Crank–Nicolson Leap-Frog Time Stepping Decoupled Scheme for the Fluid–Fluid Interaction Problems


A fully discrete Crank-Nicolson leap-frog time stepping decoupled (CNLFD) scheme is presented and studied for the fluid–fluid interaction problems. The proposed scheme deals with the spatial discretization by finite element method (FEM), treats the temporal discretization by CNLF scheme and decouples the nonlinear interface condition by using a geometric averaging of the jump. The unconditional stability and error estimate are proven. Numerical tests are performed to demonstrate the robustness of this method.

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

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11

Data Availibility Statement

I confirm that the data in this manuscript will be available and transparent.


  1. 1.

    Achdou, Y., Guermond, J.L.: Analysis of a finite element projection/Lagrange- Galerkin method for the incompressible Navier-Stokes equations. SIAM J. Numer. Anal. 37, 799–826 (2000)

    MathSciNet  MATH  Google Scholar 

  2. 2.

    Alejandro, A., Rodolfo, B.: Finite element modified of characteristics for the Navier-Stokes equations. Int. J. Numer. Method Fluids 32, 439–464 (2000)

    MATH  Google Scholar 

  3. 3.

    Asadzadeh, M., Kazemi, E., Mokhtari, R.: Discrete-ordinates and streamline diffusion methods for a flow described by BGK model. SIAM J. Sci. Comput. 36, 729–748 (2014)

    MathSciNet  MATH  Google Scholar 

  4. 4.

    Boukir, K., Maday, Y., et al.: A high-order characteristics/finite element method for the incompressible Navier-Stokes equations. Int. J. Numer. Methods Fluids 25, 1421–1454 (1997)

    MathSciNet  MATH  Google Scholar 

  5. 5.

    Brenner, S.C., Scott, L.R.: The Mathematical Theory of Finite Element Methods. Springer, Berlin (2002)

    Google Scholar 

  6. 6.

    Bresch, D., Koko, J.: Operator-splitting and Lagrange multiplier domain decomposition methods for numerical simulation of two coupled Navier-Stokes fluids. Int. J. Appl. Math. Comput. Sci. 16, 419–429 (2006)

    MathSciNet  MATH  Google Scholar 

  7. 7.

    Braack, M., Burman, E.: Local projection stabilization for the Oseen problem and its interpetation as a variational multiscale method. SIAM J. Numer. Anal. 6, 2544–2566 (2006)

    MATH  Google Scholar 

  8. 8.

    Burman, E., Fernández, M.A.: Stabilized explicit coupling for fluid-structure interaction using Nitsche’s method. C. R. Math. Acad. Sci. Paris 345, 467–472 (2007)

    MathSciNet  MATH  Google Scholar 

  9. 9.

    Causin, P., Gerbeau, J.F., Nobile, F.: Added-mass effect in the design of partitioned algorithms for fluid-structure problems. Comput. Methods Appl. Mech. Engrg. 194, 4506–4527 (2005)

    MathSciNet  MATH  Google Scholar 

  10. 10.

    Chen, G., Feng, M., Zhou, H.: Local projection stabilized method on unsteady Navier-Stokes equations with high Reynolds number using equal order interpolation. Appl. Math. Comput. 243, 465–481 (2014)

    MathSciNet  MATH  Google Scholar 

  11. 11.

    Connors, J.M., Howell, J.S., Layton, W.J.: Partitioned time stepping for a parabolic two domain problem. SIAM J. Numer. Anal. 47, 3526–3549 (2009)

    MathSciNet  MATH  Google Scholar 

  12. 12.

    Connors, J.M., Howell, J.S., Layton, W.J.: Decoupled time stepping methods for fluid-fluid interaction. SIAM J. Numer. Anal. 50, 1297–1319 (2012)

    MathSciNet  MATH  Google Scholar 

  13. 13.

    Douglas Jr., J., Russell, T.F.: Numerical methods for convection-dominated diffusion problems based on combining the method of characteristics with finite element or finite difference procedures. SIAM J. Numer. Anal. 19, 871–885 (1982)

    MathSciNet  MATH  Google Scholar 

  14. 14.

    Franz, S.: SDFEM with non-standard higher-order finite elements for a convection-diffusion problem with characteristic boundary layers. BIT Numer. Math. 51, 631–651 (2011)

    MathSciNet  MATH  Google Scholar 

  15. 15.

