Partitioned time stepping method for fully evolutionary Navier-Stokes/Darcy flow with BJS interface conditions. (English) Zbl 1488.65425
Summary: In this report, a partitioned time stepping algorithm for Navier Stokes/Darcy model is analyzed. This method requires only solving one, uncoupled Navier Stokes and Darcy problems in two different sub-domains respectively per time step. On the interface, the simplified Beavers-Joseph-Saffman conditions are imposed with an additional assumption \(\mathbf{u}\cdot\mathbf{n}_f>0\) (not hold for general case but still in many situation, such as the gentle river). Under a modest time step restriction of the form \(\Delta t\leq C\), where \(C=C\) (physical parameters), we prove stability of the method and get the error estimates. Numerical tests illustrate the validity of the theoretical results.
MSC:
| 65M60 | Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs |
| 65M55 | Multigrid methods; domain decomposition for initial value and initial-boundary value problems involving PDEs |
| 76S05 | Flows in porous media; filtration; seepage |
| 76D05 | Navier-Stokes equations for incompressible viscous fluids |
| 76M10 | Finite element methods applied to problems in fluid mechanics |
| 86A05 | Hydrology, hydrography, oceanography |
| 35Q86 | PDEs in connection with geophysics |
| 65M06 | Finite difference methods for initial value and initial-boundary value problems involving PDEs |
| 65N30 | Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs |
| 65F10 | Iterative numerical methods for linear systems |
| 65M15 | Error bounds for initial value and initial-boundary value problems involving PDEs |
| 65M12 | Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs |
Keywords:
fully evolutionary Navier-Stokes/Darcy problem; partitioned time stepping method; Beavers-Joseph-Saffman; interface conditions; error estimateSoftware:
FreeFem++References:
| [1] | M. DISCACCIATI, Domain Decomposition Methods for the Coupling of Surface and Groundwater Flows (Ph.D. Dissertation),École Polytechnique Fédérale de Lausanne., 2004. |
| [2] | G. GATICA AND F. SAYAS, Analysis of fully-mixed finite element methods for the Stokes-Darcy coupled problem, Math. Comput., 80 (2011), pp. 1911-1948. · Zbl 1301.76047 |
| [3] | M. DISCACCIATI, E. MIGLIO AND A. QUARTERONI, Mathematical and numerical models for coupling surface and groundwater flows, Appl. Numer. Math., 43 (2001), pp. 57-74. · Zbl 1023.76048 |
| [4] | W. LAYTON, F. SCHIEWECK AND I. YOTOV, Coupling fluid flow with porous media flow, SIAM J. Numer. Anal., 40(2003), pp. 2195-2218. · Zbl 1037.76014 |
| [5] | B. RIVIÈRE AND I. YOTOV, Locally conservative coupling of Stokes and Darcy flow, SIAM J. Numer. Anal., 42 (2005), pp. 1959-1977. · Zbl 1084.35063 |
| [6] | J. CAMANO, G. N. GATICA, R. OYARZÚA, R. RUIZ-BAIER AND P. VENEGAS-TAPIA, New fully-mixed finite element methods for the Stokes-Darcy coupling, Comput. Meth. Appl. Mech. Eng., 295 (2015), pp. 362-395. · Zbl 1423.74868 |
| [7] | Y. CAO, M. GUNZBURGER, F. HUA AND X. WANG, Coupled Stokes-Darcy model with Beavers-Joseph interface boundary condition, Commun. Math. Sci., 8 (2010), pp. 1-25. · Zbl 1189.35244 |
| [8] | M. MU AND X. H. ZHU, Decoupled schemes for a non-stationary mixed Stokes-Darcy model, Math. Comput., 79 (2010), pp. 707-731. · Zbl 1369.76026 |
| [9] | Y. CAO, M. GUNZBURGER, X. HE AND X. WANG, Parallel, non-iterative, multi-physics, domain decomposition methods for the time-dependent Stokes-Darcy model, Math. Comput., 83 (2014), pp. 1617-1644. · Zbl 1457.65089 |
