A study of viscous flux formulations for a p-multigrid spectral volume navier stokes solver

Journal of Scientific Computing

Volumn 41, Issue 2, 2009, Pages 165-199

  Kannan, R ^a   Wang, Z J ^a  

^a Iowa State University   (United States)

Author keywords

BR2; High order; Implicit LU SGS; LDG; P multigrid; Penalty; Spectral volume

Indexed keywords

BR2; HIGH-ORDER; IMPLICIT LU-SGS; LDG; P-MULTIGRID; PENALTY; SPECTRAL VOLUME;

COMPUTER SIMULATION; CONVERGENCE OF NUMERICAL METHODS; FOURIER ANALYSIS; GALERKIN METHODS;

NAVIER STOKES EQUATIONS;

EID: 70349880778     PISSN: 08857474     EISSN: None     Source Type: Journal    
DOI: 10.1007/s10915-009-9269-1     Document Type: Article

References (56)

1. R. Abgrall 1994 On essentially non-oscillatory schemes on unstructured meshes: analysis and implementation J. Comput. Phys. 114 45 58 10.1006/jcph.1994.1148 10.1006/jcph.1994.1148 1286187 (Pubitemid 124013471)
2. M. Aftosmis D. Gaitonde T.S. Tavares 1995 Behavior of linear reconstruction techniques on unstructured meshes AIAA J. 33 2038 2049 10.2514/3.12945 10.2514/3.12945
3. D.N. Arnold 1982 An interior penalty finite element method with discontinuous elements SIAM J. Numer. Anal. 19 742 760 10.1137/0719052 10.1137/0719052 664882
4. Barth, T.J., Frederickson, P.O.: High-order solution of the Euler equations on unstructured grids using quadratic reconstruction. AIAA Paper No. 90-0013 (1990)
5. F. Bassi S. Rebay 1997 A high-order accurate discontinuous finite element method for the numerical solution of the compressible Navier Stokes equations J. Comput. Phys. 131 267 279 10.1006/jcph.1996.5572 10.1006/jcph.1996.5572 1433934 (Pubitemid 127341397)
6. F. Bassi S. Rebay 1997 High-order accurate discontinuous finite element solution of the 2d Euler equations J. Comput. Phys. 138 251 285 10.1006/jcph.1997.5454 10.1006/jcph.1997.5454 1607481 (Pubitemid 127170411)
7. Bassi, F., Rebay, S.: GMRES discontinuous Galerkin solution of the compressible Navier-Stokes equations. In: Karniadakis, G.E., Cockburn, B., Shu, C.-W. (eds.) Discontinuous Galerkin Methods: Theory, Computation and Applications, pp. 197-208. Springer, Berlin (2000)
8. Bassi, F., Rebay, S.: Numerical solution of the Euler equations with a multiorder discontinuous Finite element method. In: Proceedings of the Second International Conference on Computational Fluid Dynamics, Sydney, Australia, 15-19 July 2002
9. Baumann, C.E.: An hp-adaptive discontinuous finite element method for computational fluid dynamics. Ph.D. dissertation, University of Texas at Austin, December 1997
10. F. Brezzi G. Manzini D. Marini P. Pietra A. Russo 2000 Discontinuous Galerkin approximations for elliptic problems Numer. Methods Partial Differ. Equ. 16 365 378 10.1002/1098-2426(200007)16:4<365::AID-NUM2>3.0.CO;2-Y 10.1002/1098-2426(200007)16:4<365::AID-NUM2>3.0.CO;2-Y 1765651
11. R.F. Chen Z.J. Wang 2000 Fast, block lower-upper symmetric Gauss Seidel scheme for arbitrary grids AIAA J. 38 12 2238 2245 10.2514/2.914
12. Q.Y. Chen 2006 Partitions of a simplex leading to accurate spectral (finite) volume reconstruction SIAM J. Sci. Comput. 27 4 1458 1470 10.1137/030601387 10.1137/030601387 2199757 (Pubitemid 44077335)
13. Q.Y. Chen 2006 Partitions for spectral finite volume reconstruction in the tetrahedron J. Sci. Comput. 29 3 299 319 10.1007/s10915-005-9009-0 10.1007/s10915-005-9009-0 2272321 (Pubitemid 44916350)
14. B. Cockburn C.-W. Shu 1998 The local discontinuous Galerkin method for time-dependent convection diffusion system SIAM J. Numer. Anal. 35 2440 2463 10.1137/S0036142997316712 10.1137/S0036142997316712 1655854
15. B. Cockburn C.W. Shu 2001 Runge-Kutta discontinuous Galerkin methods for convection-dominated problems J. Sci. Comput. 16 3 173 261 10.1023/A: 1012873910884 10.1023/A:1012873910884 1873283 (Pubitemid 33818237)
