Top

previous article On the convergence analysis of the gradient-CQ ...

next article Generation of point sets by convex optimization...

Hint

Swipe to navigate through the articles of this issue

22-08-2019 | Original Paper | Issue 3/2020

A second-order artificial compression method for the evolutionary Stokes-Darcy system

Journal:: Numerical Algorithms > Issue 3/2020

Authors:: Yi Li, Yanren Hou, Yao Rong

» Get access to the full-text

Important notes

Publisher’s note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Foundng information

This work is supported by NSFC (Grant No. 11571274) and China Scholarship Council grant (2017062 80334).

Abstract

In this paper, we present a second-order decoupled scheme based on the artificial compression method for the time-dependent Stokes-Darcy equations. This method not only uncouples the velocity and hydraulic head by implicit-explicit method but also uncouples the velocity and pressure by artificial compression method; therefore, it only requires one velocity, one pressure, and one hydraulic head problem at each time step by treating the coupling terms explicitly and relaxing the incompressibility constraint. We derive the long-time stability for the velocity and the hydraulic head and give the error analysis for the fully discrete scheme with finite element spatial discretization. Numerical tests are presented to show the accuracy and efficiency of this method.

Please log in to get access to this content

To get access to this content you need the following product:

Springer Professional "Wirtschaft"

Online-Abonnement

Mit Springer Professional "Wirtschaft" erhalten Sie Zugriff auf:

über 58.000 Bücher
über 300 Zeitschriften

aus folgenden Fachgebieten:

Bauwesen + Immobilien
Business IT + Informatik
Finance + Banking
Management + Führung
Marketing + Vertrieb
Versicherung + Risiko

Testen Sie jetzt 30 Tage kostenlos.

inform now

Springer Professional "Technik"

Online-Abonnement

Mit Springer Professional "Technik" erhalten Sie Zugriff auf:

über 50.000 Bücher
über 380 Zeitschriften

aus folgenden Fachgebieten:

Automobil + Motoren
Bauwesen + Immobilien
Business IT + Informatik
Elektrotechnik + Elektronik
Energie + Umwelt
Maschinenbau + Werkstoffe

Testen Sie jetzt 30 Tage kostenlos.

inform now

Springer Professional "Wirtschaft+Technik"

Online-Abonnement

Mit Springer Professional "Wirtschaft+Technik" erhalten Sie Zugriff auf:

über 69.000 Bücher
über 500 Zeitschriften

aus folgenden Fachgebieten:

Automobil + Motoren
Bauwesen + Immobilien
Business IT + Informatik
Elektrotechnik + Elektronik
Energie + Umwelt
Finance + Banking
Management + Führung
Marketing + Vertrieb
Maschinenbau + Werkstoffe
Versicherung + Risiko

Testen Sie jetzt 30 Tage kostenlos.

inform now

Layton, W.J., Schieweck, F., Yotov, I.: Coupling fluid flow with porous media flow. SIAM J. Numer. Anal. 40(6), 2195–2218 (2017) MATHCrossRefMathSciNet

Discacciati, M., Miglio, E., Quarteroni, A.: Mathematical and numerical models for coupling surface and groundwater flows. Appl. Numer. Math. 43(1–2), 57–74 (2002) MATHCrossRefMathSciNet

Gatica, G.N., Meddahi, S., Oyarźua, R.: A conforming mixed finite element method for the coupling of fluid flow with porous media flow. IMA J. Numer. Anal. 29(1), 86–108 (2009) MATHCrossRefMathSciNet

Cao, Y.Z., Gunzburger, M., Hu, X.L., Hua, F., Wang, X.M., Zhao, W.D.: Finite element approximations for Stokes-Darcy flow with Beavers-Joseph interface conditions. SIAM J. Numer. Anal. 47(6), 4239–4256 (2010) MATHCrossRefMathSciNet

Shan, L., Zheng, H.B.: Partitioned time stepping method for fully evolutionary Stokes-Darcy flow with Beavers-Joseph interface boundary condition. SIAM J. Numer. Anal. 51(2), 813–839 (2013) MATHCrossRefMathSciNet

Zuo, L.Y., Du, G.Z.: A parallel two-grid linearized method for the coupled Navier-Stokes-Darcy problem. Numer. Algor. 151-165, 77 (2018) MATHMathSciNet

