfram, S. (1999). The Mathematics Book, 4th ed., Wolfram Media and Cambridge University Press, New York.