Error Analysis of a New Fractional-Step Method for the Incompressible Navier–Stokes Equations with Variable Density


A new fractional-step scheme for the numerical solution to the incompressible Navier–Stokes equations with variable density is proposed. Compared with other known methods, the numerical velocity and pressure are determined by a generalized Stokes problem per time step. The stability and convergence rate \(\mathcal O(\tau )\) of the scheme are proved and the performance of the scheme is numerically illustrated.

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


  1. 1.

    Lions, P.L.: Mathematical Topics in Fluid Mechanics, Volume 1: Incompressible Models, Oxford University Press, Oxford, UK, (1996)

  2. 2.

    Antontsev, S.N., Kazhikhov, A.V., Monakhov, V.N.: Boundary Value Problems in Mechanics of Nonhomogeneous Fluids, Stud. Math. Appl. 22, North-Holland, Amsterdam. Translated from the Russian (1990)

  3. 3.

    Bell, J.B., Marcus, D.L.: A second-order projection method for variable-density flows. J. Comput. Phys. 101, 334–348 (1992)

    Article  Google Scholar 

  4. 4.

    Blasco, J., Codina, R.: Error estimates for an operator-splitting method for incompressible flows. Appl. Numer. Math. 51, 1–17 (2004)

    MathSciNet  Article  Google Scholar 

  5. 5.

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

    MathSciNet  Article  Google Scholar 

  6. 6.

    Weinan, E., Liu, J.G.: Gauge method for viscous incompressible flows. Commun. Math. Sci. 1, 317–332 (2003)

    MathSciNet  Article  Google Scholar 

  7. 7.

    Guillén-González, F., Redondo-Neble, M.V.: New error estiamtes for a viscosity-splitting scheme in time for the three-dimensional Navier–Stokes equations. IMA J. Numer. Anal. 31, 556–579 (2011)

    MathSciNet  Article  Google Scholar 

  8. 8.

    Guermond, J.L., Quartapelle, L.: A projection FEM for variable density incomressible flows. J. Comput. Phys. 165, 167–188 (2000)

    MathSciNet  Article  Google Scholar 

  9. 9.

    Guermond, J.L., Salgado, A.: A splitting method for incompressible flows with variable density based on a pressure Poisson equation. J. Comput. Phys. 228, 2834–2846 (2009)

    MathSciNet  Article  Google Scholar 

  10. 10.

    Guermond, J.L., Salgado, A.: Error analysis of a fractional time-stepping technique for incompressible flows with variable density. SIAM J. Numer. Anal. 49, 917–944 (2011)

    MathSciNet  Article  Google Scholar 

  11. 11.

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

    MathSciNet  Article  Google Scholar 

  12. 12.

    Heywood, J., Rannacher, R.: Finite-element approximation of the nonstationary Navier–Stokes problem. I. Regularity of solutions and second-order error estimates for spatial discretization. SIAM J. Numer. Anal. 19, 275–311 (1982)

    MathSciNet  Article  Google Scholar 

  13. 13.

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

    MathSciNet  Article  Google Scholar 

  14. 14.

    Li, Y., Mei, L.Q., Ge, J.T., Shi, F.: A new fractional time-stepping method for variable density incompressible flows. J. Comput. Phys 242, 124–137 (2013)

    MathSciNet  Article  Google Scholar 

  15. 15.

    Pyo, J.H., Shen, J.: Gauge–Uzawa methods for incomressible flows with variable density. J. Comput. Phys. 221, 181–197 (2007)

    MathSciNet  Article  Google Scholar 

  16. 16.

    Temam, R.: Sur l’approximation de la solution des equations de Navier–Stokes par la méthode des pas fractionnaires II. Arch. Ration. Mech. Anal. 33, 377–385 (1969)

    Article  Google Scholar 

  17. 17.

    Temam, R.: Navier–Stokes Equations. North-Holland Publishing Company, Amsterdam (1977)

    Google Scholar 

  18. 18.

    Walkington, N.J.: Convergence of the discontinuous Galerkin method for discontinuous solutions. SIAM J. Numer. Anal. 42, 180–1817 (2004)

    MathSciNet  Google Scholar 

Download references


This work was supported by National Natural Science Foundation of China with Grant No. 11771337 and by Zhejiang Provincial Natural Science Foundation with Grant Nos. LY18A010021 and LY16A010017.

Author information



Corresponding author

Correspondence to Rong An.

Additional information

Publisher's Note

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

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

An, R. Error Analysis of a New Fractional-Step Method for the Incompressible Navier–Stokes Equations with Variable Density. J Sci Comput 84, 3 (2020).

Download citation


  • Variable density flows
  • Navier–Stokes equations
  • Fractional-step method
  • Stability
  • Convergence rate

Mathematics Subject Classification

  • 65N30
  • 76M05