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

• Autorzy: Lutz Angermann , Joachim Rang
• Języki publikacji: EN

Abstrakty

EN

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.

Słowa kluczowe

EN

Partial differential-algebraic equations; perturbation index

• Wydawca: De Gruyter Open
• Czasopismo: Open Mathematics
• Rocznik: 2007
• Tom: 5
• Numer: 1
• Strony: 19-49

Daty

wydano

2007-03-01

online

2007-03-01

Twórcy

autor

Lutz Angermann

• TU Clausthal

autor

Joachim Rang

• Otto-von-Guericke Universität Magdeburg

Bibliografia

• [1] G.K. Batchelor: An introduction to fluid dynamics, 2nd ed. Cambridge University Press, Cambridge, 1999.
• [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. http://dx.doi.org/10.1007/BF01212349
• [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.
• [4] A. Favini and A. Yagi: Degenerate differential equations in Banach spaces, Marcel Dekker, New York-Basel-Hong Kong, 1999.
• [5] K.O. Friedrichs: “Symmetric positive linear differential equations”, Comm. Pure Appl. Math, Vol. 11, (1958), pp. 333–418.
• [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.
• [7] E. Griepentrog and R. März: Differential-algebraic equations and their numerical treatment, Teubner-Texte zur Mathematik, Vol. 88, Teubner, Leipzig, 1986.
• [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. http://dx.doi.org/10.1137/S1064827598349057
• [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] 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. http://dx.doi.org/10.1016/j.cma.2005.10.007
• [11] P. Kunkel and V. Mehrmann: Differential-Algebraic Equations, EMS Publishing House, Zürich, 2006.
• [12] J. Lang: Adaptive Multilevel Solution of Nonlinear Parabolic PDE Systems, Lecture Notes in Computational Science and Engineering, Vol. 16, Springer-Verlag, Berlin, 2001.
• [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. http://dx.doi.org/10.1007/BF01436628
• [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.
• [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.
• [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. http://dx.doi.org/10.1137/S1064827598332229
• [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.
• [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.
• [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. http://dx.doi.org/10.1016/j.apnum.2004.08.017
• [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. http://dx.doi.org/10.1007/s10543-005-0035-y
• [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.
• [23] J. Rang: Stability estimates and numerical methods for degenerate parabolic differential equations, PhD thesis, Technische Universität Clausthal, Clausthal, 2004.
• [24] R.E. Showalter: Monotone operators in Banach spaces and nonlinear partial differential equations, AMS, Providence, 1997.
• [25] C. Tischendorf: Coupled systems of differential algebraic and partial differential equations in circuit and device simulation Habilitation Thesis, Humboldt University at Berlin, 2003.

Identyfikatory

DOI

10.2478/s11533-006-0035-4