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Evaluating linear and nonlinear solvers for density driven flow. (English) Zbl 1423.76264
Summary: This study investigates properties of different solvers for density driven flow problems. The focus is on both non-linear and linear solvers. For the non-linear part, we compare fully coupled method using a Newton linearization and iteratively coupled versions of Jacobi and Gauss-Seidel type. Fully coupled methods require effective preconditioners for the Jacobian. To that end we present a transformation eliminating some couplings and present a strategy for employing algebraic multigrid to the transformed system as well. The work covers theoretical aspects, and provides numerical experiments. Although the primary focus is on density driven flow, we believe that the analysis may well be extended beyond to similar equations with coupled phenomena, such as geomechanics.
##### MSC:
 76M10 Finite element methods applied to problems in fluid mechanics 65M60 Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs 74F10 Fluid-solid interactions (including aero- and hydro-elasticity, porosity, etc.) 76S05 Flows in porous media; filtration; seepage
##### Software:
libDiscretization; d3f; libGrid; UG4; libAlgebra; IPARS; pcl
Full Text:
##### References:
 [1] Putti, M.; Paniconi, C., Picard and Newton linearization for the coupled model for saltwater intrusion in aquifers, Adv. Water Resour., 18, 3, 159-170 (1995) [2] Diersch, H.-J. G.; Kolditz, O., Coupled groundwater flow and transport: 2. Thermohaline and 3D convection systems, Adv. Water Resour., 21, 5, 401-425 (1998) [3] Diersch, H.-J. G.; Kolditz, O., Variable-density flow and transport in porous media: approaches and challenges, (FEFLOW© White Paper Volume II (2009), DHI-WASY GmbH: DHI-WASY GmbH Berlin) [4] Lacroix, S.; Vassilevski, Y. V.; Wheeler, M. F., Decoupling preconditioners in the implicit parallel accurate reservoir simulator (IPARS), Numer. Linear Algebra Appl., 8, 8, 537-549 (2001) · Zbl 1071.76583 [5] Lu, B.; Wheeler, M., Iterative coupling reservoir simulation on high performance computers, Pet. Sci., 6, 43-50 (2009) [6] Kim, J.; Tchelepi, H.; Juanes, R., Stability, accuracy, and efficiency of sequential methods for coupled flow and geomechanics, SPE J., 16, 2, 249-262 (2011) · Zbl 1228.74106 [7] Kim, J.; Tchelepi, H.; Juanes, R., Stability and convergence of sequential methods for coupled flow and geomechanics: fixed-stress and fixed-strain splits, Comput. Methods Appl. Mech. Engrg., 200, 13-16, 1591-1606 (2011) · Zbl 1228.74101 [8] Mikelić, A.; Wheeler, M. F., Convergence of iterative coupling for coupled flow and geomechanics, Comput. Geosci., 17, 3, 455-461 (2013) · Zbl 1392.35235 [9] Mikelić, A.; Wang, B.; Wheeler, M. F., Numerical convergence study of iterative coupling for coupled flow and geomechanics, Comput. Geosci., 1-17 (2014) [10] Langevin, C. D.; Guo, W., MODFLOW/MT3DMS-based simulation of variable-density ground water flow and transport, Ground Water, 44, 3, 339-351 (2006) [12] Johannsen, K.; Kinzelbach, W.; Oswald, S.; Wittum, G., The saltpool benchmark problem—numerical simulation of saltwater upconing in a porous medium, Adv. Water Resour., 25, 309-1708 (2002) [13] Lang, S.; Wittum, G., Large scale density driven flow simulations using parallel unstructured grid adaptation and local multigrid methods, Concurr. Comput., 17, 11, 1415-1440 (2005) [14] Grillo, A.; Lampe, M.; Wittum, G., Three-dimensional simulation of the thermohaline-driven buoyancy of a brine parcel, Comput. Vis. Sci., 13, 287-297 (2010) · Zbl 1216.76066 [15] Fein, E.; Schneider, A., $$d^3 f$$—Ein Programmpaket zur Modellierung von Dichteströmungen, Tech. Rep. GRS 139 (1999), Gesellschaft für Anlagenbau und Reaktorsicherheit (GRS) mbH, October [16] Johannsen, K., Numerische Aspekte dichtegetriebener Strömung in porösen Medien (2004), (Habilitationsschrift) [17] Hackbusch, W., Multi-Grid Methods and Applications (1985), Springer: Springer Berlin · Zbl 0585.65030 [18] Xu, J., Iterative methods by space decomposition and subspace correction, SIAM Rev., 34, 4, 581-613 (1992) · Zbl 0788.65037 [19] Ruge, J. W.; Stüben, K., Algebraic multigrid (AMG), (Multigrid Methods. Multigrid Methods, Frontiers in Applied Mathematics, vol. 3 (1987), SIAM: SIAM Philadelphia, PA), 73-130, (Chapter) [20] Stüben, K., An Introduction to Algebraic Multigrid, 413-532 (2001), Academic Press, Ch. Appendix A [21] Nägel, A., Schnelle Löser für große Gleichungssysteme mit Anwendungen in der Biophysik und den Lebenswissenschaften (2010), Universität Heidelberg, (Ph.D. thesis) [22] Bear, J.; Bachmat, Y., (Introduction to Modeling of Transport Phenomena in Porous Media. Introduction to Modeling of Transport Phenomena in Porous Media, Theory and Applications of Transport in Porous Media (1991), Kluwer Academic: Kluwer Academic Dordrecht) · Zbl 0780.76002 [23] Leijnse, A., Three-dimensional modeling of coupled flow and transport in porous media (1992), University of Notre Dame: University of Notre Dame Indiana, (Ph.D. thesis) [24] Holzbecher, E., Modeling Density-Driven Flow in Porous Media (1998), Springer: Springer Berlin, Heidelberg [25] Oberbeck, A., Über die Wärmeleitung der Flüssigkeiten bei Berücksichtigung der Strömungen infolge von Temperaturdifferenzen, Ann. Phys. Chem., 271-292 (1879) · JFM 11.0787.01 [26] Boussinesq, J., Theorie Analytique de la Chaleur, Vol. 2 (1903), Gauthier-Villlars: Gauthier-Villlars Paris [27] Johannsen, K., On the validity of the Boussinesq approximation for the Elder problem, Comput. Geosci., 7, 3, 169-182 (2003) · Zbl 1134.76471 [29] Rheinboldt, W. C., Methods for Solving Systems of Nonlinear Equations (1998), SIAM · Zbl 0906.65051 [30] Ackerer, P.; Younes, A.; Mancip, M., A new coupling algorithm for density-driven flow in porous media, Geophys. Res. Lett., 31, L12506 (2004), 1-4 [31] Stüben, K., A review of algebraic multigrid, J. Comput. Appl. Math., 128, 1-2, 281-309 (2001) · Zbl 0979.65111 [32] Vanek, P.; Mandel, J.; Brezina, M., Algebraic multigrid by smoothed aggregation for second and fourth order elliptic problems, Computing, 56, 179-196 (1996) · Zbl 0851.65087 [33] Mandel, J.; Brezina, M.; Vanek, P., Energy optimization of algebraic multigrid bases, Computing, 62, 205-228 (1999) · Zbl 0942.65034 [34] Vanek, P.; Brezina, M.; Mandel, J., Convergence of algebraic multigrid based on smoothed aggregation, Numer. Math., 88, 3, 559-579 (2001) · Zbl 0992.65139 [35] Brezina, M.; Cleary, A. J.; Falgout, R. D.; Henson, V. E.; Jones, J. E.; Manteuffel, T. A.; McCormick, S. F.; Ruge, J. W., Algebraic multigrid based on element interpolation (AMGe), SIAM J. Sci. Comput., 22, 5, 1570-1592 (2001) · Zbl 0991.65133 [36] Wagner, C., On the algebraic construction of multilevel transfer operators, Computing, 65, 73-95 (2000) · Zbl 0968.65104 [37] Naegel, A.; Falgout, R.; Wittum, G., Filtering algebraic multigrid and adaptive strategies, Comput. Vis. Sci., 11, 3, 159-167 (2008) [38] Diersch, H.-J. G., Using and testing the algebraic multigrid solver samg in FEFLOW, (FEFLOW© White Paper Volume III (2009), DHI-WASY GmbH: DHI-WASY GmbH Berlin) [39] Voss, C.; Souza, W., Variable density flow and solute transport simulation of regional aquifers containing a narrow freshwater-saltwater transition zone, Water Resour. Res., 26, 2097-2106 (1987) [40] Vogel, A.; Reiter, S.; Rupp, M.; Nägel, A.; Wittum, G., UG 4: A novel flexible software system for simulating PDE based models on high performance computers, Comput. Vis. Sci., 1-15 (2014) [41] Vogel, A.; Nägel, A.; Reiter, S., Numerical advances, (Schneider, A., Enhancement of the codes $$d^3 f$$ and $$r^3 t$$ (GRS-292) (2012), Gesellschaft für Anlagen- und Reaktorsicherheit (GRS) mbH), 156-196 [42] Behie, A.; Vinsome, P., Block iterative methods for fully implicit reservoir simulation, Soc. Pet. Eng. J., 22, 658-668 (1982) [44] Lacroix, S.; Vassilevski, Y.; Wheeler, J.; Wheeler, M., Iterative solution methods for modeling multiphase flow in porous media fully implicitly, SIAM J. Sci. Comput., 25, 3, 905-926 (2003) · Zbl 1163.65310 [45] Scheichl, R.; Masson, R.; Wendebourg, J., Decoupling and block preconditioning for sedimentary basin simulations, Comput. Geosci., 7, 295-318 (2003) · Zbl 1076.76070 [46] Klíe, H.; Wheeler, M. F., Advanced solver methods for subsurface environmental problems, Tech. Rep. (2005), The University of Texas at Austin: The University of Texas at Austin Austin, TX, 78712 [49] Heil, M., An efficient solver for the fully coupled solution of large-displacement fluid-structure interaction problems, Comput. Methods Appl. Mech. Engrg., 193, 1-2, 1-23 (2004) · Zbl 1137.74439 [50] Matthies, H.; Niekamp, R.; Steindorf, J., Algorithms for strong coupling procedures, Comput. Methods Appl. Mech. Engrg., 195, 17-18, 2028-2049 (2006) · Zbl 1142.74050
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