Shock capturing with higher-order, PDE-based artificial viscosity

Collection of Technical Papers - 18th AIAA Computational Fluid Dynamics Conference

Volumn 1, Issue , 2007, Pages 34-47

  Barter, Garrett E ^a,b   Darmofal, David L ^a,b  

^a MASSACHUSETTS INSTITUTE OF TECHNOLOGY   (United States)

^b AIAA   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

APPROXIMATION THEORY; GALERKIN METHODS; MATHEMATICAL MODELS; PARTIAL DIFFERENTIAL EQUATIONS; VISCOSITY MEASUREMENT;

DOWNSTREAM FLOW; STATE-BASED ARTIFICIAL VISCOSITY;

FLUID DYNAMICS;

EID: 35648995851     PISSN: None     EISSN: None     Source Type: Conference Proceeding    
DOI: 10.2514/6.2007-3823     Document Type: Conference Paper

References (30)

1. Cockburn, B., Karniadakis, G. E., and Shu, C. W., "The Development of discontinuous Galerkin methods," Discontinuous Galerkin Methods: Theory, Computation and Applications, edited by K. Cockburn and Shu, Springer, Berlin, 2000, pp. 3-50.
2. Dong, S. and Karniadakis, G. E., "Multilevel Parallelization Models for High-Order CFD," AIAA Paper 2004-1087, 2004.
3. Karniadakis, W. L. D. S., "Galerkin and Discontinuous Galerkin spectral/lip Methods," Comput. Methods Appl. Mech. Engrg., Vol. 175, 1999, pp. 343-359.
4. Sherwin, S. J. and Karniadakis, G. E., "A new triangular and tetrahedral basis for high-order finite element methods," Int. J. Num. Meth. Eng., Vol. 38, 1995, pp. 3775-3802.
5. Cockburn, B. and Shu, C. W., "Runge-Kutta Discontinuous Galerkin Methods for Convection-Dominated Problems," J. of Sci. Comput., Vol, 16, No. 3, September 2001, pp. 173-261.
6. 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, December 1998, pp. 2440-2463.
7. 1-discontinuous Galerkin method for scalar conservation laws," RAIRO Model Math. Anal Numer., Vol. 25, 1991, pp. 337-361.
8. Cockburn, B. and Shu, C. W., "TVB Runge-Kutta local projection discontinuous Galerkin finite element method for scalar conservation laws H: General framework," Math. Comp., Vol. 52, 1989, pp. 411-435.
9. Cockburn, B., Lin, S. Y., and Shu, C. W., "TVB Runge-Kutta local projection discontinuons Galerkin finite element method for conservation laws III: One dimensional systems," J. Comput. Phys., Vol. 84, 1989, pp. 90-113.
10. Cockburn, B., Hou, S., , and Shu, C. W., "TVB Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws IV: The multidimensional case," Math. Comp., Vol. 54, 1990, pp. 545-581.
11. Cockburn, B. and Shu, C. W., "The Runge-Kutta discontinuous Galerkin finite element method for conservation laws II: Multidimensional systems," J. Comput. Phys., Vol. 141, 1998, pp. 199-224.
12. Bassi, F. and Rebay, S., "A High-Order Accurate Discountinuous Finite Element Method for the Numerical Solution of the Compressible Navier-Stokes Equations," J. Comput. Phys., Vol. 131, 1997, pp. 267-279.
13. Bassi, F. and Rebay, S., "GMRES Discontinuous Galerkin Solution of the Compressible Navier-Stokes Equations," Discontinuous Galerkin Methods: Theory, Computation and Applications, edited by K. Cockburn and Shu, Springer, Berlin, 2000, pp. 197-208.
14. Bassi, F. and Rebay, S., "An Implicit high-order discontinuous Galerkin method for the steady state compressible NavierStokes Equations," Computational Fluid Dynamics 98, Proceedings of the Fourth European Computational Fluid Dynamics Conference, edited by P. T. P. Hirsh; and Pandolfi, Wiley, New York, 1998, pp. 1227-1233.
15. Bey, K. and Oden, J. T., "A Runge-Kutta discontinuous finite element method for high speed flows," AIAA Paper 91-1575, June 1991.
16. Bey, K. and Oden, J. T., "hp-version discontinuous Galerkin methods for hyperbolic conservation laws," Comput. Methods Appl. Mech. Engrg., Vol. 133, 1996, pp. 259-286.
17. Persson, P.-O. and Peraire, J., "Sub-Cell Shock Capturing for Discontinuous Galerkin Methods," AIAA Paper 2006-112, 2006.
18. Nguyen, N. C., Persson, P.-O., and Peraire, J., "RANS Solutions Using High Order Discontinuous Galerkin Methods," AIAA Paper 2007-0914, 2007.
19. Fidkowski, K. J., Oliver, T. A., Lu, J., and Darmofal, D. L., "p-Multigrid Solution of High-Order Discontinuous Galerkin Discretizations of the Compressible Navier-Stokes Equations," Journal of Computational Physics, Vol. To appear, 2005, pp. 1-2.
20. Nastase, C. R. and Mavriplis, D. J., "Discontinuous Galerkin Methods Using an hp-Multigrid Solver for Inviscid Compressible Flows on Three-dimensional Unstructured Meshes," AIAA Paper 2006-0107, 2006.
21. Wang, L. and Mavriplis, D. J., "Implicit Solution of the Unsteady Euler Equations for High-Order Accurate Discontinuous Galerkin Discretizations," AIAA Paper 2006-0109, 2006.
22. Persson, P.-O. and Peraire, J., "An Efficient Low Memory Implicit DC Algorithm for Time Dependent Problems," AIAA Paper 2006-0113, 2006.
23. Luo, H., Baum, J. D., and Lohner, R., "A p-multigrid discontinuous Galerkin method for the Euler equations on unstructured grids [application to compressible flows]," J. Comput. Phys., Vol. 211, 2006, pp. 767-783.
24. von Neumann, J. and Richtmyer, R., "A method for the numerical calculation of hydrodynamic shocks," Journal of Applied Physics. Vol. 21, 1950, pp. 232-237.
25. Krivodonova, L., Xin, J., Remade, J.-F., Chevaugeon, N., and Flaherty, J. E., "Shock detection and limiting with discontinuous Galerkin methods for hyperbolic conservation laws," Applied Numerical Mathematics, Vol. 48, No. 3, 2004, pp. 323-338.
26. Adjerid, S., Devine, K. D., Flaherty, J. E., and Krivodonova, L., "A posteriori error estimation for discontinuous Galerkin solutions of hyperbolic problems," Comput. Methods Appl. Mech. Eng., Vol. 191, 2002, pp. 1097-1112.
27. Jameson, A., "Analysis and Design of Numerical Schemes for Gas Dynamics, 2: Artificial Diffusion and Discrete Shock Structure," International Journal of Computational Fluid Dynamics, Vol. 5, 1995, pp. 1-38.
28. Roe, P. L., "Approximate Riemann Solvers, Parameter Vectors, and Difference Schemes," J. Comput. Phys., Vol. 43, 1981, pp. 357-372.
29. Hänel, D., Schwane, R., and Seider, G., "On the Accuracy of Upwind Schemes for the Solution of the Navier-Stokes Equations," AIAA Paper 87-1105, 1987.
30. Quattrochi, D. J., Hypersonic heat transfer and anisotropic visualization with a higher-order discontinuous Galerkin finite element method, Masters thesis, Massachusetts Institute of Technology, Department of Aeronautics and Astronautics, 2006.