×

An efficient S-DDM iterative approach for compressible contamination fluid flows in porous media. (English) Zbl 1305.76074

Summary: We develop an efficient splitting domain decomposition method (S-DDM) for compressible contamination fluid flows in porous media over multiple block-divided sub-domains by combining the non-overlapping domain decomposition, splitting, linearization and extrapolation techniques. The proposed S-DDM iterative approach divides the large domain into multiple block sub-domains. In each time interval, the S-DDM scheme is applied to solve the water head equation, in which an efficient local multilevel scheme is used for computing the values of water head on the interfaces of sub-domains, and the splitting implicit scheme is used for computing the interior values of water head in sub-domains; and the S-DDM scheme is then proposed to solve the concentration equation by combining the upstream volume technique. Numerical experiments are performed and analyzed to illustrate the efficiency of the S-DDM iterative approach for simulating compressible contamination fluid flows in porous media. The developed method takes the excellent attractive advantages of both the non-overlapping domain decomposition and the splitting technique, and reduces computational complexities, large memory requirements and long computational durations.

MSC:

76M25 Other numerical methods (fluid mechanics) (MSC2010)
76S05 Flows in porous media; filtration; seepage
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Aziz, K.; Settari, A., Petroleum Reservoir Simulation (1979), Applied Science Publisher LTD: Applied Science Publisher LTD London
[2] Batu, V., Applied Flow and Solute Transport Modeling in Aquifers (2005), CRC Press
[3] Bear, J., Hydraulics of Groundwater (1978), McGraw-Hill: McGraw-Hill New York
[4] Chen, W.; Li, X.; Liang, D., Energy-conserved splitting FDTD methods for Maxwell’s equations, Numer. Math., 108, 3, 445-485 (2008) · Zbl 1185.78020
[5] Chen, W.; Li, X.; Liang, D., Symmetric energy-conserved splitting FDTD scheme for the Maxwell’s equations, Commun. Comput. Phys., 6, 804-825 (2009) · Zbl 1364.78035
[6] Chen, Z.; Huan, G.; Ma, Y., Computational Methods for Multiphase Flows in Porous Media. Computational Methods for Multiphase Flows in Porous Media, Computational Science and Engineering Series, vol. 2 (2006), SIAM: SIAM Philadelphia · Zbl 1092.76001
[7] Clavero, C.; Jorge, J. C.; Lisbona, F.; Shishkin, G. I., A fractional step method on a special mesh for the resolution of multidimensional evolutionary convection-diffusion problems, Appl. Numer. Math., 27, 211-231 (1998) · Zbl 0929.65058
[8] Dai, W.; Raja, N., A new ADI scheme for solving three-dimensional parabolic equations with first-order derivatives and variable coefficients, J. Comput. Anal. Appl., 122, 223-250 (1995)
[9] Dawson, C. N.; Du, Q.; Dupont, T. F., A finite difference domain decomposition algorithm for numerical solution of the heat equation, Math. Comp., 57, 63-71 (1991) · Zbl 0732.65091
[10] Diersch, H.-J.; Kolditz, O., Variable-density flow and transport in porous media: approaches and challenges, Adv. Water Resour., 25, 899-944 (2002)
[11] Douglas, J.; Peaceman, D., Numerical solution of two dimensional heat flow problem, Am. Inst. Chem. Eng. J., 1, 505-512 (1955)
[12] Douglas, J.; Kim, S., Improved accuracy for locally one-dimensional methods for parabolic equations, Math. Model Meth. Appl. Sci., 11, 1563-1579 (2001) · Zbl 1012.65095
[13] Du, Q.; Mu, M.; Wu, Z. N., Efficient parallel algorithms for parabolic problems, SIAM J. Numer. Anal., 39, 1469-1487 (2001) · Zbl 1013.65090
[14] Ewing, R. E., The Mathematics of Reservoir Simulation, Frontiers in Applied mathematics, vol. 1 (1983), SIAM: SIAM Philadelphia · Zbl 0533.00031
[15] Feistauer, M.; Felcman, J.; Straskraba, I., Mathematical and Computational Methods for Compressible Flow (2003), Oxford University Press · Zbl 1028.76001
