Fast and robust solvers for pressure-correction in bubbly flow problems

Journal of Computational Physics

Volumn 227, Issue 23, 2008, Pages 9742-9761

  MacLachlan, S P ^a   Tang, J M ^a   Vuik, C ^a  

^a DELFT UNIVERSITY OF TECHNOLOGY   (Netherlands)

Author keywords

Bubbly flows; Conjugate gradient; Deflation; Discontinuous coefficients; Multigrid; Poisson equation

Indexed keywords

CONJUGATE GRADIENT METHOD; NAVIER STOKES EQUATIONS; POISSON EQUATION; TWO PHASE FLOW;

BUBBLY FLOW; CONJUGATE GRADIENT; CORRECTION APPROACHES; DEFLATION; DISCONTINUOUS COEFFICIENTS; FLOW PROBLEMS; MULTIGRIDS; NAVIER-STOKES EQUATION; PRESSURE CORRECTION; TWO-PHASE FLUID FLOW;

LINEAR SYSTEMS;

EID: 53549099921     PISSN: 00219991     EISSN: 10902716     Source Type: Journal    
DOI: 10.1016/j.jcp.2008.07.022     Document Type: Article

References (69)

1. Adams J.C. MUDPACK: Multigrid portable FORTRAN software for the efficient solution of linear elliptic partial differential equations. Appl. Math. Comput. 34 2 (1989) 113-146
2. Alber D., and Olson L. Parallel coarse-grid selection. Numer. Linear Algebra Appl. 14 8 (2007) 611-643
3. Alcouffe R.E., Brandt A., Dendy J.E., and Painter J.W. The multigrid method for the diffusion equation with strongly discontinuous coefficients. SIAM J. Sci. Stat. Comput. 2 (1981) 430-454
4. T. Austin, M. Berndt, B.K. Bergen, J.E. Dendy, J.D. Moulton, Parallel, scalable, and robust multigrid on structured grids, T-7 Research Highlight LA-UR-03-9167, Theoretical Division, Los Alamos National Laboratory, Los Alamos, NM, USA, 2003.
5. Behie A., and Forsyth P.A. Multi-grid solution of three-dimensional problems with discontinuous coefficients. Appl. Math. Comput. 13 (1983) 229-240
6. Bell J.B., Colella P., and Glaz H.M. A second-order projection method for the incompressible Navier-Stokes equations. J. Comput. Phys. 85 2 (1989) 257-283
7. D. Braess, On the combination of the multigrid method and conjugate gradients, Multigrid Methods, II (Cologne, 1985), Lecture Notes in Math., vol. 1228, Springer, Berlin, 1986, pp. 52-64.
8. Brandt A., McCormick S.F., and Ruge J.W. Algebraic multigrid (AMG) for sparse matrix equations. In: Evans D.J. (Ed). Sparsity and Its Applications (1984), Cambridge University Press, Cambridge 257-284
9. Bunner B., and Tryggvason G. Dynamics of homogeneous bubbly flows. part 1. rise velocity and microstructure of the bubbles. J. Fluid Mech. 466 (2002) 17-52
10. Chorin A.J. Numerical solution of the Navier-Stokes equations. Math. Comp. 22 (1968) 745-762
11. Chorin A.J. On the convergence of discrete approximations to the Navier-Stokes equations. Math. Comp. 23 (1969) 341-353
12. de Sousa F.S., Mangiavacchi N., Nonato L.G., Castelo A., Tomé; M.F., Ferreira V.G., Cuminato J.A., and McKee S. A front-tracking/front-capturing method for the simulation of 3d multi-fluid flows with free surfaces. J. Comput. Phys. 198 2 (2004) 469-499
13. Dendy J.E. Black box multigrid. J. Comput. Phys. 48 3 (1982) 366-386
14. Dendy J.E. Two multigrid methods for three-dimensional equations with highly discontinuous coefficients. SIAM J. Sci. Stat. Comput. 8 (1987) 673-685
15. Dendy J.E. Black box multigrid for periodic and singular problems. Appl. Math. Comput. 25 1, part I (1988) 1-10
16. Erlangga Y., and Nabben R. Multilevel projection-based nested Krylov iteration for boundary value problems. SIAM J. Sci. Comput. 30 3 (2008) 1572-1595
17. Esmaeeli A., and Tryggvason G. Direct numerical simulations on bubbly flows. Part 1. Low Reynolds number arrays. J. Fluid Mech. 377 (1998) 313-345
18. Esmaeeli A., and Tryggvason G. Direct numerical simulations on bubbly flows. Part 2. Moderate Reynolds number arrays. J. Fluid Mech. 385 (1999) 325-358
19. R.D. Falgout, J.E. Jones, Multigrid on massively parallel architectures, in: Multigrid methods, VI (Gent, 1999), of Lect. Notes Comput. Sci. Eng., vol. 14, Springer, Berlin, 2000, pp. 101-107.
20. Fortin M. Old and new finite elements for incompressible flows. Internat. J. Numer. Meth. Fluids 1 4 (1981) 347-364
