Skip to main content
Log in

A stabilized method for transient transport equations

  • Original Paper
  • Published:
Computational Mechanics Aims and scope Submit manuscript

Abstract

We present a stabilized method for the transient advective-reactive-diffusive equation. The discretization is based on Rothe’s method, which discretizes in time before the spatial discretization. The resulting steady state advective-reactive-diffusive equation is approximated by the Unusual Stabilized Finite Element Method (USFEM).

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

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Similar content being viewed by others

References

  1. Valli A, Carey G, Coutinho A (2005) Control strategies for timestep selection in finite element simulation of incompressible flows and coupled reaction-convection-diffusion processes. Int J Numer Methods Fluids 47(3): 201–231

    Article  MATH  MathSciNet  Google Scholar 

  2. Rothe E (1930) Zweidimensionale parabolische Randwertaufgaben als Grenzfall eindimensionaler Randwertaufgaben. Math Ann 102: 650–670

    Article  MATH  MathSciNet  Google Scholar 

  3. Harari I (2004) Stability of semidiscrete formulations for parabolic problems at small time steps. Comput Methods Appl Mech Eng 193(15–16): 1491–1516

    Article  MATH  MathSciNet  Google Scholar 

  4. Harari I, Hauke G (2007) Semidiscrete formulations for transient transport at small time steps. Int J Numer Methods Fluids 54(6–8): 731–743

    Article  MATH  MathSciNet  Google Scholar 

  5. de Sampaio P (2006) A stabilized finite element method for incompressible flow and heat transfer: a natural derivation based on the use of local time-steps. Comput Methods Appl Mech Eng 195(44–47): 6177–6190

    Article  MATH  Google Scholar 

  6. Forster C (2007) Robust methods for fluid–structure interaction with stabilized finite element methods. Ph.D. thesis, Institut fur Baustatik und Baudynamik, Universitat Stuttgart. ISBN 978-3-00-022267-2

  7. Forster C, Wall WA, Ramm E (2009) On residual based stabilisation methods for transient flow problems at small time increments (preprint)

  8. Hsu MC, Bazilevs Y, Calo VM, Tezduyar TE, Hughes TJR (2009) Improving stability of stabilized and multiscale formulations in flow simulations at small time steps. Comp Methods Appl Mech Eng. doi:10.1016/j. cma.2009.06.019

  9. Bochev PB, Gunzburger MD, Lehoucq RB (2007) On stabilized finite element methods for the Stokes problem in the small time step limit. Int J Numer Meth Fluids 53: 573–597

    Article  MATH  MathSciNet  Google Scholar 

  10. Krause R, Walloth M (2009) A time discretization scheme based on Rothe’s method for dynamical contact problems with friction. Comp Methods Appl Mech Eng 199(1–4):1–19

    Article  MathSciNet  Google Scholar 

  11. Franca LP, Farhat C (1995) Bubble functions prompt unusual stabilized finite element methods. Comput Methods Appl Mech Eng 123: 299–308

    Article  MATH  MathSciNet  Google Scholar 

  12. Harari I, Hughes TJR (1994) Stabilized finite element methods for steady advection-diffusion with production. Comput Methods Appl Mech Eng 115(1–2): 165–191

    Article  MathSciNet  Google Scholar 

  13. Valentin F, Franca LP (1995) Combining stabilized finite element methods. Comput Appl Math 14: 285–300

    MATH  MathSciNet  Google Scholar 

  14. Franca LP, Valentin F (2000) On an improved unusual stabilized finite element method for the advective-reactive-diffusive equation. Comput Meth Appl Mech Eng 190: 1785–1800

    Article  MATH  MathSciNet  Google Scholar 

  15. Tezduyar TE, Osawa Y (2000) Finite element stabilization parameters computed from element matrices and vectors. Comput Methods Appl Mech Eng 190: 411–430

    Article  MATH  Google Scholar 

  16. Hauke G, Sangalli G, Doweidar ME (2007) Combining adjoint stabilized methods for the advection-diffusion-reaction problem. Math Model Methods Appl Sci 17(2): 305–326

    Article  MATH  MathSciNet  Google Scholar 

  17. Asensio M et al (2007) Coupling stabilized finite element methods with finite difference time integration for advection-diffusion-reaction problems. Comput Meth Appl Mech Eng 196: 3475–3491

    Article  MATH  MathSciNet  Google Scholar 

  18. Barrenechea G, Blasco J (2007) Pressure stabilization of finite element approximations of time-dependent incompressible flow problems. Comput Meth Appl Mech Eng 197: 219–231

    Article  MATH  MathSciNet  Google Scholar 

  19. Badia S, Codina R (2009) On a multiscale approach to the transient stokes problem: dynamic subscales and anisotropic space–time discretization. Appl Math Comput 207: 415–433

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Alvaro L. G. A. Coutinho.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Henao, C.A.A., Coutinho, A.L.G.A. & Franca, L.P. A stabilized method for transient transport equations. Comput Mech 46, 199–204 (2010). https://doi.org/10.1007/s00466-010-0465-5

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00466-010-0465-5

Keywords

Navigation