Skip to main content
Log in

Perturbation index of linear partial differential-algebraic equations with a hyperbolic part

  • Research Article
  • Published:
Central European Journal of Mathematics

Abstract

This paper deals with linear partial differential-algebraic equations (PDAEs) which have a hyperbolic part. If the spatial differential operator satisfies a Gårding-type inequality in a suitable function space setting, a perturbation index can be defined. Theoretical and practical examples are considered.

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. G.K. Batchelor: An introduction to fluid dynamics, 2nd ed. Cambridge University Press, Cambridge, 1999.

    MATH  Google Scholar 

  2. J.T. Beale, T. Kato and A. Majda: “Remarks on the breakdown of smooth solutions for the 3-D Euler equations”, Comm. Math. Phys., Vol. 94(1), (1984), pp. 61–66.

    Article  MathSciNet  Google Scholar 

  3. K.E. Brenan, S.L. Campbell and L.R. Petzold: Numerical Solution of Initial-Value Problems in DAEs, Classics In Applied Mathematics, Vol. 14 SIAM, Philadelphia, 1996.

    Google Scholar 

  4. A. Favini and A. Yagi: Degenerate differential equations in Banach spaces, Marcel Dekker, New York-Basel-Hong Kong, 1999.

    MATH  Google Scholar 

  5. K.O. Friedrichs: “Symmetric positive linear differential equations”, Comm. Pure Appl. Math, Vol. 11, (1958), pp. 333–418.

    MathSciNet  Google Scholar 

  6. E. Griepentrog, M. Hanke and R. März: Toward a better understanding of differential-algebraic equations (Introductory survey), Seminarberichte Nr. 92-1, Humboldt-Universität zu Berlin, Fachbereich Mathematik, Berlin, 1992.

    Google Scholar 

  7. E. Griepentrog and R. März: Differential-algebraic equations and their numerical treatment, Teubner-Texte zur Mathematik, Vol. 88, Teubner, Leipzig, 1986.

    MATH  Google Scholar 

  8. M. Günther and Y. Wagner: “Index concepts for linear mixed systems of Differential-algebraic and hyperbolic-type equations”, SIAM J. Sci. Comput., Vol. 22(5), (2000), pp. 1610–1629.

    Article  MathSciNet  Google Scholar 

  9. E. Hairer and G. Wanner: Solving ordinary differential equations II: Stiff and differential-algebraic problems, Springer Series in Computational Mathematics, Vol. 14, 2nd edition, Springer-Verlag, Berlin, 1996.

    MATH  Google Scholar 

  10. V. John, G. Matthies and J. Rang: “A comparison of time-discretization/linearization approaches for the incompressible Navier-Stokes equations”, Comput. Methods Appl. Mech. Engrg., Vol. 195(44–47), (2006), pp. 5995–6010.

    Article  MathSciNet  Google Scholar 

  11. P. Kunkel and V. Mehrmann: Differential-Algebraic Equations, EMS Publishing House, Zürich, 2006.

    MATH  Google Scholar 

  12. J. Lang: Adaptive Multilevel Solution of Nonlinear Parabolic PDE Systems, Lecture Notes in Computational Science and Engineering, Vol. 16, Springer-Verlag, Berlin, 2001.

    MATH  Google Scholar 

  13. L. Landau and E. Lifschitz: Fluid mechanics, Addison-Wesley, 1953.

  14. P. Lesaint: Finite element methods for symmetric hyperbolic systems, Numer. Math., Vol. 21, (1973), pp. 244–255.

    Article  MathSciNet  Google Scholar 

  15. A. Majda: Compressible fluid flow and systems of conservation laws in several space variables, Applied Mathematical Sciences, Vol. 53, Springer-Verlag, New York, 1984.

    MATH  Google Scholar 

  16. A. Majda: The interaction of nonlinear analysis and modern applied mathematics, In: Proceedings of the International Congress of Mathematicians, Tokyo, Math. Soc. Japan., (1990), pp. 175–191.

    Google Scholar 

  17. W.S. Martinson and P.I. Barton: A Differentiation Index for Partial Differential Equations, SIAM J. Sci. Comput., Vol. 21(6), (2000), pp. 2295–2315.

    Article  MathSciNet  Google Scholar 

  18. M. Marion and R. Temam: Navier-Stokes equations: theory and approximation, In: P.G. Ciarlet and J.L. Lions (Eds.): Handbook of numerical analysis, Handb. Numer. Anal., Vol. 6, North-Holland, Amsterdam, 1998, pp. 503–688.

    Google Scholar 

  19. J. Rang and L. Angermann: The perturbation index of linearized problems in porous media, Mathematik-Bericht Nr. 2004/1, Institut für Mathematik, TU Clausthal, Clausthal, 2004.

    Google Scholar 

  20. J. Rang and L. Angermann: “The perturbation index of linear partial differential algebraic equations”, Appl. Numer. Math., Vol. 53(2–4), (2005), pp. 437–456.

    Article  MathSciNet  Google Scholar 

  21. J. Rang and L. Angermann: “New Rosenbrock W-methods of order 3 for PDAEs of index 1”, BIT, Vol. 45(4), (2005), pp. 761–787.

    Article  MathSciNet  Google Scholar 

  22. J. Rang and L. Angermann: Remarks on the differentiation index and on the perturbation index of non-linear differential algebraic equations, Mathematik-Bericht Nr. 2005/3, Institut für Mathematik, TU Clausthal, Clausthal, 2005.

    Google Scholar 

  23. J. Rang: Stability estimates and numerical methods for degenerate parabolic differential equations, PhD thesis, Technische Universität Clausthal, Clausthal, 2004.

    Google Scholar 

  24. R.E. Showalter: Monotone operators in Banach spaces and nonlinear partial differential equations, AMS, Providence, 1997.

    Google Scholar 

  25. C. Tischendorf: Coupled systems of differential algebraic and partial differential equations in circuit and device simulation Habilitation Thesis, Humboldt University at Berlin, 2003.

Download references

Author information

Authors and Affiliations

Authors

About this article

Cite this article

Angermann, L., Rang, J. Perturbation index of linear partial differential-algebraic equations with a hyperbolic part. centr.eur.j.math. 5, 19–49 (2007). https://doi.org/10.2478/s11533-006-0035-4

Download citation

  • Received:

  • Accepted:

  • Issue Date:

  • DOI: https://doi.org/10.2478/s11533-006-0035-4

Keywords

MSC (2000)

Navigation