[16] Forsyth, P. A., A control volume finite element approach to NAPL groundwater contamination, SIAM J. Sci. Stat. Comp., 12, 1029-1057 (1991) · Zbl 0725.76087
[17] Frolkovic, P.; De Schepper, H., Numerical modelling of convection dominated transport coupled with density driven flow in porous media, Adv. Water Resourc., 24, 1, 63-72 (2001)
[18] S. Gaiffe, R. Glowinski, R. Lemonnier, Domain decomposition and splitting methods for parabolic problems via a mixed formulation, in: The 12th International Conference on Domain Decomposition, Chiba, Japan, 1999.; S. Gaiffe, R. Glowinski, R. Lemonnier, Domain decomposition and splitting methods for parabolic problems via a mixed formulation, in: The 12th International Conference on Domain Decomposition, Chiba, Japan, 1999.
[19] Gao, L. P.; Zhang, B.; Liang, D., Splitting finite difference methods on staggeres grids for three-dimensional time-dependent Maxwell equations, Commun. Comput. Phys., 4, 405-432 (2008) · Zbl 1364.78036
[20] Karlsen, K. H.; Lie, K.-A., An unconditionally stable splitting scheme for a class of nonlinear parabolic equations, IMA J. Numer. Anal., 19, 1-28 (1999)
[21] Kuznetsov, Y. A., New algorithm for approximate realization of implicit difference schemes, Soviet J. Numer. Anal. Math. Model., 3, 99-114 (1988) · Zbl 0825.65066
[22] Liang, D.; Zhao, W., An optimal weighted upwinding covolume method on non-standard grids for convection-diffusion problems in 2D, Int. J. Numer. Meth. Eng., 67, 4, 553-577 (2006) · Zbl 1110.76321
[23] Lions, P. L., On the Schwarz alternating method I, (Glowinskin, R.; Golub, G. H.; Meurant, G. A.; Periaux, J., Proceedings of the First International Symposium on Domain Decomposition Methods for Partial Differential Equations, Paris, 1987 (1988), SIAM: SIAM Philadelphia, USA), 1-42 · Zbl 0658.65090
[24] Lions, P. L., On the Schwarz alternating method II, (Chan, T.; Glowinskin, R.; Periaux, J.; Wildlund, O., Proceedings of the Second International Symposium on Domain Decomposition Methods for Partial Differential Equations, Los Angeles, 1988 (1989), SIAM: SIAM Philadelphia, USA), 47-70 · Zbl 0681.65072
[25] Mazzia, A.; Putti, M., Three-dimensional mixed finite element-finite volume approach for the solution of density-dependent flow in porous media, J. Comp. Appl. Math., 185, 347-359 (2006) · Zbl 1075.76041
[26] Narayan, K. A.; Schleeberger, C.; Bristow, K. L., Modelling seawater intrusion in the Burdekin Delta Irrigation Area, North Queensland, Australia, Agric. Water Manage., 89, 3, 217-228 (2007)
[27] Park, Y. J.; Sudicky, E. A.; Panday, S.; Sykes, J. F.; Guvanasen, V., Application of implicit sub-time stepping to simulate flow and transport in fractured porous media, Adv. Water Resourc., 31, 995-1003 (2008)
[28] Portero, L.; Jorge, J. C., A generalization of Peaceman-Rachford fractional step method, J. Comp. Appl. Math., 189, 1, 676-688 (2006) · Zbl 1086.65093
[29] Smith, B. F.; Bjost, P. E.; Gropp, W. D., Domain Decomposition Methods for Partial Differential Equations (1996), Cambridge University Press · Zbl 0857.65126
[30] Strang, G., On the construction and comparison of difference schemes, SIAM J. Numer. Anal., 5, 506-517 (1968) · Zbl 0184.38503
[31] Yanenko, N., The Method of Fractional Steps (1971), Springer-Verlag: Springer-Verlag Berlin, Heidelberg, New York · Zbl 0209.47103
[32] Yuan, Y.; Liang, D.; Rui, H., Characteristics-finite element methods for seawater intrusion numerical simulation and theoretical analysis, Acta Math. Appl. Sinica, 14, 12-23 (1998) · Zbl 0968.76567
[33] Zhang, H.; Schwartz, F. W., Multispecies contaminant plumes in variable density flow systems, Water Resourc. Res., 31, 4, 837-847 (1995)
[34] Zimmermann, S.; Bauer, P.; Held, R.; Kinzelbach, W.; Walther, J. H., Salt transport on islands in the Okavango Delta: numerical investigations, Adv. Water Resourc., 29, 11-29 (2006)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.