Skip to main content
SpringerLink
Log in

On the Long-Time H 2-Stability of the Implicit Euler Scheme for the 2D Magnetohydrodynamics Equations

  • Published:
Journal of Scientific Computing Aims and scope Submit manuscript

Abstract

Pursuing our work in Tone (Asymptot. Analysis 51:231–245, 2007) and Tone and Wirosoetisno (SIAM J. Number. Analysis 44:29–40, 2006), we consider in this article the two-dimensional magnetohydrodynamics equations, we discretize these equations in time using the implicit Euler scheme and with the aid of the classical and uniform discrete Gronwall lemma, we prove that the scheme is H 2-uniformly stable in time.

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

Access this article

Log in via an institution

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

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Bernardi, C., Maday, Y.: Approximations Spectrales de Problèmes aux Limites Elliptiques. Mathématiques & Applications, vol. 10, Springer, Berlin (1992)

    MATH  Google Scholar 

  2. Girault, V., Raviart, P.-A.: Finite Element Approximation of the Navier-Stokes Equations. Lecture Notes in Mathematics, vol. 749, Springer, Berlin (1979)

    MATH  Google Scholar 

  3. Gottlieb, D., Orszag, S.: Numerical Analysis of Spectral Methods, Theory and Applications, no. 26 in CBMS-NSF Regional Conference Series in Applied Mathematics, SIAM, Philadelphia (1977)

  4. Heywood, J.G., 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. Analysis 19, 275–311 (1982)

    Article  MATH  MathSciNet  Google Scholar 

  5. Landau, L., Lifschitz, E.: Electrodynamique des Milieux Continus, Physique Théorique. Moscow (1969)

  6. Marion, M., Temam, R.: Navier–Stokes Equations. Theory and Approximation. Handbook of Numerical Analysis, vol. VI, pp. 503–688. North-Holland, Amsterdam (1998)

    Google Scholar 

  7. Shen, J.: Long time stabilities and convergences for the fully discrete nonlinear Galerkin methods. Appl. Anal. 38, 201–229 (1990)

    Article  MATH  MathSciNet  Google Scholar 

  8. Temam, R.: Infinite-Dimensional Dynamical Systems in Mechanics and Physics. 2nd edn. Applied Mathematical Sciences, vol. 68. Springer, New York (1997)

    MATH  Google Scholar 

  9. Temam, R.: Navier–Stokes Equations. Theory and Numerical Analysis, AMS Chelsea, Providence, RI, 2001. First published by North-Holland in 1977

  10. Tone, F.: The H 2-stability for all time of the 2d periodic Navier–Stokes equations. Asymptot. Analysis 51, 231–245 (2007)

    MATH  MathSciNet  Google Scholar 

  11. Tone, F., Wirosoetisno, D.: On the long-time stability of the implicit Euler scheme for the 2D Navier–Stokes equations. SIAM J. Numer. Analysis 44, 29–40 (2006)

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics and Statistics, University of West Florida, Pensacola, FL, 32514, USA

    Florentina Tone

Authors
  1. Florentina Tone
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Florentina Tone.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Tone, F. On the Long-Time H 2-Stability of the Implicit Euler Scheme for the 2D Magnetohydrodynamics Equations. J Sci Comput 38, 331–348 (2009). https://doi.org/10.1007/s10915-008-9236-2

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10915-008-9236-2

Keywords

Search

Navigation