21. Frank J., and Vuik C. On the construction of deflation-based preconditioners. SIAM J. Sci. Comput. 23 (2001) 442-462
22. V. Girault, P.-A. Raviart, Finite element approximation of the Navier-Stokes equations, Lecture Notes in Mathematics, vol. 749, Springer-Verlag, Berlin, 1979.
23. Greenbaum A. Iterative methods for solving linear systems. Frontiers in Applied Mathematics vol. 17 (1997), SIAM, Philadelphia, PA
24. Henson V.E., and Yang U.M. Boomer AMG: a parallel algebraic multigrid solver and preconditioner. Appl. Numer. Math. 41 1 (2002) 155-177
25. Hua J., and Lou J. Numerical simulation of bubble rising in viscous liquid. J. Comput. Phys. 222 2 (2007) 769-795
26. Inamuro T., Ogata T., Tajima S., and Konishi N. A lattice boltzmann method for incompressible two-phase flows with large density differences. J. Comp. Phys. 198 (2004) 628-644
27. Jones J.E., and McCormick S.F. Parallel multigrid methods. Parallel Numerical Algorithms (Hampton, VA, 1994)of ICASE/LaRC Interdiscip. Ser. Sci. Eng. vol. 4 (1997), Kluwer Acad. Publ., Dordrecht 203-224
28. Kaasschieter E.F. Preconditioned conjugate gradients for solving singular systems. J. Comput. Appl. Math. 24 1-2 (1988) 265-275
29. R. Kettler, J. Meijerink, A multigrid method and a combined multigrid-conjugate gradient method for elliptic problems with strongly discontinuous coefficients in general domains, Tech. Rep. 604, Shell Oil Company, 1981.
30. Khalil M., and Wesseling P. Vertex-centered and cell-centered multigrid for interface problems. J. Comput. Phys. 98 (1992) 1-20
31. Kilmer M., and de Sturler E. Recycling subspace information for diffuse optical tomography. SIAM J. Sci. Comput. 27 6 (2006) 2140-2166
32. D. Kwak, C. Kiris, J. Dacles-Mariani, An assessment of artificial compressibility and pressure projection methods for incompressible flow simulations, in: 16th International Conference on Numerical Methods in Fluid Dynamics, Lecture Notes in Physics, vol. 515, Springer, 1998, pp. 177-182.
33. Mansfield L. Damped Jacobi preconditioning and coarse-grid deflation for conjugate gradient iteration on parallel computers. SIAM J. Sci. Stat. Comput. 12 6 (1991) 1314-1323
34. Marchandise E., Geuzaine P., Chevaugeon N., and Remacle J. A stabilized finite element method using a discontinuous level set approach for the computation of bubble dynamics. J. Comput. Phys. 225 1 (2007) 949-974
35. Meijerink J.A., and van der Vorst H.A. An iterative solution method for linear systems of which the coefficient matrix is a symmetric M-matrix. Math. Comp. 31 137 (1977) 148-162
36. Mohr M., and Wienands R. Cell-centred multigrid revisited. Comput. Vis. Sci. 7 3-4 (2004) 129-140
37. Moulton J.D., Dendy J.E., and Hyman J.M. The black box multigrid numerical homogenization algorithm. J. Comput. Phys. 141 (1998) 1-29
38. Nabben R., and Vuik C. A comparison of deflation and coarse grid correction applied to porous media flow. SIAM J. Numer. Anal. 42 (2004) 1631-1647
39. 2 convergence of an algorithm for solving finite element equations. Math. Comp. 31 (1977) 892-906
40. Nicolaides R.A. Deflation of conjugate gradients with applications to boundary value problems. SIAM J. Numer. Anal. 24 2 (1987) 355-365
41. Padiy A., Axelsson O., and Polman B. Generalized augmented matrix preconditioning approach and its application to iterative solution of ill-conditioned algebraic systems. SIAM J. Matrix Anal. Appl. 22 3 (2000) 793-818
42. Parks M., de Sturler E., Mackey G., Johnson D., and Maiti S. Recycling Krylov subspaces for sequences of linear systems. SIAM J. Sci. Comput. 28 5 (2006) 1651-1674
43. Puckett E., Almgren A., Bell J., Marcus D., and Rider W. A high-order projection method for tracking fluid interfaces in variable density incompressible flows. J. Comp. Phys. 130 (1997) 269-282
44. M. Raw, Robustness of coupled algebraic multigrid for the Navier-Stokes equations, Technical Paper 96-0297, AIAA Press, Washington, DC, 1996.
45. Ruge J.W., and Stüben K. Algebraic multigrid (AMG). In: McCormick S.F. (Ed). Multigrid Methods. Frontiers in Applied Mathematics vol. 3 (1987), SIAM, Philadelphia, PA 73-130