Li, R., Li, J., He, X.M., Chen, Z.X.: A stabilized finite volume element method for a coupled Stokes-Darcy problem. Appl. Numer. Math. https://doi.org/10.1016/j.apnum.2017.09.013 (2017)

Li, R., Gao, Y.L., Li, J., Chen, Z.X.: A weak Galerkin finite element method for a coupled Stokes-Darcy problem on general meshes. J. Comput. Appl. Math. 111-127, 334 (2018) MATHMathSciNet

Riviére, B.: Analysis of a discontinuous finite element method for the coupled Stokes and Darcy problems. J. Sci. Comput. 22/23(1), 479–500 (2005) MATHCrossRefMathSciNet

10.

Wang, G., He, Y.N., Li, R.: Discontinuous finite volume methods for the stationary Stokes-Darcy problem. Int. J. Numer. Meth. Eng. 107(5), 395–418 (2016) MATHCrossRefMathSciNet

11.

Galvis, J., Sarkis, M.: Non-matching mortar discretization analysis for the coupling Stokes-Darcy equations. Electron. Trans. Numer. Anal. 26(29), 350–384 (2007) MATHMathSciNet

12.

Discacciati, M., Quarteroni, A., Valli, A.: Robin-Robin domain decomposition methods for the Stokes-Darcy coupling. SIAM J. Numer. Anal. 45(3), 1246–1268 (2007) MATHCrossRefMathSciNet

13.

Zuo, L.Y., Hou, Y.R.: A two-grid decoupling method for the mixed Stokes-Darcy model. J. Comput. Appl. Math. 275(275), 139–147 (2015) MATHCrossRefMathSciNet

14.

Mu, M., Xu, J.C.: A two-grid method of a mixed Stokes-Darcy model for coupling fluid flow with porous media flow. SIAM J. Numer. Anal. 45(5), 1801–1813 (2007) MATHCrossRefMathSciNet

15.

Hou, Y.R: Optimal error estimates of a decoupled scheme based on two-grid finite element for mixed Stokes-Darcy model. Appl. Math. Lett. 57, 90–96 (2016) MATHCrossRefMathSciNet

16.

Cai, M.C., Mu, M.: A multilevel decoupled method for a mixed Stokes/Darcy model. J. Comput. Appl. Math. 236(9), 2452–2465 (2012) MATHCrossRefMathSciNet

17.

Mu, M., Zhu, X.H.: Decoupled schemes for a non-stationary mixed Stokes-Darcy model. Math. Comput. 79(270), 707–731 (2010) MATHCrossRefMathSciNet

18.

Layton, W., Tran, H., Trenchea, C.: Analysis of long time stability and errors of two partitioned methods for uncoupling evolutionary groundwater-surface water flows. SIAM J. Numer. Anal. 51(51), 248–272 (2013) MATHCrossRefMathSciNet

19.

Layton, W., Tran, H., Xiong, X.: Long time stability of four methods for splitting the evolutionary Stokes-Darcy problem into Stokes and Darcy subproblems. J. Comput. Appl. Math. 236(13), 3198–3217 (2012) MATHCrossRefMathSciNet

20.

Li, Y., Hou, Y.R., Li, R.: A stabilized finite volume method for the evolutionary Stokes-Darcy system. Computers and Mathematics with Applications 75(2), 596–613 (2018) MATHCrossRefMathSciNet

21.

Li, R., Gao, Y.L., Li, J., Chen, Z.X.: Discontinuous finite volume element method for a coupled non-stationary Stokes-Darcy problem. J Sci Comput. 74(2), 693–727 (2018) MATHCrossRefMathSciNet

22.

Chen, W.B., Gunzburger, M., Sun, D., Wang, X.M.: Efficient and long-time accurate second-order methods for Stokes-Darcy System. SIAM J. Numer. Anal. 51(5), 2563–2584 (2013) MATHCrossRefMathSciNet

23.

Chen, W.B., Gunzburger, M., Sun, D., Wang, X.M.: Efficient and long-time accurate third-order methods for Stokes-Darcy system. Numerische Mathematik 134 (4), 857–879 (2016) MATHCrossRefMathSciNet

24.

