[go: up one dir, main page]
More Web Proxy on the site http://driver.im/ Skip to main content
Springer Nature Link
Log in

A second-order decoupled algorithm with different subdomain time steps for the non-stationary Stokes/Darcy model

  • Original Paper
  • Published:
Numerical Algorithms Aims and scope Submit manuscript

Abstract

In this paper, we propose and analyze a second-order decoupled algorithm with different subdomain time steps for the non-stationary Stokes/Darcy model. It is based on the second-order spectral deferred correction method in time and the finite element method in space. We provide the stability and convergence results of our decoupled scheme. Last, some numerical experiments are given to illustrate the accuracy and effectiveness of our decoupled scheme.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
£29.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price includes VAT (United Kingdom)

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3

Similar content being viewed by others

References

  1. Bernardi, C., Orfi, A.Y.: A priori error analysis of the fully discretized time-dependent coupled Darcy and Stokes equations. Sema J. 38(3), 1–23 (2004)

    MATH  Google Scholar 

  2. Bernardi, C., Orfi, A.Y.: A priori error analysis of the fully discretized time-dependent coupled Darcy and Stokes equations. SeMA J. 73(2), 97–119 (2016)

    Article  MathSciNet  Google Scholar 

  3. Bernardi, C., Orfi, A.Y.: A posteriori error analysis of the fully discretized time-dependent coupled Darcy and Stokes equations. Comput. Math. Appl. 76(2), 340–360 (2018)

    Article  MathSciNet  Google Scholar 

  4. Bourlioux, A., Layton, A. T., Minion, M. L.: High-order multi-implicit spectral deferred correction methods for problems of reactive flow. J. Comput. Phys. 189(2), 651–675 (2003)

    Article  MathSciNet  Google Scholar 

  5. Causley, M., Seal, D.: On the convergence of spectral deferred correction methods. Commun. Appl. Math. Comput. Sci. 14(1), 33–64 (2019)

    Article  MathSciNet  Google Scholar 

  6. Connors, J. M.: Partitioned time discretization for atmosphere-ocean interaction. Ph.D. thesis, University of Pittsburgh (2010)

  7. Connors, J. M., Howell, J. S.: A fluid-fluid interaction method using decoupled subproblems and differing time steps. Numer. Methods Partial Differ. Equ. 28(4), 1283–1308 (2012)

    Article  MathSciNet  Google Scholar 

  8. Discacciati, M.: Domain decomposition methods for the coupling of surface and groundwater flows. Tech. rep. EPFL (2004)

  9. 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. Visual. Sci. 6(2–3), 93–103 (2004)

    Article  MathSciNet  Google Scholar 

  10. Du, G., Li, Q., Zhang, Y: A two-grid method with backtracking for the mixed Navier-Stokes/Darcy model. Numer. Methods Partial Differ. Equ. 36 (6), 1601–1610 (2020)

    Article  MathSciNet  Google Scholar 

  11. Dutt, A., Greengard, L., Rokhlin, V.: Spectral deferred correction methods for ordinary differential equations. Bit Numer. Math. 40(2), 241–266 (2000)

    Article  MathSciNet  Google Scholar 

  12. Girault, V., Kanschat, G., Rivire, B.: Error analysis for a monolithic discretization of coupled Darcy and Stokes problems. J. Numer. Math. 22(2), 109–142 (2014)

    Article  MathSciNet  Google Scholar 

  13. Gunzburger, M., Labovsky, A.: High accuracy method for turbulent flow problems. Math. Models Methods Appl. Sci. 22(6), 1250005 (2012)

    Article  MathSciNet  Google Scholar 

  14. Guo, R., Xu, Y. : High order numerical simulations for the binary fluid-surfactant system using the discontinuous Galerkin and spectral deferred correction methods. SIAM J. Sci. Comput. 42(2), B353–B378 (2020)

    Article  MathSciNet  Google Scholar 

  15. Hamon, F. P., Day, M. S., Minion, M.L.: Concurrent implicit spectral deferred correction scheme for low-Mach number combustion with detailed chemistry. Combust. Theory Model. 23(2), 279–309 (2019)

    Article  MathSciNet  Google Scholar 

  16. Heywood, J. G., Rannacher, R.: Finite-element approximation of the nonstationary Navier–Stokes problem. Part IV: Error analysis for second-order time discretization. SIAM J. Numer. Anal. 27(2), 353–384 (1990)

