On the Numerical Analysis of Overdetermined Linear Partial Differential Systems

  • Marcus Hausdorf
  • Werner M. Seiler
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 2630)


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.


Algebraic Equation Computer Algebra Triangular Form Underlying System Multi Index 
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  1. 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. 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.zbMATHGoogle Scholar
  3. 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. 4.
    S.L. Campbell and C.W. Gear. The index of general nonlinear DAEs. Numer. Math., 72:173–196, 1995.zbMATHCrossRefMathSciNetGoogle Scholar
  5. 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. 6.
    V.P. Gerdt and Yu.A. Blinkov. Involutive bases of polynomial ideals. Math. Comp. Simul., 45: 519–542, 1998.zbMATHCrossRefMathSciNetGoogle Scholar
  7. 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. 8.
    E. Hairer and G. Wanner. Solving Ordinary Differential Equations II. Springer Series in Computational Mathematics 14. Springer-Verlag, Berlin, 1996.zbMATHGoogle Scholar
  9. 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.zbMATHCrossRefMathSciNetGoogle Scholar
  10. 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. 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.zbMATHCrossRefGoogle Scholar
  12. 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.zbMATHGoogle Scholar
  13. 13.
    P. Kunkel and V. Mehrmann. Canonical forms for linear differential-algebraic equations with variable coefficients. J. Comp. Appl. Math., 56:225–251, 1994.zbMATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    G. Le Vey. Some remarks on solvability and various indices for implicit differential equations. Num. Algo., 19:127–145, 1998.zbMATHCrossRefGoogle Scholar
  15. 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.zbMATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    Y.O. Macutan and G. Thomas. Theory of formal integrability and DAEs: Effective computations. Num. Algo., 19:147–157, 1998.zbMATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    J.F. Pommaret. Systems of Partial Differential Equations and Lie Pseudogroups. Gordon & Breach, London, 1978.zbMATHGoogle Scholar
  18. 18.
    P.J. Rabier and W.C. Rheinboldt. A geometric treatment of implicit differential algebraic equations. J. Diff. Eq., 109:110–146, 1994.zbMATHCrossRefMathSciNetGoogle Scholar
  19. 19.
    S. Reich. On an existence and uniqueness theory for nonlinear differential-algebraic equations. Circ. Sys. Sig. Proc., 10:343–359, 1991.zbMATHCrossRefGoogle Scholar
  20. 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.zbMATHCrossRefMathSciNetGoogle Scholar
  21. 21.
    M. Renardy and R.C. Rogers. An Introduction to Partial Differential Equations. Texts in Applied Mathematics 13. Springer-Verlag, New York, 1993.zbMATHGoogle Scholar
  22. 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. 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. 24.
    W.M. Seiler. Completion to involution and semi-discretisations. Appl. Num. Math., 42:437–451, 2002.zbMATHCrossRefMathSciNetGoogle Scholar
  25. 25.
    J. Tuomela and T. Arponen. On the numerical solution of involutive ordinary differential systems. IMA J. Num. Anal., 20:561–599, 2000.zbMATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2003

Authors and Affiliations

  • Marcus Hausdorf
    • 1
  • Werner M. Seiler
    • 1
  1. 1.Lehrstuhl für Mathematik IUniversität MannheimMannheimGermany

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