    Franz, S., Kellogg, R., Stynes, M.: Galerkin and streamline diffusion finite element methods on a Shishkin mesh for a convection-diffusion problem with corner singularities. Math. Comput. 81, 661–685 (2012)

    MathSciNet  MATH  Google Scholar 

  16. 16.

    Hansbo, P., Szepessy, A.: A velocity-pressure streamline diffusion finite element method for the incompressible Navier-Stokes equation. Comput. Methods Appl. Mech. Engrg. 84, 175–192 (1990)

    MathSciNet  MATH  Google Scholar 

  17. 17.

    He, Y.N.: Two-level methods based on finite element and Crank-Nicolson extrapolation for the time-dependent Navier-Stokes equations. SIAM J. Numer. Anal. 41, 1263–1285 (2003)

    MathSciNet  MATH  Google Scholar 

  18. 18.

    He, Y.N., Sun, W.W.: Stability and convergence of the Crank-Nicolson/Adams-Bashforth scheme for the time-dependent Navier-Stokes equations. SIAM J. Numer. Anal. 45, 837–869 (2007)

    MathSciNet  MATH  Google Scholar 

  19. 19.

    He, Y.N., Sun, W.W.: Stabilized finite element methods based on Crank-Nicolson extrapolation scheme for the time-dependent Navier-Stokes equations. Math. Comput. 76, 115–136 (2007)

    MathSciNet  MATH  Google Scholar 

  20. 20.

    Hecht, F.: New development in Freefem++. J. Numer. Math. 20, 251–265 (2012)

    MathSciNet  MATH  Google Scholar 

  21. 21.

    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, 353–384 (1990)

    MathSciNet  MATH  Google Scholar 

  22. 22.

    Hurl, N., Layton, W., Li, Y., Trenchea, C.: Stability analysis of the Crank-Nicolson-Leapfrog method with the Robert-Asselin-Williams time filter. BIT Numer. Math. 54, 1009–1021 (2014)

    MathSciNet  MATH  Google Scholar 

  23. 23.

    Huang, Y.Q., Li, J.C., Lin, Q.: Superconvergence analysis for time-dependent Maxwells equations in metamaterials. Numer. Methods for PDEs 28, 1794–1816 (2012)

    MathSciNet  MATH  Google Scholar 

  24. 24.

    Huang, Y.Q., Li, J.C., Yang, W.: Modeling backward wave propagation in metamaterials by the finite element time-domain method. SIAM J. Sci. Comput. 35, B248–B274 (2013)

    MathSciNet  MATH  Google Scholar 

  25. 25.

    Huang, Y.Q., Li, J.C., Yang, W.: Theoretical and numerical analysis of a non-local dispersion model for light interaction with metallic nanostructures. Comput. Math. Appl. 72, 921–932 (2016)

    MathSciNet  MATH  Google Scholar 

  26. 26.

    Huang, Y.Q., Li, J.C., Yang, W., Sun, S.Y.: Superconvergence of mixed finite element approximations to 3-D Maxwells equations in metamaterials. J. Comput. Phy. 230, 8275–8289 (2011)

    MathSciNet  MATH  Google Scholar 

  27. 27.

    Jia, H., Li, K., Liu, S.: Characteristic stabilized finite element method for the transient Navier-Stokes equations. Comput. Methods Appl. Mech. Engrg. 199, 2996–3004 (2010)

    MathSciNet  MATH  Google Scholar 

  28. 28.

    Jiang, N., Kubacki, M., Layton, W., Trenchea, C.: A Crank-Nicolson Leapfrog stabilization: unconditional stability and two applications. J. Comput. Appl. Math. 281, 263–276 (2015)

    MathSciNet  MATH  Google Scholar 

  29. 29.

    Johnson, C., Navert, U., Pitkaranta, J.: Finite element methods for linear hyperbolic problems. Comput. Methods Appl. Mech. Engrg. 45, 285–312 (1985)

    MathSciNet  MATH  Google Scholar 

  30. 30.

    Johnson, C., Saranen, J.: Streamline diffusion methods for the incompressible Euler and Navier-Stokes equations. Math. Comp. 47, 1–18 (1986)

    MathSciNet  MATH  Google Scholar 

  31. 31.

    Kubacki, M.: Uncoupling evolutionary groundwater-surface water flows using the CrankNicolson Leapfrog method. Numer. Methods for PDEs. 29, 1192–1216 (2013)

    MathSciNet  MATH  Google Scholar 

  32. 32.

    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)

    MathSciNet  MATH  Google Scholar 

  33. 33.

