Extension of fractional step techniques for incompressible flows: The preconditioned Orthomin(1) for the pressure Schur complement

Computers and Fluids

Volumn 44, Issue 1, 2011, Pages 297-313

  Houzeaux, G ^a   Aubry, R ^a   Vazquez M ^a  

^a BARCELONA SUPERCOMPUTING CENTER   (Spain)

Author keywords

Fractional step methods; Incompressible flow; Navier Stokes equations; Orthomin(1) iteration; Pressure Schur complement system; Richardson iteration

Indexed keywords

ALGEBRAIC SOLVER; FRACTIONAL STEP; FRACTIONAL STEP METHODS; GLOBAL CONVERGENCE; GLOBAL MATRIX; INCOMPRESSIBLE NAVIER STOKES EQUATIONS; MASS CONSERVATION; MASSIVELY PARALLEL COMPUTERS; MOMENTUM PRESERVING; NUMERICAL EXAMPLE; ORTHOMIN(1) ITERATION; PRECONDITIONERS; PRESSURE SCHUR COMPLEMENT; PRESSURE SCHUR COMPLEMENT SYSTEM; PROJECTION SCHEMES; RICHARDSON; RICHARDSON ITERATION; SCHUR COMPLEMENT;

ALGEBRA; INCOMPRESSIBLE FLOW; ITERATIVE METHODS; MATHEMATICAL OPERATORS; NUMERICAL METHODS; VISCOUS FLOW;

NAVIER STOKES EQUATIONS;

EID: 79952616913     PISSN: 00457930     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.compfluid.2011.01.017     Document Type: Article

References (44)