16. Delanaye, M., Liu, Y.: Quadratic reconstruction finite volume schemes on 3D arbitrary unstructured polyhedral grids. AIAA Paper No. 99-3259-CP (1999)
17. K.J. Fidkowski T.A. Oliver J. Lu D.L. Darmofal 2005 p-multigrid solution of high-order discontinuous Galerkin discretizations of the compressible Navier-Stokes equations J. Comput. Phys. 207 92 113 10.1016/j.jcp.2005.01.005 10.1016/j.jcp.2005.01.005 (Pubitemid 40640296)
18. S. Gottlieb 2005 On high-order strong stability preserving Runge-Kutta and multi step time discretizations J. Sci. Comput. 25 1/2 105 128 2231945
19. S. Gottlieb C.W. Shu 1998 Total variation diminishing Runge-Kutta schemes Math. Comput. 67 73 85 10.1090/S0025-5718-98-00913-2 10.1090/S0025-5718-98- 00913-2 1443118 (Pubitemid 128027423)
20. Haga, T., Ohnishi, N., Sawada, K., Masunaga, A.: Spectral volume computation of flowfield in aerospace application using Earth Simulator. AIAA Paper, 2006-2823
21. A. Harten B. Engquist S. Osher S. Chakravarthy 1987 Uniformly high order essentially non-oscillatory schemes III J. Comput. Phys. 71 231 10.1016/0021-9991(87)90031-3 10.1016/0021-9991(87)90031-3 897244
22. B.T. Helenbrook H.L. Atkins 2006 Application of p-multigrid to discontinuous Galerkin formulations of the Poisson equation AIAA J. 44 566 575 10.2514/1.15497 10.2514/1.15497
23. A. Jameson S. Yoon 1987 Lower-upper implicit schemes with multiples grids for the Euler equations AIAA J. 25 7 929 935 10.2514/3.9724 10.2514/3.9724
24. C. Liang R. Kannan Z.J. Wang 2009 A p-multigrid spectral difference method with explicit and implicit smoothers on unstructured grids Comput. Fluids 38 2 254 265 10.1016/j.compfluid.2008.02.004
25. M.-S. Liou C. Steffen 1993 A New Flux Splitting Scheme J. Comput. Phys. 107 23 39 10.1006/jcph.1993.1122 10.1006/jcph.1993.1122 1226371
26. Y. Liu M. Vinokur Z.J. Wang 2006 Spectral (finite) volume method for conservation laws on unstructured grids V: extension to three-dimensional systems J. Comput. Phys. 212 454 472 10.1016/j.jcp.2005.06.024 10.1016/j.jcp.2005.06.024 2187900 (Pubitemid 41799933)
27. H. Luo J.D. Baum R. Löhner 2006 A p-multigrid discontinuous Galerkin method for the Euler equations on unstructured grids J. Comput. Phys. 211 767 783 10.1016/j.jcp.2005.06.019 10.1016/j.jcp.2005.06.019 2173405 (Pubitemid 41472450)
28. Maday, Y., Munoz, R.: Spectral element multigrid, Part 2: Theoretical justification. Tech. Rep. 88-73, ICASE (1988)
29. D.J. Mavriplis 1998 Multigrid strategies for viscous flow solvers on anisotropic unstructured meshes J. Comput. Phys. 145 141 165 10.1006/jcph.1998.6036 10.1006/jcph.1998.6036 1640162 (Pubitemid 128341069)
30. Mavriplis, D.J., Jameson, A., Martinelli, L.: Multigrid solution of the Navier-Stokes equations on triangular meshes. AIAA Paper 89-0120 (1989)
31. C.R. Nastase D.J. Mavriplis 2006 High-order discontinuous Galerkin methods using an hp-multigrid approach J. Comput. Phys. 213 330 357 10.1016/j.jcp.2005.08.022 10.1016/j.jcp.2005.08.022 (Pubitemid 43174878)
32. Radespiel, R., Swanson, R.C.: An investigation of cell-centered and cell vertex multigrid schemes for Navier-Stokes equations. AIAA Paper No. 89-0543 (1989)
33. P. Rasetarinera M.Y. Hussaini 2001 An efficient implicit discontinuous spectral Galerkin method J. Comput. Phys. 172 718 738 10.1006/jcph.2001.6853 10.1006/jcph.2001.6853 (Pubitemid 33082560)
34. P.L. Roe 1981 Approximate Riemann solvers, parameter vectors and difference schemes J. Comput. Phys. 43 357 372 10.1016/0021-9991(81)90128-5 10.1016/0021-9991(81)90128-5 640362
35. E.M. Ronquist A.T. Patera 1987 Spectral element multigrid I. Formulation and numerical results. J. Sci. Comput. 2 4 389 406 10.1007/BF01061297 939914
36. V.V. Rusanov 1961 Calculation of interaction of non-steady shock waves with obstacles J. Comput. Math. Phys. USSR 1 267 279 147083
37. Y. Saad M.H. Schultz 1986 GMRES: A generalized minimal residual algorithm for solving nonsymmetric linear systems SIAM J. Sci. Stat. Comput. 7 865 848568
