Semi-Implicit interior penalty discontinuous galerkin method for viscous compressible flows

Communications in Computational Physics

Volumn 4, Issue 2, 2008, Pages 231-274

  Dolejší, Vit ^a  

^a CHARLES UNIVERSITY   (Czech Republic)

Author keywords

Backward difference formulae; Compressible Navier Stokes equations; Discontinuous Galerkin finite element method; Linearization

Indexed keywords

EID: 50149117609     PISSN: 18152406     EISSN: None     Source Type: Journal    
DOI: None     Document Type: Article

References (60)

1. ADIGMA. Adaptive higher-order variational methods for aerodynamic applications in industry, Specific Targeted Research Project no. 30719 supported by European Ccommison. URL: http://www.dlr.de/as/en/ Desktopdefault.aspx/tabid-2035/2979_read-4582/.
2. D. N. Arnold. An interior penalty finite element method with discontinuous elements. SIAM J. Numer. Anal., 19(4):742-760, 1982.
3. D. N. Arnold, F. Brezzi, B. Cockburn, and L. D. Marini. Unified analysis of discontinuous Galerkin methods for elliptic problems. SIAM J. Numer. Anal., 39(5):1749-1779, 2002.
4. I. Babuška, C. E. Baumann, and J. T. Oden. A discontinuous hp finite element method for diffusion problems: 1-d analysis. Computers and Mathematics with Applications, 37:103-122, 1999.
5. F. Bassi, A. Crivellini, S. Rebay, and M. Savini. Discontinuous Galerkin solution of the Reynolds averaged Navier-Stokes and k-ω turbulence model equations. Comput. Fluids, 34:507-540, 2005.
6. F. Bassi and S. Rebay. A high-order accurate discontinuous finite element method for the numerical solution of the compressible Navier-Stokes equations. J. Comput. Phys., 131:267-279, 1997.
7. F. Bassi and S. Rebay. High-order accurate discontinuous finite element solution of the 2D Euler equations. J. Comput. Phys., 138:251-285, 1997.
8. F. Bassi and S. Rebay. A high order discontinuous Galerkin method for compressible turbulent flow. In B. Cockburn, G. E. Karniadakis, and C.-W. Shu, editors, Discontinuous Galerkin Method: Theory, Computations and Applications, Lecture Notes in Computational Science and Engineering 11, pages 113-123. Springer-Verlag, 2000.
9. F. Bassi and S. Rebay. Numerical evaluation of two discontinuous Galerkin methods for the compressible Navier-Stokes equations. Int. J. Numer Methods Fluids, 40:197-207, 2002.
10. C. E. Baumann and J. T. Oden. A discontinuous hp finite element method for the Euler and Navier-Stokes equations. Int. J. Numer. Methods Fluids, 31(1):79-95, 1999.
11. P. Birken, J. D. Tebbens, A. Meister, and M. Tůma. Preconditioner updates applied to CFD model problems. Mathematische schriften Kassel preprint-reihe fachbereich mathematik/informatik, Universität Kassel, 2007.
12. P. G. Ciarlet. The Finite Elements Method for Elliptic Problems. North-Holland, Amsterdam, New York, Oxford, 1979.
13. B. Cockburn. Discontinuous Galerkin methods for convection dominated problems. In T. J. Barth and H. Deconinck, editors, High-Order Methods for Computational Physics, Lecture Notes in Computational Science and Engineering 9, pages 69-224. Springer, Berlin, 1999.
14. B. Cockburn, S. Hou, and C. W. Shu. TVB Runge-Kutta local projection discontinuous Galerkin finite element for conservation laws IV: The multi-dimensional case. Math. Comp., 54:545-581, 1990.
15. B. Cockburn, G. E. Karniadakis, and C.-W. Shu, editors. Discontinuous Galerkin Methods. Springer, Berlin, 2000.
16. M. Crouzeix. Une méthode multipas implicit-explicit pour l'approximation des équations d'évolutions paraboliques. Numer. Math., 35:27-276, 1980.
17. C. N. Dawson, S. Sun, and M. F. Wheeler. Compatible algorithms for coupled flow and transport. Comput. Meth. Appl. Mech. Engng., 193:2565-2580., 2004.
18. V. Dolejší. Anisotropic mesh adaptation for finite volume and finite element methods on triangular meshes. Comput. Vis. Sci., 1(3):165-178, 1998.
