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International Conference on Symbolic and Numerical Scientific Computation

SNSC 2001: Symbolic and Numerical Scientific Computation pp 152–167Cite as

On the Numerical Analysis of Overdetermined Linear Partial Differential Systems

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Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 2630))

Abstract

We discuss the use of the formal theory of differential equations in the numerical analysis of general systems of partial differential equations. This theory provides us with a very powerful and natural framework for generalising many ideas from differential algebraic equations to partial differential equations. We study in particular the existence and uniqueness of (formal) solutions, the method of an underlying system, various index concepts and the effect of semi-discretisations.

Supported by Deutsche Forschungsgemeinschaft, Landesgraduiertenförderung Baden-Württemberg and INTAS grant 99-1222.

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References

  1. J. Belanger, M. Hausdorf, and W.M. Seiler. A MuPAD library for differential equations. In V.G. Ghanza, E.W. Mayr, and E.V. Vorozhtsov, editors, Computer Algebra in Scientific Computing — CASC 2001, pages 25–42. Springer-Verlag, Berlin, 2001.

    Google Scholar 

  2. K.E. Brenan, S.L. Campbell, and L.R. Petzold. Numerical Solution of Initial-Value Problems in Differential-Algebraic Equations. Classics in Applied Mathematics 14. SIAM, Philadelphia, 1996.

    MATH  Google Scholar 

  3. J. Calmet, M. Hausdorf, and W.M. Seiler. A constructive introduction to involution. In R. Akerkar, editor, Proc. Int. Symp. Applications of Computer Algebra — ISACA 2000, pages 33–50. Allied Publishers, New Delhi, 2001.

    Google Scholar 

  4. S.L. Campbell and C.W. Gear. The index of general nonlinear DAEs. Numer. Math., 72:173–196, 1995.

    Article  MATH  MathSciNet  Google Scholar 

  5. S.L. Campbell and W. Marszalek. The index of an infinite dimensional implicit system. Math. Model. Syst., 1:1–25, 1996.

    Google Scholar 

  6. V.P. Gerdt and Yu.A. Blinkov. Involutive bases of polynomial ideals. Math. Comp. Simul., 45: 519–542, 1998.

    Article  MATH  MathSciNet  Google Scholar 

  7. E. Hairer, C. Lubich, and M. Roche. The Numerical Solution of Differential-Algebraic Equations by Runge-Kutta Methods. Lecture Notes in Mathematics 1409. Springer-Verlag, Berlin, 1989.

    Google Scholar 

  8. E. Hairer and G. Wanner. Solving Ordinary Differential Equations II. Springer Series in Computational Mathematics 14. Springer-Verlag, Berlin, 1996.

    MATH  Google Scholar 

  9. M. Hausdorf and W.M. Seiler. An efficient algebraic algorithm for the geometric completion to involution. Appl. Alg. Eng. Comm. Comp., 13:163–207, 2002.

    Article  MATH  MathSciNet  Google Scholar 

  10. M. Hausdorf and W.M. Seiler. Perturbation versus differentiation indices. In V.G. Ghanza, E.W. Mayr, and E.V. Vorozhtsov, editors, Computer Algebra in Scientific Computing — CASC 2001, pages 323–337. Springer-Verlag, Berlin, 2001.

    Google Scholar 

  11. B.N. Jiang, J. Wu, and L.A. Povelli. The origin of spurious solutions in computational electrodynamics. J. Comp. Phys., 125:104–123, 1996.

    Article  MATH  Google Scholar 

  12. H.-O. Kreiss and J. Lorenz. Initial-Boundary Value Problems and the Navier-Stokes Equations. Pure and Applied Mathematics 136. Academic Press, Boston, 1989.

    MATH  Google Scholar 

  13. P. Kunkel and V. Mehrmann. Canonical forms for linear differential-algebraic equations with variable coefficients. J. Comp. Appl. Math., 56:225–251, 1994.

    Article  MATH  MathSciNet  Google Scholar 

  14. G. Le Vey. Some remarks on solvability and various indices for implicit differential equations. Num. Algo., 19:127–145, 1998.

    Article  MATH  Google Scholar 

  15. W. Lucht, K. Strehmel, and C. Eichler-Liebenow. Indexes and special discretization methods for linear partial differential algebraic equations. BIT, 39:484–512, 1999.

    Article  MATH  MathSciNet  Google Scholar 

  16. Y.O. Macutan and G. Thomas. Theory of formal integrability and DAEs: Effective computations. Num. Algo., 19:147–157, 1998.

    Article  MATH  MathSciNet  Google Scholar 

  17. J.F. Pommaret. Systems of Partial Differential Equations and Lie Pseudogroups. Gordon & Breach, London, 1978.

    MATH  Google Scholar 

  18. P.J. Rabier and W.C. Rheinboldt. A geometric treatment of implicit differential algebraic equations. J. Diff. Eq., 109:110–146, 1994.

    Article  MATH  MathSciNet  Google Scholar 

  19. S. Reich. On an existence and uniqueness theory for nonlinear differential-algebraic equations. Circ. Sys. Sig. Proc., 10:343–359, 1991.

    Article  MATH  Google Scholar 

  20. G.J. Reid, P. Lin, and A.D. Wittkopf. Differential elimination-completion algorithms for DAE and PDAE. Stud. Appl. Math., 106:1–45, 2001.

    Article  MATH  MathSciNet  Google Scholar 

  21. M. Renardy and R.C. Rogers. An Introduction to Partial Differential Equations. Texts in Applied Mathematics 13. Springer-Verlag, New York, 1993.

    MATH  Google Scholar 

  22. W.M. Seiler. Indices and solvability for general systems of differential equations. In V.G. Ghanza, E.W. Mayr, and E.V. Vorozhtsov, editors, Computer Algebra in Scientific Computing — CASC 1999, pages 365–385. Springer-Verlag, Berlin, 1999.

    Google Scholar 

  23. W.M. Seiler. Involution — the formal theory of differential equations and its applications in computer algebra and numerical analysis. Habilitation thesis, Dept. of Mathematics, Universität Mannheim, 2001.

    Google Scholar 

  24. W.M. Seiler. Completion to involution and semi-discretisations. Appl. Num. Math., 42:437–451, 2002.

    Article  MATH  MathSciNet  Google Scholar 

  25. J. Tuomela and T. Arponen. On the numerical solution of involutive ordinary differential systems. IMA J. Num. Anal., 20:561–599, 2000.

    Article  MATH  MathSciNet  Google Scholar 

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Author information

Authors and Affiliations

  1. Lehrstuhl für Mathematik I, Universität Mannheim, 68131, Mannheim, Germany

    Marcus Hausdorf & Werner M. Seiler

Authors
  1. Marcus Hausdorf
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  2. Werner M. Seiler
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Editor information

Editors and Affiliations

  1. RISC-Linz, Johannes Kepler University Linz, Altenberger Str. 69, 4040, Linz, Austria

    Franz Winkler

  2. Institute of Computational Mathematics, Johannes Kepler University Linz, Altenberger Str. 69, 4040, Linz, Austria

    Ulrich Langer

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Hausdorf, M., Seiler, W.M. (2003). On the Numerical Analysis of Overdetermined Linear Partial Differential Systems. In: Winkler, F., Langer, U. (eds) Symbolic and Numerical Scientific Computation. SNSC 2001. Lecture Notes in Computer Science, vol 2630. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-45084-X_6

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  • DOI: https://doi.org/10.1007/3-540-45084-X_6

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  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-40554-2

  • Online ISBN: 978-3-540-45084-9

  • eBook Packages: Springer Book Archive

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