Michaela, K., Marina, M.: Analysis of a second-order, unconditionally stable, partitioned method for the evolutionary Stokes-Darcy model. Int. J. Numer. Anal. Model. 12(4), 704–730 (2015) MathSciNet

25.

Shan, L., Zheng, H.B., Layton, W.: A decoupling method with different sub-domain time steps for the nonstationary Stokes-Darcy model. Numer. Methods Partial Differential Equations 29(2), 549–583 (2013) MATHCrossRefMathSciNet

26.

Rybak, I., Magiera, J.: A multiple-time-step technique for coupled free flow and porous medium systems. J. Comput. Phys. 272(5), 327–342 (2014) MATHCrossRefMathSciNet

27.

Li, Y., Hou, Y.R.: A second-order partitioned method with different subdomain time steps for the evolutionary Stokes-Darcy system. Math. Methods Appl. Sci. 41 (5), 2178–2208 (2018) MATHCrossRefMathSciNet

28.

Shen, J.: On error estimates of projection methods for the Navier-Stokes equations: first order schemes. SIAM J. Numer. Anal. 29(1), 57–77 (1992) MATHCrossRefMathSciNet

29.

Shen, J.: On error estimates of the projection method for the Navier-Stokes equations: second order schemes. Math. Comp. 65, 1039–1065 (1996) MATHCrossRefMathSciNet

30.

Guermond, J.L., Minev, P., Shen, J.: An overview of projection methods for incompressible flows. Comput. Methods Appl. Mech. Engrg. 195, 6011–6045 (2006) MATHCrossRefMathSciNet

31.

Shen, J.: On error estimates of the penalty method for unsteady Navier-Stokes equations. SIAM J. Numer. Anal. 32(2), 386–403 (1995) MATHCrossRefMathSciNet

32.

Kan, J.: A second order accurate pressure-correction scheme for viscous incompressible flow. SIAM J. Sci. Computing. 7, 870–891 (1986) MATHCrossRefMathSciNet

33.

Chorin, A.J.: Numerical solution of the Navier-Stokes equations. Math. Comp. 22, 745–762 (1968) MATHCrossRefMathSciNet

34.

Chen, Z.X.: Finite Element Methods and their Applications. Scientific Computation. Springer, Berlin (2005)

35.

Hecht, F., Pironneau, O., Ohsuka, K.: FreeFEM++. Available at: http://www.freefem.org/ff++/ (2010)

36.

Shen, J.: On a new pseudocompressibility method for the incompressible Navier-Stokes equations. Applied Numerical mathematics 21, 71–90 (1996) MATHCrossRefMathSciNet

37.

DeCaria, V., Layton, W., McLaughlin, M.: A conservative, second order, unconditionally stable artificial compression method. Comput. Methods Appl. Mech. Engrg. 733-747, 325 (2017) MATHMathSciNet

38.

Kubacki, M., Moraiti, M.: Analysis of a second-order, unconditionally stable, partitioned method for the evolutionary Stokes-Darcy model. Int. J. Numer. Anal. Model. 12(4), 704–730 (2015) MathSciNet

Title: A second-order artificial compression method for the evolutionary Stokes-Darcy system
Authors:: Yi Li
Yanren Hou
Yao Rong
Publication date: 22-08-2019
DOI: https://doi.org/10.1007/s11075-019-00791-x
Publisher: Springer US
Journal: Numerical Algorithms
Issue 3/2020
Print ISSN: 1017-1398
Electronic ISSN: 1572-9265

Premium Partner

Image Credits

Springer Professional

Hint

Swipe to navigate through the articles of this issue

Publisher’s note

Foundng information

Abstract

Please log in to get access to this content

To get access to this content you need the following product:

Springer Professional "Wirtschaft"

Springer Professional "Technik"

Springer Professional "Wirtschaft+Technik"

Other articles of this Issue 3/2020

Generation of point sets by convex optimization for interpolation in reproducing kernel Hilbert spaces

On the convergence analysis of the gradient-CQ algorithms for the split feasibility problem

Reconstruction algorithms of an inverse source problem for the Helmholtz equation

An adaptive local discontinuous Galerkin method for nonlinear two-point boundary-value problems

An explicit six-step singularly P-stable Obrechkoff method for the numerical solution of second-order oscillatory initial value problems

Condition numbers of the multidimensional total least squares problems having more than one solution

Premium Partner