19. V. Dolejší. Anisotropic mesh adaptation technique for viscous flow simulation. East-West J. Numer. Math., 9(1):1-24, 2001.
20. V. Dolejší. On the discontinuous Galerkin method for the numerical solution of the Navier-Stokes equations. Int. J. Numer. Methods Fluids, 45:1083-1106, 2004.
21. V. Dolejší1. An efficient implementation of the semi-implicit discontinuous galerkin method for compressible flow simulation. In T. Vejchodský J. Chleboun, K. Segeth, editor, Programs and Algorithms of Numerical Mathematics 13, pages 74-79. Programs and Algorithms of Numerical Mathematics 13, Academy of Science of the Czech Republic, Prague, 2006.
22. V. Dolejší. Analysis and application of IIPG method to quasilinear nonstationary convection-diffusion problems. J. Comp. Appl. Math., in press.
23. V. Dolejší and M. Feistauer. On the discontinuous Galerkin method for the numerical solution of compressible high-speed flow. In F. Brezzi, A. Buffa, S. Corsaro, and A. Murli, editors, Numerical Mathematics and Advanced Applications, ENUMATH 2001, pages 65-84. Springer-Verlag, Italia, Milano, 2003.
24. V. Dolejší and M. Feistauer. Error estimates of the discontinuous Galerkin method for nonlinear nonstationary convection-diffusion problems. Numer. Funct. Anal. Optim., 26(25-26): 2709-2733, 2005.
25. V. Dolejší, M. Feistauer, and J. Hozman. Analysis of semi-implicit DGFEM for nonlinear convection-diffusion problems. Comput. Methods Appl. Mech. Engrg., 196:2813-2827, 2007.
26. 2)-error estimate of the discontinuous galerkin method for a nonlinear nonstationary convection-diffusion problem. IMA J. Numer. Anal., in press.
27. V. Dolejší, M. Feistauer, and V. Sobotíková. A discontinuous Galerkin method for nonlinear convection-diffusion problems. Comput. Methods Appl. Mech. Engrg., 194:2709-2733, 2005.
28. V. Dolejší and F. Kůs. Adaptive backward difference formula - discontinuous Galerkin finite element method for the solution of conservation laws. Int. J. Numer. Methods Eng., 73(12):1739-1766, 2008.
29. V. Dolejší and M. Vlasák. Analysis of a BDF-DGFE scheme for nonlinear convection-diffusion problems. Preprint, 2007, www.karlin.mff.cuni.cz/ms-preprints.
30. V. Dolejší. Discontinuous Galerkin method for the numerical simulation of unsteady compressible flow. WSEAS Transactions on Systems, 5(5):1083-1090, 2006.
31. V. Dolejší and M. Feistauer. Semi-implicit discontinuous Galerkin finite element method for the numerical solution of inviscid compressible flow. J. Comput. Phys., 198(2):727-746, 2004.
32. M. Dumbser and C.-D. Munz. Building blocks for arbitrary high-order discontinuous Galerkin methods. J. Sci. Comput., 27:215-230, 2006.
33. M. Feistauer. Mathematical Methods in Fluid Dynamics. Longman Scientific & Technical, Harlow, 1993.
34. M. Feistauer, J. Felcman, M. Lukáčová-Medvidová, and G. Warnecke. Error estimates of a combined finite volume-finite element method for nonlinear convection-diffusion problems. SIAM J. Numer. Anal., 36(5):1528-1548, 1999.
35. M. Feistauer, J. Felcman, and I. Straškraba. Mathematical and Computational Methods for Compressible Flow. Oxford University Press, Oxford, 2003.
36. E. Hairer, S. P. Norsett, and G. Wanner. Solving ordinary differential equations I, Nonstiff problems. Number 8 in Springer Series in Computational Mathematics. Springer Verlag, 2000.
37. R. Hartmann. Adaptive discontinuous galerkin methods with shock-capturing for the compressible navier-stokes equations. Int. J. Numer. Meth. Fluids, 51:1131-1156, 2006.
38. R. Hartmann and P. Houston. Adaptive discontinuous Galerkin finite element methods for the compressible Euler equations. J. Comput. Phys., 183(2):508-532, 2002.
39. R. Hartmann and P. Houston. Symmetric interior penalty DG methods for the compressible Navier-Stokes equations I: Method formulation. Int. J. Numer. Anal. Model., 1:1-20, 2006.