38. Sharov, D., Nakahashi, K.: Low speed preconditioning and LUSGS scheme for 3D viscous low computations on unstructured grids. AIAA Paper No. 98-0614, January 1998
39. C.-W. Shu 1988 Total-variation-diminishing time discretizations SIAM J. Sci. Stat. Comput. 9 1073 1084 10.1137/0909073
40. C.-W. Shu 1988 Navier-Stokes equations on triangular meshes AIAA J. 10.1137/0909073
41. C.-W. Shu S. Osher 1988 Efficient implementation of essentially non-oscillatory shock-capturing schemes J. Comput. Phys. 77 439 471 10.1016/0021-9991(88)90177-5 10.1016/0021-9991(88)90177-5 954915
42. Spiteri, R.J., Ruuth, S.J.: A new class of optimal high-order strong-stability preserving time discretization methods
43. Sun, Y., Wang, Z.J.: Efficient implicit non-linear LU-SGS approach for compressible flow computation using high-order spectral difference method. Commun. Comput. Phys. (submitted)
44. Y. Sun Z.J. Wang 2004 Evaluation of discontinuous Galerkin and spectral volume methods for scalar and system conservation laws on unstructured grid Int. J. Numer. Methods Fluids 45 8 819 838 10.1002/fld.726 10.1002/fld.726 (Pubitemid 38904950)
45. Y. Sun Z.J. Wang Y. Liu 2006 Spectral (finite) volume method for conservation laws on unstructured grids VI: Extension to viscous flow J. Comput. Phys. 215 41 58 10.1016/j.jcp.2005.10.019 10.1016/j.jcp.2005.10.019 2215651
46. Van Den Abeele, K.V., Broeckhoven, T., Lacor, C.: Dispersion and dissipation properties of the 1d spectral volume method and application to a p-multigrid algorithm. J. Comput. Phys. (2007, in press)
47. K.V. Van Den Abeele C. Lacor 2007 An accuracy and stability study of the 2D spectral volume method J. Comput. Phys. 226 1 1007 1026 10.1016/j.jcp.2007. 05.004 10.1016/j.jcp.2007.05.004 2356866
48. B. Van Leer 1974 Towards the ultimate conservative difference scheme II. Monotonicity and conservation combined in a second order scheme J. Comput. Phys. 14 361 10.1016/0021-9991(74)90019-9 10.1016/0021-9991(74)90019-9
49. B. Van Leer 1979 Towards the ultimate conservative difference scheme V. A second order sequel to Godunov's method J. Comput. Phys. 32 101 10.1016/0021-9991(79)90145-1 10.1016/0021-9991(79)90145-1
50. Z.J. Wang 2002 Spectral (finite) volume method for conservation laws on unstructured grids: basic formulation J. Comput. Phys. 178 210 10.1006/jcph.2002.7041 10.1006/jcph.2002.7041 1899140
51. Z.J. Wang Y. Liu 2002 Spectral (finite) volume method for conservation laws on unstructured grids II: extension to two-dimensional scalar equation J. Comput. Phys. 179 665 10.1006/jcph.2002.7082 10.1006/jcph.2002.7082 1911379
52. Z.J. Wang Y. Liu 2004 Spectral (finite) volume method for conservation laws on unstructured grids III: extension to one-dimensional systems J. Sci. Comput. 20 137 10.1023/A:1025896119548 10.1023/A:1025896119548 2032660
53. Z.J. Wang Y. Liu 2004 Spectral (finite) volume method for conservation laws on unstructured grids IV: extension to two-dimensional Euler equations J. Comput. Phys. 194 716 10.1016/j.jcp.2003.09.012 10.1016/j.jcp.2003.09.012 2034862 (Pubitemid 38284242)
54. Z.J. Wang Y. Liu 2006 Extension of the spectral volume method to high-order boundary representation J. Comput. Phys. 211 154 178 10.1016/j.jcp.2005.05.022 10.1016/j.jcp.2005.05.022 (Pubitemid 41363979)
55. M. Zhang C.W. Shu 2003 An analysis of three different formulations of the discontinuous Galerkin method for diffusion equations Math. Models Methods Appl. Sci. 13 395 413 10.1142/S0218202503002568 10.1142/S0218202503002568 1977633 (Pubitemid 36403876)
56. M. Zhang C.W. Shu 2005 An analysis and a comparison between the discontinuous Galerkin method and the spectral finite volume methods Comput. Fluids 34 4-5 581 592 10.1016/j.compfluid.2003.05.006 10.1016/j.compfluid.2003. 05.006 (Pubitemid 40032125)