| [10] | L. SHAN, H. ZHENG AND W. J. LAYTON, A decoupling method with different subdomain time steps for the nonstationary Stokes-Darcy model, Numer. Meth. PDE., 29 (2013), pp. 549-583. · Zbl 1364.76096 |
| [11] | L. SHAN, AND H. ZHENG, Partitioned time stepping method for fully evolutionary Stokes-Darcy flow with the Beavers-Joseph interface conditions, SIAM J. Numer. Anal. 51 (2013), pp. 813-839. · Zbl 1268.76035 |
| [12] | Y. HOU, Optimal error estimates of a decoupled schemes based on two-grid finite element for mixed Stokes-Darcy model, Appl. Math. Lett., 57(2016), pp. 90-96. · Zbl 1408.76367 |
| [13] | L. ZOU AND Y. HOU, A two-grid decoupling method for the mixed Stokes-Darcy model, J. Com-put. Appl. Math., 275 (2015), pp. 139-147. · Zbl 1334.76157 |
| [14] | A. CESMELIOGLU AND B. RIVIERE, Existence of a weak solution for the fully coupled Navier-Stokes/Darcy-transport problem, J. Diff. Eqn., 252 (2012), pp. 4138-4175. · Zbl 1234.35178 |
| [15] | V. GIRAULT AND B. RIVIÈRE, DG approximation of coupled Navier-Stokes and Darcy equations by Beaver-Joseph-Saffman interface condition, SIAM J. Numer. Anal., 47 (2009), pp. 2052-2089. · Zbl 1406.76082 |
| [16] | L. BADEA, M. DISCACCIATI AND A. QUARTERONI, Mathematical analysis of the Navier-Stokes/Darcy coupling, Technical report, Politecnico di Milano, Milan, 2006. |
| [17] | M. DISCACCIATI AND A. QUARTERONI, Navier-Stokes/Darcy coupling: modeling, analysis and numerical approximation, Rev. Mat. Complut., 22 (2009), pp. 315-426. · Zbl 1172.76050 |
| [18] | L. BADEA, M. DISCACCIATI AND A. QUARTERONI., Numerical analysis of the Navier-Stokes/Darcy coupling, Numer. Math., 115 (2010), pp. 195-227. · Zbl 1423.35304 |
| [19] | M. CAI, M. MU AND J. XU, Numerical solution to a mixed Navier-Stokes/Darcy model by the two-grid approach, SIAM J. Numer. Anal., 47 (2009), pp. 3325-3338. · Zbl 1213.76131 |
| [20] | Z. SI, Y. WANG AND S. LI, Decoupled modified characteristics finite element method for the time dependent Navier-Stokes/Darcy problem, Math. Meth. Appl. Sci., 37 (2014), pp. 1392-1404. · Zbl 1426.76309 |
| [21] | X. HE, J. LI, Y. LIN AND J. MING, A domain decomposition method for the steady-state Navier-Stokes-Darcy model with Beavers-Joseph interface condition, SIAM J. Sci. Comput., 37 (2015), pp. S264-S290. · Zbl 1325.76119 |
| [22] | H. JIA, P. SHI, K. LI AND H. JIA, A decoupling method with different subdomain time steps for the non-stationary Navier-Stokes/Darcy model, J. Comput. Math., 34 (2016), pp. 1-25. |
| [23] | H. JIA, H. JIA AND Y. HUANG, A modified two-grid decoupling method for the mixed Navier-Stokes Darcy model, Comput. Math. Appl., 72 (2016), pp. 1142-1152. · Zbl 1416.76115 |
| [24] | R. TEMAM, Navier-Stokes Equations: Theory and Numerical Analysis, North-Holland, Am-sterdam, 1984. · Zbl 0568.35002 |
| [25] | R. LI, J. LI, Z. CHEN AND Y. GAO, A stabilized finite element method based on two local Guass integrations for a coupled Stokes-Darcy problem, J. Comput. Appl. Math., 292 (2016), pp. 92-104. · Zbl 1329.76181 |
| [26] | T. ZHANG, Z. SI AND Y. HE, A stabilised characteristic finite element method for transient Navier-Stokes equations, Int. J. Comput. Fluid Dyn., 24 (2010), pp. 369-381. · Zbl 1271.76177 |
| [27] | F. Hecht, FreeFEM++, J. Numer. Math., 20 (2012), pp. 251-265. · Zbl 1266.68090 |
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.