40. R. Hartmann and P. Houston. Symmetric interior penalty DG methods for the compressible Navier-Stokes equations II: Goal-oriented a posteriori error estimation. Int. J. Numer. Anal. Model., 3:141-162, 2006.
41. P. Houston, J. Robson, and E. Süli. Discontinuous Galerkin finite element approximation of quasilinear elliptic boundary value problems I: The scalar case. IMA J. Numer. Anal., 25:726-749, 2005.
42. P. Houston, C. Schwab, and E. Süli. Discontinuous hp-finite element methods for advection-diffusion problems. SIAM J. Numer. Anal., 39(6):2133-2163, 2002.
43. C. M. Klaij, J.J.W. van der Vegt, and H. Van der Ven. Pseudo-time stepping for space-time discontinuous Galerkin discretizations of the compressible Navier-Stokes equations. J. Comput. Phys., 219(2):622-643, 2006.
44. C. M. Klaij, J.J.W. van der Vegt, and H. Van der Ven. Space-time discontinuous Galerkin method for the compressible Navier-Stokes equations, J. Comput. Phys., 217(2):589-611, 2006.
45. I. Lomtev, C. B.Quillen, and G. E. Karniadakis. Spectral/hp methods for viscous compressible flows on unstructured 2d meshes. J. Comput. Phys., 144(2):325-357, 1998.
46. F. Lörcher, G. Gassner, and C.-D. Munz. A discontinuous Galerkin scheme based on a spacetime expansion. I. Inviscid compressible flow in one space dimension. J. Sci, Comput., 32(2):175-199, 2007.
47. S. Mittal. Finite element computation of unsteady viscous compressible flows. Comput. Methods Appl. Mech. Eng., 157:151-175, 1998.
48. R. Radespiel and R. C. Swanson. An investigation of cell centered and cell vertex multigrid schemes for the Navier-Stokes equations. AIAA paper No. 89-0453, 1989.
49. B. Rivièe and M. F. Wheeler. A discontinuous Galerkin method applied to nonlinear parabolic equations. In B. Cockburn, G. E. Karniadakis, and C.-W. Schu, editors, Discontinuous Galerkin methods. Theory, computation and applications., volume 11 of Lect. Notes Comput. Sci. Eng., pages 231-244. Berlin, Springer, 2000.
50. B. Rivière, M. F. Wheeler, and V. Girault. Improved energy estimates for interior penalty, constrained and discontinuous Galerkin methods for elliptic problems. I. Comput. Geosci., 3(3-4):337-360, 1999.
51. B. Rivière, M. F. Wheeler, and V. Girault. A priori error estimates for finite element methods based on discontinuous approximation spaces for elliptic problems. SIAM J. Numer. Anal., 39(3):902-931, 2001.
52. P. L. Roe. Approximate Riemann solvers, parameter vectors, and difference schemes. J. Comput. Phys., 43(2):357-372, 1981.
53. S. Sun. Discontinuous Galerkin methods for reactive transport in porous media. PhD thesis, The University of Texas, Austin, 2003.
54. S. Sun and M. F. Wheeler. Symmetric and nonsymmetric discontinuous Galerkin methods for reactive transport in porous media. SIAM J. Numer. Anal., 43(1):195-219, 2005.
55. R. Temam. Navier-Stokes Equations. Theory and numerical analysis. North-Holland, Amsterdam-New York-Oxford, 1977.
56. J. J. W. van der Vegt and H. van der Ven. Space-time discontinuous Galerkin finite element method with dynamic grid motion for inviscid compressible flows. I: General formulation. J. Comput. Phys., 182(2):546-585, 2002.
57. H. van der Ven and J. J. W. van der Vegt. Space-time discontinuous Galerkin finite element method with dynamic grid motion for inviscid compressible flows II. efficient flux quadrature. Comput. Methods Appl. Mech. Engrg., 191:4747-4780, 2002.
58. J. M. Varah. Stability restrictions on second order, three level finite difference scheme for parabolic equations. SIAM J. Numer. Anal., 17(2):300-309, 1980.
59. G. Vijayasundaram. Transonic flow simulation using upstream centered scheme of Godunov type in finite elements. J. Comput. Phys., 63:416-433, 1986.
60. P. Wesseling. Principles of Computational Fluid Dynamics. Springer, Berlin, 2001.