    Lehrenfeld, C., Reusken, A.: Nitsche-XFEM with streamline diffusion stabilization for a two-phase mass transport problem. SIAM J. Sci. Comput. 34, 2740–2759 (2012)

    MathSciNet  MATH  Google Scholar 

  34. 34.

    Li, J.C.: Numerical convergence and physical fidelity analysis for Maxwells equations in metamaterials. Comput. Methods Appl. Mech. Engrg. 198, 3161–3172 (2009)

    MathSciNet  MATH  Google Scholar 

  35. 35.

    Li, J.C., Huang, Y.Q., Lin, Y.P.: Developing finite element methods for Maxwells equations in a Cole-Cole dispersive medium. SIAM J. Sci. Comput. 33, 3153–3174 (2011)

    MathSciNet  MATH  Google Scholar 

  36. 36.

    Li, J.C., Waters, J.W., Machorro, E.A.: An implicit leap-frog discontinuous Galerkin method for the time-domain Maxwells equations in metamaterials. Comput. Methods Appl. Mech. Engrg. s223–224, 43–54 (2012)

    MathSciNet  MATH  Google Scholar 

  37. 37.

    Müller, P.: The Equations of Oceanic Motions. Cambridge University Press, Cambridge (2006)

    Google Scholar 

  38. 38.

    Pironneau, O.: On the transport-diffusion algorithm and its application to the Navier-Stokes equations. Numer. Math. 38, 309–332 (1982)

    MathSciNet  MATH  Google Scholar 

  39. 39.

    Qian, L.C., Feng, X.L., He, Y.N.: The characteristic finite difference streamline diffusion method for convection dominated diffusion problems. Appl. Math. Model. 36, 561–572 (2012)

    MathSciNet  MATH  Google Scholar 

  40. 40.

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

    MathSciNet  MATH  Google Scholar 

  41. 41.

    Sun, T.J., Ma, K.Y.: The finite difference streamline-diffusion methods for the incompressible Navier-Stokes equations. Appl. Math. Comput. 149, 493–505 (2004)

    MathSciNet  MATH  Google Scholar 

  42. 42.

    Sun, T.J., Yang, D.P.: The finite difference streamline-diffusion methods for Sobolev equation with convection-dominated term. Appl. Math. Comput. 125, 325–345 (2002)

    MathSciNet  MATH  Google Scholar 

  43. 43.

    Tang, Q.L., Huang, Y.Q.: Stability and convergence analysis of a Crank-Nicolson leap-frog scheme for the unsteady incompressible Navier-Stokes equations. Appl. Numer. Math. 124, 110–129 (2018)

    MathSciNet  MATH  Google Scholar 

  44. 44.

    Zhang, G.D., He, Y.N., Zhang, Y.: Streamline diffusion finite element method for stationary incompressible magnetohydrodynamics. Numer. Methods for Part. Diff. Eqs. 30, 1877–1901 (2014)

    MathSciNet  MATH  Google Scholar 

  45. 45.

    Zhang, Q., Sun, C.: Finite difference-streamline diffusion method for nonlinear convection-diffusion equation. Math. Numer. Sin. 20, 211–224 (1998)

    MathSciNet  MATH  Google Scholar 

  46. 46.

    Zhang, Y.H., Shan, L., Hou, Y.R.: New approach to prove the stability of a decoupled algorithm for a fluid-fluid interaction problem. J. Comput. Appl. Math. 371, (2020)

Download references


The authors sincerely thank the anonymous reviewers and editor for their helpful suggestions.

Author information



Corresponding author

Correspondence to Yinnian He.

Additional information

Publisher's Note

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

This work is in part supported by the NSF of China (Nos. 11861054, U19A2079, 11671345 and 11771348)), Scientific research starting foundation for the high level talents of shihezi university (No. RCSX201732) and the Xinjiang Provincial University Research Foundation of China (No. XJEDU2020I001, XJEDU2020Y001).

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Qian, L., Feng, X. & He, Y. Crank–Nicolson Leap-Frog Time Stepping Decoupled Scheme for the Fluid–Fluid Interaction Problems. J Sci Comput 84, 4 (2020).

Download citation


  • Fluid–fluid interaction problems
  • Nonlinear interface condition
  • Crank-Nicolson leap-frog
  • Decoupled scheme
  • Stability analysis

Mathematics Subject Classification

  • 65N12
  • 65N30
  • 65M15
  • 65M60