Skip to content
BY-NC-ND 3.0 license Open Access Published by De Gruyter Open Access March 1, 2007

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

  • Lutz Angermann EMAIL logo and Joachim Rang
From the journal Open Mathematics


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.

[1] G.K. Batchelor: An introduction to fluid dynamics, 2nd ed. Cambridge University Press, Cambridge, 1999. Search in 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. in 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. 10.1137/1.9781611971224Search in Google Scholar

[4] A. Favini and A. Yagi: Degenerate differential equations in Banach spaces, Marcel Dekker, New York-Basel-Hong Kong, 1999. 10.1201/9781482276022Search in Google Scholar

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

[7] E. Griepentrog and R. März: Differential-algebraic equations and their numerical treatment, Teubner-Texte zur Mathematik, Vol. 88, Teubner, Leipzig, 1986. Search in 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. in 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. 10.1007/978-3-642-05221-7_1Search in 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. in Google Scholar

[11] P. Kunkel and V. Mehrmann: Differential-Algebraic Equations, EMS Publishing House, Zürich, 2006. 10.4171/017Search in 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. 10.1007/978-3-662-04484-1Search in Google Scholar

[13] L. Landau and E. Lifschitz: Fluid mechanics, Addison-Wesley, 1953. Search in Google Scholar

[14] P. Lesaint: Finite element methods for symmetric hyperbolic systems, Numer. Math., Vol. 21, (1973), pp. 244–255. in 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. 10.1007/978-1-4612-1116-7Search in 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. Search in 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. in 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. Search in 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. Search in 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. in 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. in 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. Search in Google Scholar

[23] J. Rang: Stability estimates and numerical methods for degenerate parabolic differential equations, PhD thesis, Technische Universität Clausthal, Clausthal, 2004. Search in Google Scholar

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

Published Online: 2007-3-1
Published in Print: 2007-3-1

© 2007 Versita Warsaw

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

Downloaded on 29.11.2023 from
Scroll to top button