Skip to main content
SpringerLink
Log in

Convergence of a Time Discretisation for Doubly Nonlinear Evolution Equations of Second Order

  • Published:
Foundations of Computational Mathematics Aims and scope Submit manuscript

Abstract

The convergence of a time discretisation with variable time steps is shown for a class of doubly nonlinear evolution equations of second order. This also proves existence of a weak solution. The operator acting on the zero-order term is assumed to be the sum of a linear, bounded, symmetric, strongly positive operator and a nonlinear operator that fulfils a certain growth and a Hölder-type continuity condition. The operator acting on the first-order time derivative is a nonlinear hemicontinuous operator that fulfils a certain growth condition and is (up to some shift) monotone and coercive.

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. G. Andreassi, G. Torelli, Si una equazione di tipo iperbolico non lineare, Rend. Sem. Mat. Univ. Padova 34, 224–241 (1964).

    MathSciNet  Google Scholar 

  2. D. Bahuguna, Application of Rothe’s method to semilinear hyperbolic equations, Appl. Anal. 33, 233–242 (1989).

    Article  MATH  MathSciNet  Google Scholar 

  3. V. Barbu, Nonlinear Semigroups and Differential Equations in Banach Spaces (Noordhoff, Leyden, 1976).

    MATH  Google Scholar 

  4. P. Colli, A. Favini, Time discretization of nonlinear Cauchy problems applying to mixed hyperbolic-parabolic equations, Int. J. Math. Math. Sci. 19(3), 481–494 (1996).

    Article  MATH  MathSciNet  Google Scholar 

  5. E. Emmrich, Two-step BDF time discretisation of nonlinear evolution problems governed by monotone operators with strongly continuous perturbations, Comput. Methods Appl. Math. 6(4), 827–843 (2008).

    MATH  MathSciNet  Google Scholar 

  6. E. Emmrich, Variable time-step ϑ-scheme for nonlinear evolution equations governed by a monotone operator, Calcolo 46(3), 187–210 (2009).

    Article  MATH  MathSciNet  Google Scholar 

  7. E. Emmrich, M. Thalhammer, Stiffly accurate Runge–Kutta methods for nonlinear evolution problems governed by a monotone operator, Math. Comput. Published on line first (DOI:10.1090/S0025-5718-09-02285-6).

  8. H.O. Fattorini, Second Order Linear Differential Equations in Banach Spaces, North-Holland Mathematics Studies, vol. 108 (Elsevier, Amsterdam, 1985).

    MATH  Google Scholar 

  9. A. Friedman, J. Nečas, Systems of nonlinear wave equations with nonlinear viscosity, Pac. J. Math. 135(1), 29–55 (1988).

    MATH  Google Scholar 

  10. H. Gajewski, K. Gröger, K. Zacharias, Nichtlineare Operatorgleichungen und Operatordifferentialgleichungen (Akademie, Berlin, 1974).

    MATH  Google Scholar 

  11. J.M. Greenberg, On the existence, uniqueness, and stability of solutions of the equation ρ 0 X tt =E(X x )X xx +λ X xxt , J. Math. Anal. Appl. 25, 575–591 (1969).

    Article  MATH  MathSciNet  Google Scholar 

  12. J.M. Greenberg, R.C. MacCamy, V.J. Mizel, On the existence, uniqueness, and stability of solutions of the equation σ′(u x )u xx +λ u xtx =ρ 0 u tt , J. Math. Mech. 17, 707–728 (1967/1968).

    MathSciNet  Google Scholar 

  13. J.-L. Lions, Quelques Méthodes de Résolution des Problèmes aux Limites Non Linéaires (Dunod, Gauthier-Villars, Paris, 1969).

    MATH  Google Scholar 

  14. J.-L. Lions, W.A. Strauss, Some non-linear evolution equations, Bull. SMF 93, 43–96 (1965).

    MATH  MathSciNet  Google Scholar 

  15. P.A. Raviart, J.M. Thomas, Introduction à l’Analyse Numérique des Équations aux Dérivées Partielles (Masson, Paris, 1983).

    MATH  Google Scholar 

  16. T. Roubíček, Nonlinear Partial Differential Equations with Applications (Birkhäuser, Basel, 2005).

    MATH  Google Scholar 

  17. R.D. Skeel, Variable step size destabilizes the Störmer/leapfrog/Verlet method, BIT 33(1), 172–175 (1993).

    Article  MATH  MathSciNet  Google Scholar 

  18. E. Zeidler, Nonlinear Functional Analysis and Its Applications, II/B: Nonlinear Monotone Operators (Springer, New York, 1990).

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Fakultät für Mathematik, Universität Bielefeld, Postfach 100131, 33501, Bielefeld, Germany

    Etienne Emmrich

  2. Institut für Mathematik, Leopold-Franzens-Universität Innsbruck, Technikerstraße 13/VII, 6020, Innsbruck, Austria

    Mechthild Thalhammer

Authors
  1. Etienne Emmrich
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Mechthild Thalhammer
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Etienne Emmrich.

Additional information

Communicated By Douglas Arnold.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Emmrich, E., Thalhammer, M. Convergence of a Time Discretisation for Doubly Nonlinear Evolution Equations of Second Order. Found Comput Math 10, 171–190 (2010). https://doi.org/10.1007/s10208-010-9061-5

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10208-010-9061-5

Keywords

Mathematics Subject Classification (2000)

Search

Navigation