Discontinuous Galerkin methods using an hp-multigrid solver for inviscid compressible flows on three-dimensional unstructured meshes

Collection of Technical Papers - 44th AIAA Aerospace Sciences Meeting

Volumn 2, Issue , 2006, Pages 1332-1348

  Nastase, Cristian R ^a,b   Mavriplis, Dimitri J ^a,b  

^a UNIVERSITY OF WYOMING   (United States)

^b AIAA   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

DISCRETIZATIONS; GALERKIN DISCRETIZATIONS; HYBRID MESHES;

ALGORITHMS; COMPRESSIBLE FLOW; EULER EQUATIONS; HIERARCHICAL SYSTEMS; NONLINEAR SYSTEMS; THREE DIMENSIONAL;

GALERKIN METHODS;

EID: 34250706800     PISSN: None     EISSN: None     Source Type: Conference Proceeding    
DOI: None     Document Type: Conference Paper

References (28)

1. Lesaint, P. and Raviart, P. A., On a Finite Element Method to Solve the Neutron Transport Equation, Mathematical Aspects of Finite Elements in Partial Differential Equations, C. de Boor, Academic Press, New York, NY, 1974.
2. Bassi, F. and Rebay, S., "High-Order Accurate Discontinuous Finite Element Solution of the 2D Euler Equations," J. Comput. Phys., Vol. 138, 1997, pp. 251-285.
3. Cockburn, B. and Shu, C.-W., "The Local Discontinuous Galerkin Method for Time-Dependent Convection-Diffusion Systems," SIAM J. Numer. Anal., Vol. 35, No. 6, 1998, pp. 2440-2463.
4. Warburton, T. C., Lomtev, I., Du, Y, Sherwin, S. J., and Karniadakis, G. E., "Galerkin and Discontinuous Galerkin Spectral/hp Methods," Comput. Methods Appl. Mech. Engrg., Vol. 175, 1999, pp. 343-359.
5. Cockburn, B., Karniadakis, G., and Shu, C.-W., "Discontinuous Galerkin Methods: Theory, Computation and Applications," Lecture Notes in Computational Science A, Springer-Verlag, New York, NY, 1999, pp. 69-224.
6. Cockburn, B. and Shu, C.-W., "Runge-Kutta Discontinuous Galerkin Methods for Convection-Dominated Problems," SIAM J. Sci. Comput., Vol. 16, No. 3, 2001, pp. 173-261.
7. Hughes, T. J. R. and Brooks, A., "Streamline upwind-Petrov-Galerkin formulations for convection dominated flows with particular emphasis on the incompressible Navier-Stokes equations," Comput. Meth. Appl. Mech. Engrg., Vol. 32, 1982, pp. 199-259.
8. Helenbrook, B., Mavriplis, D. J., and Atkins, H., "Analysis of "p"-Multigrid for Continuous and Discontinuous Finite Element Discretizations," Proceedings of the 16th AIAA Computational Fluid Dynamics Conference, 2003, AIAA Paper 2003-3989.
9. Ronquist, E. M. and Patera, A. T., "Spectral Element Multigrid I: Formulation and Numerical Results," SIAM J. Sci. Comput., Vol. 2, No. 4, 1987, pp. 389-406.
10. Fidkowski, K. J. and Darmofal, D. L., "Development of a Higher-Order Solver for Aerodynamic Applications," Proceedings of the 42nd Aerospace Sciences Meeting and Exihibit, Reno NV, 2004, AIAA Paper 2004-0436.
11. Fidkowski, K. J., Oliver, T. A., J.Lu, and Darmofal, D. L., "p-Multigrid Solution of High-Order Discontinuous Galerkin Discretizations of the Compressible Navier-Stokes Equations," submitted to J. Comput. Phys.
12. Lottes, J. W. and Fischer, P. F., "Hybrid Multigrid/Schwarz Algorithms for the Spectral Element Method," SIAM J. Sci. Comput., 2004, To appear.
13. Nastase, C. R. and Mavriplis, D. J., "High-Order Discontinuous Galerkin Methods using a Spectral Multigrid Approach," Proceedings of the 43rd Aerospace Sciences Meeting and Exihibit, Reno NV, 2005, AIAA Paper 2005-1268.
14. Davis, S. F., "Simplified Second-Order Godunov-Type Methods," SIAM J. Sci. Statist. Comput., Vol. 9, No. 3, 1988, pp. 445-473.
15. Roe, P. L., "Approximate Riemann Solvers, Parameter vectors, and Difference Schemes," J. Comput. Phys., Vol. 43, 1981, pp. 357-372.
16. Harten, A., Lax, P. D., and Van Leer, B., "On Upstream Differencing and Godunov-Type Schemes for Hyperbolic Conservation Laws," SIAM Review, Vol. 25, No. 1, 1983, pp. 35-61.
17. Toro, F. E, Riemann Solvers and Numerical Methods for Fluid Dynamics, Applied Mechanics, Springer-Verlag, New York, NY, 1999.
18. Batten, P., Clarke, N., Lambert, C., and Causon, D. M., "On the Choice of Wavespeeds for the HLLC Riemann Solver," SIAM J. Sci. Comput., Vol. 18, No. 2, 1997, pp. 1553-1570.
19. Batten, P., Leschiner, M. A., and Goldberg, U. C., "Average-State Jacobians and Implicit Methods for Compressible Viscous and Turbulent Flows," J. Comput. Phys., Vol. 137, 1997, pp. 38-78.
20. Solin, P., Segeth, P., and Zel, I., High-Order Finite Element Methods, Studies in Advanced Mathematics, Chapman and Hall, 2003.
21. Szabo, B. and Babuska, I., Finite Element Analysis, John Wiley & Sons, Inc., New York, NY, 1991.
22. Dunavant, D. A., "Highr Degree Efficient Symmetrical Gaussian Quadrature Rules for the Triangle," Int. J. Numer. Meth. Engng., Vol. 21, 1985, pp. 1129-1148.
23. Dunavant, D. A., "Economical Symmetrical Quadrature Rules for Complete Polynomials Over a Square Domain," Int. J. Numer. Meth. Engng., Vol. 21, 1985, pp. 1777-1784.
24. Cockburn, B., Hou, S., and Shu, C.-W., "The Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws IV: The multidimensional case," Math. Comput., Vol. 54, No. 545, 1990.
25. Mavriplis, D. J., "An Assessment of Linear versus Non-Linear Multigrid Methods for Unstructured Mesh Solvers," J. Comput. Phys., Vol. 175, 2002, pp. 302-325.
26. Mavriplis, D. J., "Unstructured Mesh Generation and Adaptivity," VKI Lecture Series VKI-LS 1995-02, mar 1995.
27. Trottenberg, U., Schuller, A., and Oosterlee, C., Multigrid, Academic Press, London, UK, 2000.
28. Mavriplis, D. J. and Venkatakrishnan, V., "Agglomeration Multigrid for Two Dimensional Viscous Flows," Computers and Fluids, Vol. 24, No. 5, may 1995, pp. 553-570.