## Long time stability of four methods for splitting the evolutionary Stokes-Darcy problem into Stokes and Darcy subproblems.(English)Zbl 1238.76030

Summary: This report analyzes the long time stability of four methods for non-iterative, sub-physics, uncoupling for the evolutionary Stokes-Darcy problem. The four methods uncouple each timestep into separate Stokes and Darcy solves using ideas from splitting methods. Three methods uncouple sequentially while one is a parallel uncoupling method. We prove long time stability of four splitting based partitioned methods under timestep restrictions depending on the problem parameters. The methods include those that are stable uniformly in $$S_{0}$$, the storativity coefficient, for moderate $$k_{\min}$$, the minimum hydraulic conductivity, uniformly in $$k_{\min}$$ for moderate $$S_{0}$$ and with no coupling between the timestep and the spacial meshwidth.

### MSC:

 76M10 Finite element methods applied to problems in fluid mechanics 76Dxx Incompressible viscous fluids

### Keywords:

Stokes; Darcy coupling; partitioned methods; splitting methods

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 [1] Bear, J., Hydraulics of groundwater, (1979), McGraw-Hill New York [2] A. Johnson, Compilation of specific yields for various materials, US Geological Survey Water Supply Paper 1667-D, 1967. [3] Mu, M.; Zhu, X., Decoupled schemes for a non-stationary mixed stokes – darcy model, Math. comp., 79, 707-731, (2010) · Zbl 1369.76026 [4] Layton, W.; Trenchea, C., Stability of two IMEX methods, CNLF and BDF2-AB2, for uncoupling systems of evolution equations, Appl. numer. math., 62, 112-120, (2012) · Zbl 1237.65101 [5] W. Layton, H. Tran, C. Trenchea, Analysis of long time stability and errors of two partitioned methods for uncoupling evolutionary groundwater—surface water flows, Tech. Report, 2011 (submitted for publication). www.mathematics.pitt.edu/research/technical-reports.php. · Zbl 1370.76173 [6] Li Shan, Haibiao Zheng, Partitioned time stepping method for fully evolutionary Stokes-Darcy flow with Beavers-Joseph interface, Tech. Report, 2011. www.mathematics.pitt.edu/research/technical-reports.php. · Zbl 1268.76035 [7] Beavers, G.S.; Joseph, D., Boundary conditions at a naturally permeable wall, J. fluid mech., 30, 197-207, (1967) [8] Saffman, P.G., On the boundary condition at the interface of a porous medium, Stud. appl. math., 1, 93-101, (1971) · Zbl 0271.76080 [9] Jaeger, W.; Mikelic, A., On the interface boundary conditions of Beavers, Joseph and Saffman, SIAM J. appl. math., 60, 1111-1127, (2000) · Zbl 0969.76088 [10] Cao, Y.; Gunzburger, M.; Hua, F.; Wang, X., Coupled stokes – darcy model with beavers – joseph interface boundary condition, Commun. math. sci., 8, 1-25, (2010) · Zbl 1189.35244 [11] Cao, Y.; Gunzburger, M.; Hu, X.; Hua, F.; Wang, X.; Zhao, W., Finite element approximations for stokes – darcy flow with beavers – joseph interface conditions, SIAM J. numer. anal. (SINUM), 47, 4239-4256, (2010) · Zbl 1252.76040 [12] Y. Cao, M. Gunzburger, X.-M. He, X. Wang, Parallel, non-iterative multi-physics domain decomposition methods for the time-dependent Stokes-Darcy problem, 2011 (submitted for publication). · Zbl 1457.65089 [13] W. Layton, Fluid – porous interface conditions with the “Inertia Term” $$1 / 2$$$$\mid U_{F L U I D} \mid^2$$ are not Galilean invariant, Tech. Report, 2009. [14] Pinder, G.F.; Celia, M.A., Subsurface hydrology, (2006), John Wiley and Sons Hoboken, New Jersey [15] Layton, W.; Schieweck, F.; Yotov, I., Coupling fluid flow with porous media flow, SIAM J. numer. anal. (SINUM), 40, 2195-2218, (2003) · Zbl 1037.76014 [16] Discacciati, M.; Miglio, E.; Quarteroni, A., Mathematical and numerical models for coupling surface and groundwater flows, Appl. numer. math., 43, 57-74, (2001) · Zbl 1023.76048 [17] Payne, L.E.; Straughan, B., Analysis of the boundary condition at the interface between a viscous fluid and a porous medium and related modeling questions, J. math. pures appl., 77, 1959-1977, (1998) · Zbl 0906.35067 [18] Payne, L.E.; Song, J.C.; Straughan, B., Continuous dependence and convergence results for Brinkman and forcheimer models with variable viscosity, Proc. R. soc. lond. ser. A, 455, 2173-2190, (1999) · Zbl 0933.76091 [19] Arbogast, T.; Brunson, D.S., A computational method for approximating a darcy – stokes system governing a vuggy porous medium, Comput. geosci., 11, 207-218, (2007) · Zbl 1186.76660 [20] Badia, S.; Codina, R., Unified stabilized finite element formulations for the Stokes and the Darcy problems, SIAM J. numer. anal., 47, 3, 1971-2000, (2009) · Zbl 1406.76047 [21] Burman, E.; Hansbo, P., A unified stabilized method for Stokes and darcy’s equations, J. comput. appl. math., 198, 35-51, (2007) · Zbl 1101.76032 [22] Chrispell, J.C.; Ervin, V.J.; Jenkins, E.W., Coupled generalized non-linear Stokes flow with flow through a porous media, SIAM J. numer. anal., 47, 929-952, (2009) · Zbl 1279.76032 [23] Gatica, G.N.; Meddahi, S.; Oyarzúa, R., A conforming mixed finite-element method for the coupling of fluid flow with porous media flow, IMA J. numer. anal., 29, 86-108, (2009) · Zbl 1157.76025 [24] Karper, T.; Mardal, K.A.; Winther, R., Unifed fnite element discretizations of coupled darcy – stokes flow, Numer. methods partial differential equations, 25, 2, 311-326, (2009) · Zbl 1157.76026 [25] Mardal, K.A.; Tai, X.C.; Winther, R., A robust finite element method for darcy – stokes flow, SIAM J. numer. anal., 40, 5, 1605-1631, (2002) · Zbl 1037.65120 [26] Urquiza, J.J.M.; N’Dri, D.; Garon, A.; Delfour, M.C., Coupling Stokes and Darcy equations, Appl. numer. math., 58, 525-538, (2008) · Zbl 1134.76033 [27] Zhang, S.; Xie, X.; Chen, Y., Low order nonconforming rectangular finite element methods for darcy – stokes problems, J. comput. math., 27, 2-3, 400-424, (2009) · Zbl 1212.65464 [28] Cao, Y.; Gunzburger, M.; He, X.-M.; Wang, X., Robin – robin domain decomposition methods for the steady-state stokes – darcy system with the beavers – joseph interface condition, Numer. math., 117, 4, 601-629, (2011) · Zbl 1305.76072 [29] M. Discacciati, Domain decomposition methods for the coupling of surface and groundwater flows, Ph.D. Thesis, Ecole Polytechnique Federale de Lausanne, Swizerland, 2004. [30] Discacciati, M.; Quarteroni, A., Convergence analysis of a subdomain iterative method for the finite element approximation of the coupling of Stokes and Darcy equations, Comput. vis. sci., 6, 93-103, (2004) · Zbl 1299.76252 [31] Discacciati, M.; Quarteroni, A., Analysis of a domain decomposition method for the coupling of Stokes and Darcy equations, (), 3-20 · Zbl 1254.76051 [32] Hoppe, R.H.W.; Porta, P.; Vassilevski, Y., Computational issues related to iterative coupling of subsurface and channel flows, Calcolo, 44, 1-20, (2007) · Zbl 1150.76028 [33] Cai, M.; Mu, M.; Xu, J., Preconditioning techniques for a mixed Stokes/Darcy model in porous media applications, J. comput. appl. math., 233, 2, 346-355, (2009) · Zbl 1172.76023 [34] Mu, M.; Xu, J., A two-grid method of a mixed stokes – darcy model for coupling fluid flow with porous media flow, SIAM J. numer. anal. (SINUM), 45, 1801-1813, (2007) · Zbl 1146.76031 [35] Jiang, B., A parallel domain decomposition method for coupling of surface and groundwater flows, Comput. methods appl. mech. engrg., 198, 9-12, 947-957, (2009) · Zbl 1229.76048 [36] Miglio, E.; Quarteroni, A.; Saleri, F., Coupling of free surface flow and groundwater flows, Comput. & fluids, 23, 73-83, (2003) · Zbl 1035.76051 [37] D. Vassilev, I. Yotov, Domain decomposition for coupled Stokes and Darcy flows, Technical Report, Univ. of Pittsburgh, 2011. · Zbl 1295.76036 [38] J. Verwer, Convergence and component splitting for the Crank-Nicolson Leap-Frog integration method, CWI Report MAS-E0902, 2009. [39] Varah, J.M., Stability restrictions on a second order, three level finite difference schemes for parabolic equations, SIAM J. numer. anal. (SINUM), 17, 300-309, (1980) · Zbl 0426.65048 [40] Asher, U.; Ruuth, S.; Wetton, B., Implicit – explicit methods for time dependent partial differential equations, SIAM J. numer. anal. (SINUM), 32, 797-823, (1995) · Zbl 0841.65081 [41] Crouzeix, M., Une méthode multipas implicite-explicite pour l’approximation des équationes d’évolution paraboliques, Numer. math., 35, 257-276, (1980) · Zbl 0419.65057 [42] J. Frank, W. Hundsdorfer, J. Verwer, Stability of implicit – explicit linear multistep methods, CWI Report 1996. · Zbl 0887.65094 [43] Hundsdorfer, W.; Verwer, J., Numerical solution of time dependent advection diffusion reaction equations, (2003), Springer Berlin · Zbl 1030.65100 [44] Marchuk, G.I., Splitting methods, (1988), Nauka Moscow · Zbl 0653.65065 [45] Marchuk, G.I., Splitting and alternate direction methods, (), 197-464 · Zbl 0875.65049 [46] Yanenko, N.N., The method of fractional steps, (1971), Springer Berlin · Zbl 0209.47103 [47] Anitescu, M.; Layton, W.; Pahlevani, F., Implicit for local effects, explicit for nonlocal is unconditionally stable, Electron. trans. numer. anal., 18, 174-187, (2004) · Zbl 1085.76046 [48] J.M. Connors, J.S. Howell, A fluid – fluid interaction method using decoupled subproblems and differing time steps, Numer. Methods Partial Differential Equations (2011) in press (doi:10.1002/num.20681). · Zbl 1345.86005 [49] F. Hua, Modeling, analysis and simulation of Stokes-Darcy system with Beavers-Joseph interface condition, Ph.D. Dissertation, The Florida State University, 2009. [50] Hundsdorfer, W., Accuracy and stability of splitting with stabilizing corrections, Appl. numer. math., ISSN: 0168-9274, 42, 1-3, 213-233, (2002), Numerical Solution of Differential and Differential-Algebraic Equations, 4-9 September 2000, Halle, Germany · Zbl 1004.65095 [51] X. Wang, On the coupled continuum pipe flow model (CCPF) for flows in karst aquifer, DCDS-B, vol. 13, 2010, pp. 489-501. · Zbl 1184.86006 [52] Holden, H.; Karlsen, K.N.; Lie, K.-A.; Risebro, N.H., Splitting methods for partial differential equations with rough solutions, (2010), European Math., Soc. Zurich [53] Gunzburger, M.D., Finite element methods for viscous incompressible flows—A guide to theory, practices, and algorithms, (1989), Academic Press [54] Girault, V.; Raviart, P.A., Finite element methods for Navier Stokes equations, (1986), Springer-Verlag · Zbl 0396.65070 [55] Layton, W., Introduction to the numerical analysis of incompressible, viscous flows, (2007), SIAM Philadelphia [56] Lesaint, P., FEM for symmetric hyperbolic equations, Numer. math., 21, 244-255, (1973) · Zbl 0283.65061 [57] M. Moraiti, On the quasi-static approximation in the Stokes-Darcy model of groundwater-surfacewater flows, Technical Report, 2011. [58] Lube, G.; Olshanskii, M., Stable finite element calculations of incompressible flows using the rotation form of convection, IMA J. numer. anal., 22, 437-461, (2002) · Zbl 1016.76051 [59] Olshanskii, M.A.; Reusken, A., Navier – stokes equations in rotation form: a robust multigrid solver for the velocity problem, SIAM J. sci. comput., 23, 1682-1706, (2002) · Zbl 1020.65095 [60] Olshanskii, M.A.; Reusken, A., Grad-div stabilization for the Stokes equations, Math. comp., 73, 1699-1718, (2004) · Zbl 1051.65103 [61] F. Hecht, O. Pironneau, FreeFem$$+ +$$ webpage. http://www.freefem.org. [62] Çeşmelioǧlu, A.; Rivière, B., Analysis of time-dependent navier – stokes flow coupled with Darcy flow, J. numer. math., 16, 249-280, (2008) · Zbl 1159.76010
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