On the Design of Algebraic Fractional Step Methods for Viscoelastic Incompressible Flows

  • Ramon CodinaEmail author
Part of the SEMA SIMAI Springer Series book series (SEMA SIMAI, volume 17)


Classical fractional step methods for viscous incompressible flows aim to uncouple the calculation of the velocity and the pressure. In the case of viscoelastic flows, a new variable appears, namely, a stress, which has an elastic and a viscous contribution. The purpose of this article is to present two families of fractional step methods for the time integration of this type of flows whose objective is to permit the uncoupled calculation of velocities, stresses and pressure, both families designed at the algebraic level. This means that the splitting of the equations is introduced once the spatial and the temporal discretizations have been performed. The first family is based on the extrapolation of the pressure and the stress in order to predict a velocity, then the calculation of a new stress, the pressure and then a correction to render the scheme stable. The second family has a discrete pressure Poisson equation as starting point; in this equation, velocities and stresses are extrapolated to compute a pressure, and from this pressure stresses and velocities can then be computed. This work presents an overview of methods previously proposed in our group, as well as some new schemes in the case of the second family.


Fractional step schemes Viscoelastic flows Pressure extrapolation Velocity extrapolation Inexact factorization 



The author wishes to acknowledge the financial support received from project Elastic-Flow, ref. DPI2015-67857-R, from the Retos Investigación program of the Spanish Ministerio de Economía y Competitividad.


  1. 1.
    Badia, S., Codina, R.: Algebraic pressure segregation methods for the incompressible Navier-Stokes equations. Arch. Comput. Methods Eng. 15, 1–52 (2007)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Badia, S., Codina, R.: Pressure segregation methods based on a discrete pressure Poisson equation. An algebraic approach. Int. J. Numer. Methods Fluids 56, 351–382 (2008)MathSciNetzbMATHGoogle Scholar
  3. 3.
    Blasco, J., Codina, R.: Error estimates for an operator splitting method for incompressible flows. Appl. Numer. Math. 51, 1–17 (2004)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Castillo, E., Codina, R.: Variational multi-scale stabilized formulations for the stationary three-field incompressible viscoelastic flow problem. Comput. Methods Appl. Mech. Eng. 279, 579–605 (2014)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Castillo, E., Codina, R.: First, second and third order fractional step methods for the three-field viscoelastic flow problem. J. Comput. Phys. 296, 113–137 (2015)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Chorin, A.J.: A numerical method for solving incompressible viscous flow problems. J. Comput. Phys. 2, 12–26 (1967)CrossRefGoogle Scholar
  7. 7.
    Codina, R.: Pressure stability in fractional step finite element methods for incompressible flows. J. Comput. Phys. 170, 112–140 (2001)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Codina, R.: Finite element approximation of the three-field formulation of the Stokes problem using arbitrary interpolations. SIAM J. Numer. Anal. 47, 699–718 (2009)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Codina, R., Badia, S.: On some pressure segregation methods of fractional-step type for the finite element approximation of incompressible flow problems. Comput. Methods Appl. Mech. Eng. 195, 2900–2918 (2006)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Fernández-Cara, E., Guillén, F., Ortega, R.R.: Mathematical modeling and analysis of viscoelastic fluids of the Oldroyd kind. In: Handbook of Numerical Analysis, VIII. North-Holland, Amsterdam (2002)Google Scholar
  11. 11.
    Guermond, J.L., Minev, P., Shen, J.: An overview of projection methods for incompressible flows. Comput. Methods Appl. Mech. Eng. 195, 6011–6045 (2006)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Owen, H., Codina, R.: A third-order velocity correction scheme obtained at the discrete level. Int. J. Numer. Methods Fluids 69, 57–72 (2012)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Perot, J.B.: An analysis of the fractional step method. J. Comput. Phys. 108, 51–58 (1993)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Quarteroni, A., Saleri, F., Veneziani, A.: Factorization methods for the numerical approximation of Navier-Stokes equations. Comput. Methods Appl. Mech. Eng. 188, 505–526 (2000)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Saramito, P.: A new θ-scheme algorithm and incompressible FEM for viscoelastic fluid flows. ESAIM Math. Model. Numer. Anal. Modél. Math. Anal. Numér. 28, 1–35 (1994)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Temam, R.: Sur l’approximation de la solution des équations de Navier-Stokes par la méthode des pas fractionaires (I). Arch. Ration. Mech. Anal. 32, 135–153 (1969)CrossRefGoogle Scholar

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© Springer Nature Switzerland AG 2018

Authors and Affiliations

  1. 1.Universitat Politècnica de CatalunyaBarcelonaSpain

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