    Article  MathSciNet  Google Scholar 

  17. Hoppe, R. H., Porta, P., Vassilevski, Y.: Computational issues related to iterative coupling of subsurface and channel flows. Calcolo 44(1), 1–20 (2007)

    Article  MathSciNet  Google Scholar 

  18. Hou, Y.: 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)

    Article  MathSciNet  Google Scholar 

  19. Jia, J., Hill, J. C., Evans, K. J., Fann, G. I., Taylor, M. A.: A spectral deferred correction method applied to the shallow water equations on a sphere. Mon. Weather Rev. 141(10), 3435–3449 (2013)

    Article  Google Scholar 

  20. Layton, W. J., Schieweck, F., Yotov, I.: Coupling fluid flow with porous media flow. SIAM J. Numer. Anal. 40(6), 2195–2218 (2002)

    Article  MathSciNet  Google Scholar 

  21. Liu, F., Shen, J.: Stabilized semi-implicit spectral deferred correction methods for Allen-Cahn and Cahn-Hilliard equations. Math. Methods Appl. Sci. 38(18), 4564–4575 (2015)

    Article  MathSciNet  Google Scholar 

  22. Minion, M. L.: Semi-implicit spectral deferred correction methods for ordinary differential equations. Commun. Math. Sci. 1(3), 471–500 (2003)

    Article  MathSciNet  Google Scholar 

  23. Minion, M. L.: Semi-implicit projection methods for incompressible flow based on spectral deferred corrections. Appl. Numer. Math. 48(3), 369–387 (2004)

    Article  MathSciNet  Google Scholar 

  24. Minion, M. L.: A hybrid parareal spectral deferred corrections method. Commun. Appl. Math. Comput. Sci. 5(5), 265–301 (2011)

    MathSciNet  MATH  Google Scholar 

  25. 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. 45 (5), 1801–1813 (2007)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  27. Peszyńska, M., Wheeler, M. F., Yotov, I.: Mortar upscaling for multiphase flow in porous media. Comput. Geosci. 6(1), 73–100 (2002)

    Article  MathSciNet  Google Scholar 

  28. Rong, Y., Hou, Y., Zhang, Y.: Numerical analysis of a second order algorithm for simplified magnetohydrodynamic flows. Adv. Comput. Math. 43(4), 823–848 (2017)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  30. Rybak, I., Magiera, J., Helmig, R., Rohde, C.: Multirate time integration for coupled saturated/unsaturated porous medium and free flow systems. Comput. Geosci. 19(2), 299–309 (2015)

    Article  MathSciNet  Google Scholar 

  31. Shan, L., Zheng, H., Layton, W.J.: A decoupling method with different subdomain time steps for the nonstationary Stokes-Darcy model. Numer. Methods Partial Differ. Equ. 29(2), 549–583 (2013)

    Article  MathSciNet  Google Scholar 

  32. Shevchenko, I., Kaltenbacher, M., Wohlmuth, B.: A multi-time stepping integration method for the ultrasound heating problem. ZAMM-J. Appl. Math. Mech./Z. Angew. Math. Mech. 92(11–12), 869–881 (2012)

    Article  MathSciNet  Google Scholar 

  33. Xue, D., Hou, Y.: Numerical analysis of a second order algorithm for a non-stationary Navier-Stokes/Darcy model. J. Comput. Appl. Math. 369, 112579 (2020)

    Article  MathSciNet  Google Scholar 

  34. Zuo, L., Du, G.: A multi-grid technique for coupling fluid flow with porous media flow. Comput. Math. Appl. 75(11), 4012–4021 (2018)

    Article  MathSciNet  Google Scholar 

Download references

Funding

This work is supported by the National Natural Science Foundation of China (Grant No. 11971378 & 11571274).

Author information

Authors and Affiliations

  1. School of Mathematics and Statistics, Xi’an Jiaotong University, Xi’an, Shaanxi, 710049, China

    Dandan Xue & Yanren Hou

Authors
  1. Dandan Xue
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Yanren Hou
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Dandan Xue.

Additional information

Publisher’s note

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

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Xue, D., Hou, Y. A second-order decoupled algorithm with different subdomain time steps for the non-stationary Stokes/Darcy model. Numer Algor 88, 1137–1182 (2021). https://doi.org/10.1007/s11075-021-01070-4

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11075-021-01070-4

Keywords

Search

Navigation