46. Saad Y., Yeung M., Erhel J., and Guyomarc'h F. A deflated version of the conjugate gradient algorithm. SIAM J. Sci. Comput. 21 5 (2000) 1909-1926
47. Singh R., and Shyy W. Three-dimensional adaptive cartesian grid method with conservative interface restructuring and reconstruction. J. Comput. Phys. 224 1 (2007) 150-167
48. Stüben K. An introduction to algebraic multigrid. In: Trottenberg U., Oosterlee C., and Schüller A. (Eds). Multigrid (2001), Academic Press, San Diego, CA 413-528
49. Sussman M., and Puckett E. A coupled level set and volume-of-fluid method for computing 3d and axisymmetric incompressible two-phase flows. J. Comput. Phys. 162 2 (2000) 301-337
50. Sussman M., Smith K.M., Hussaini M.Y., Ohta M., and Zhi-Wei R. A sharp interface method for incompressible two-phase flows. J. Comput. Phys. 221 2 (2007) 469-505
51. Sweet R.A. Vectorization and parallelization of FISHPAK. Parallel Processing for Scientific Computing (Houston, TX, 1991) (1992), SIAM, Philadelphia, PA 637-642
52. J.M. Tang, S.P. MacLachlan, R. Nabben, C. Vuik, Theoretical comparison of two-level preconditioners based on multigrid and deflation, DIAM Report 08-05, Delft University of Technology, Delft, 2008.
53. J.M. Tang, R. Nabben, C. Vuik, Y.A. Erlangga, Theoretical and numerical comparison of various projection methods derived from deflation, domain decomposition and multigrid methods, DIAM Report 07-04, Delft University of Technology, Delft, 2007.
54. Tang J.M., and Vuik C. Efficient deflation methods applied to 3D bubbly flow problems. Elec. Trans. Num. Anal. 26 (2007) 330-349
55. Tang J.M., and Vuik C. New variants of deflation techniques for bubbly flow problems. J. Numer. Anal. Indust. Appl. Math. 2 3-4 (2007) 227-249
56. Tang J.M., and Vuik C. On deflation and symmetric positive semi-definite matrices. J. Comput. Appl. Math. 206 2 (2007) 603-614
57. O. Tatebe, The multigrid preconditioned conjugate gradient method, in: N.D. Melson, T.A. Manteuffel, S.F. McCormick (Eds.), Sixth Copper Mountain Conference on Multigrid Methods, vol. CP 3224, NASA, Hampton, VA, 1993, pp. 621-634.
58. Trottenberg U., Oosterlee C.W., and Schüller A. Multigrid (2001), Academic Press, London
59. Tryggvason G., Bunner B., Esmaeeli A., Juric D., Al-Rawahi N., Tauber W., Han J., Nas S., and Jan Y.-J. A front-tracking method for the computations of multiphase flow. J. Comput. Phys. 169 (2001) 708-759
60. Tryggvason G., Esmaeeli A., Lu J., and Biswas S. Direct numerical simulations of gas/liquid multiphase flows. J. Comput. Phys. 38 9 (2006) 660-681
61. van der Pijl S.P., Segal A., Vuik C., and Wesseling P. A mass-conserving level-set method for modelling of multi-phase flows. Int. J. Numer. Meth. Fluids 47 (2005) 339-361
62. van der Pijl S.P., Segal A., Vuik C., and Wesseling P. Computing three-dimensional two-phase flows with a mass-conserving level set method. Comput. Vis. Sci. 11 (2008) 221-235
63. Van Kan J. A second-order accurate pressure-correction scheme for viscous incompressible flow. SIAM J. Sci. Stat. Comput. 7 3 (1986) 870-891
64. Van Kan J., Vuik C., and Wesseling P. Fast pressure calculation for 2D and 3D time dependent incompressible flow. Num. Linear Algebra Appl. 7 (2000) 429-447
65. Vuik C., Segal A., Meijerink J., and Wijma G. The construction of projection vectors for a Deflated ICCG method applied to problems with extreme contrasts in the coefficients. J. Comp. Phys. 172 (2001) 426-450
66. Vuik C., Segal A., and Meijerink J.A. An efficient preconditioned CG method for the solution of a class of layered problems with extreme contrasts in the coefficients. J. Comp. Phys. 152 (1999) 385-403
67. Wen X.-H., and Gómez-Hernández J.J. Upscaling hydraulic conductivities in heterogeneous media: an overview. J. Hydrol. 183 (1996) ix-xxxii
68. Wesseling P. Cell-centered multigrid for interface problems. J. Comput. Phys. 79 (1988) 85-91
69. Wesseling P. An introduction to multigrid methods. Pure and Applied Mathematics (New York) (1992), John Wiley & Sons Ltd., Chichester