1. Chorin A. Numerical solution of the Navier-Stokes equations. Math Comput 1968, 22:745-762.
2. Temam R. Sur l'approximation de la solution des équations de Navier-Stokes par la méthode des pas fractionaires (I). Arch Rat Mech Anal 1969, 32:135-153.
3. Zienkiewicz O.C., Szmelter J., Peraire J. Compressible and incompressible flow; an algorithm for all seasons. Comp Meth Appl Mech Eng 1990, 78:105-121.
4. Codina R., Vázquez M., Zienkiewicz O.C. A general algorithm for compressible and incompressible flow - Part III. The semi-implicit form. Int J Numer Meth Fluids 1998, 27:13-32.
5. Quarteroni A., Saleri F., Veneziani A. Factorization methods for the numerical approximation of Navier-Stokes equations. Comp Meth Appl Mech Eng 2000, 188:505-526.
6. Blair Perot J. An analysis of the fractional step method. J Comput Phys 1993, 108:51-58.
7. van Kan J. A second-order accurate pressure correction scheme for viscous incompressible flow. SIAM J Sci Stat Comput Arch 1986, 7:870-891.
8. Turek S. Efficient solvers for incompressible flow problems - an algorithmic and computational approach 1999, Springer, Berlin, Heidelberg, New York.
9. Cahouet J., Chabard J.P. Some fast 3D finite element solvers for the generalized Stokes problem. Int J Numer Meth Fluids 1988, 8:869-895.
10. Mardal K.A., Winther R. Uniform preconditioners for the time dependent Stokes problem. Numerische Mathematik 2004, 98(2):305-327.
11. Saleri F., Veneziani A. Pressure correction algebraic splitting methods for the incompressible Navier-Stokes equations. SIAM J Numer Anal 2005, 43:174-194.
12. Gauthier F., Saleri F., Veneziani A. A fast preconditioner for the incompressible Navier-Stokes equations. Comput Vis Sci 2004, 6:105-112.
13. Wathen A.J., Silvester D.J. Fast iterative solution of stabilised Stokes systems. Part I: Using simple diagonal preconditioners. SIAM J Numer Anal 1993, 30:630-649.
14. Silvester D.J., Wathen A.J. Fast iterative solution of stabilised Stokes systems. Part II: Using general block preconditioners. SIAM J Numer Anal 1994, 31:1352-1367.
15. Wathen A., Fischer B., Silvester D. The convergence rate of the minimal residual method for the Stokes problem. Numer Math 1995, 71(1):121-134.
16. Rusten T., Winther R. A preconditioned iterative method for saddle point problems. SIAM J Matrix Anal Appl 1992, 13:887-904.
17. Atanga J., Silvester D. Iterative methods for stabilized mixed velocity-pressure finite elements. Int J Numer Meth Fluids 1992, 14:71-81.
18. Ramage A., Wathen A. Iterative solution techniques for the Stokes and Navier-Stokes equations. Int J Numer Meth Fluids 1994, 19:67-83.
19. Saad Y. Iterative methods for sparse linear systems 2003, SIAM.
20. Arrow K., Hurwicz L., Uzawa H. Studies in linear and nonlinear programming 1958, Stanford University Press.
21. Elman H.C., Golub G.H. Inexact and preconditioned Uzawa algorithms for saddle point problems. SIAM J Numer Anal 1994, 31:1645-1661.
22. Bramble J.H., Pasciak J.E., Vassilev A.T. Analysis of the inexact Uzawa algorithm for saddle point problems. SIAM J Numer Anal 1997, 34:1072-1092.
23. Bertrand F., Tanguy P.A. Krylov-based Uzawa algorithms for the solution of the Stokes equations using discontinuous-pressure tetrahedral finite elements. J Comput Phys 2002, 181(2):617-638.
24. Bramble J.H., Pasciak J.E. Iterative techniques for time dependent Stokes problems. Comp Math Appl 1997, 33:13-30.
25. Aubry R., Idelsohn S.R., Oñate E. Fractional step like schemes for free surface problems with thermal coupling using the Lagrangian PFEM. Comput Mech 2006, 38(4):249-309.
26. Elman H.C., Silvester D.J., Wathen A.J. Finite elements and fast iterative solvers 2005, Oxford University Press.
27. Loghin D., Wathen A.J. Schur complement preconditioners for the Navier-Stokes equations. Int J Numer Meth Fluids 2002, 40:403-412.
28. Kay D., Loghin D., Wathen A.J. A preconditioner for the steady-state Navier-Stokes equations. SIAM J Sci Comput 2002, 24:237-256.
29. Houzeaux G., Vázquez M., Aubry R., Cela J.M. A massively parallel fractional step solver for incompressible flows. J Comput Phys 2009, 228:6316-6332.
30. Batchelor G.K. An introduction to fluid dynamics 1970, Cambridge University Press.
31. Houzeaux G., Principe J. A variational subgrid scale model for transient incompressible flows. Int J Comput Fluid Dyn 2008, 22(3):135-152.
32. Golub G.H., Van Loan C.F. Matrix computations 1996, The Johns Hopkins University Press.
33. Greenbaum A. Iterative methods for solving linear systems. SIAM series in frontiers in applied mathematics; 1997.
34. Vinsome PKW. Orthomin, an iterative method for solving sparse sets of simultaneous linear equations. In: 4th Symposium on reservoir simulations. Society of Petroleum Engineers of AIME; 1976. p. 149-59.
35. Codina R. Stabilized finite element approximation of transient incompressible flows using orthogonal subscales. Comput Meth Appl Mech Eng 2002, 191:4295-4321.
36. Codina R. Comparison of some finite element methods for solving the diffusion-convection-reaction equation. Comput Meth Appl Mech Eng 1998, 156:185-210.
37. Shakib F., Hughes T.J.R., Johan Z. A new finite element formulation for computational fluid dynamics: X. The compressible Euler and Navier-Stokes equations. Comput Meth Appl Mech Eng 1991, 89:141-219.
38. Hauke G., Garci{dotless}́a-Olivares A. Variational subgrid scale formulations for the advection-diffusion-reaction equation. Comput Meth Appl Mech Eng 2001, 190:6847-6865.
39. Houzeaux G., Eguzkitza B., Vázquez M. A variational multiscale model for the advection-diffusion-reaction equation. Commun Numer Meth Engn 2008, 25:787-809.
40. Wolfe WP, Ochs SS. CFD calculations of S809 aerodynamic characteristics, 1007. AIAA Paper 97-0973.
41. Betts PL, Bokhari H. New experiments on natural convection of air in a tall cavity. In 4th UK national conference on heat transfer, IMechE conference. Mechanical Engineering Publications for the Institution of Mechanical Engineers; 26-27 September 1995. p. 213-7.
42. Laurence D., Uribe J.C., Utyuzhnikov S.V. A robust formulation of the v2-f model. Flow Turb Comb 2004, 73:169-185.
43. Mut F, Wright S, Putman CM, Ascoli G, Cebral JR. Image-based modeling of hemodynamics in cerebral arterial trees. In: SPIE medical imaging. Paper No. 7262-17, Orlando (USA); February 7-11 2009.
44. Wright SN, Brown KM, Cebral JR, Kochunov P, Mazziota JC, Senft SL, et al. Digital reconstruction and morphometric analysis of human brain vasculature from magnetic resonance angiography. In: Society of Neurosciences annual meeting, San Diego (USA); November 3-7 2007.