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On the History of Differential-Algebraic Equations

A Retrospective with Personal Side Trips
Chapter
Part of the Differential-Algebraic Equations Forum book series (DAEF)

Abstract

The present article takes an off-the-wall approach to the history of Differential-Algebraic Equations and uses personal side trips and memories of conferences, workshops, and summer schools to highlight some of the milestones in the field. Emphasis is in particular placed on the application fields that set the ball rolling and on the development of numerical methods.

Keywords

Differential-algebraic equations Historical remarks Index notions BDF methods Runge–Kutta methods Partial differential-algebraic equations Constrained mechanical system Electric circuit analysis 

Mathematics Subject Classification (2010):

34A09 65L80 65M20 01-02 34-03 

Notes

Acknowledgements

Over the years, I have had the privilege to meet so many colleagues working in the field of differential-algebraic equations. Our discussions and the stories that were told are an integral part of this survey article, and I would like to thank them all for their invisible but highly acknowledged contribution. It was my academic teacher Peter Rentrop who gave me the chance to do a PhD in this exciting field and who made my participation at various conferences and summer schools possible during the Boom Days. I am more than grateful for his inspiration and support.

Moreover, I wish to sincerely thank all my master and PhD students who worked in this or related fields for their collaboration, their effort, and their patience. Special thanks, finally, goes to Ernst Hairer who read an early version of this manuscript, and to Achim Ilchmann who always encouraged me to continue with this effort.

References

  1. 1.
    Abraham, R., Marsden, J.E., Ratiu, T.: Manifolds, Tensor Analysis, and Applications. Springer, New York (1988)CrossRefzbMATHGoogle Scholar
  2. 2.
    Ali, G., Bartel, A., Günther, M., Tischendorf, C.: Elliptic partial differential-algebraic multiphysics models in electrical network design. Math. Models Methods Appl. Sci. 13 (09), 1261–1278 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Altmann, R.: Index reduction for operator differential-algebraic equations in elastodynamics. Z. Angew. Math. Mech. 93 (9), 648–664 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Andrews, G.C., Ormrod, M.K.: Advent: a simulation program for constrained planar kinematic and dynamic systems. In: Presented at the Design Engineering Technical Conference, Columbus, Ohio, 5–8 October 1986. Departement of Mechanical Engineering, University of Waterloo, Ontario, Canada, N2L 3G1 (1986)Google Scholar
  5. 5.
    Arnold, V.I.: Ordinary Differential Equations. MIT Press, Cambridge (1981)Google Scholar
  6. 6.
    Arnold, M., Murua, A.: Non-stiff integrators for differential–algebraic systems of index 2. Numer. Algorithms 19 (1–4), 25–41 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Arnold, M., Simeon, B.: Pantograph and catenary dynamics: a benchmark problem and its numerical solution. Appl. Numer. Math. 34, 345–362 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Ascher, U., Lin, P.: Sequential regularization methods for nonlinear higher index DAEs. SIAM J. Sci. Comput. 18, 160–181 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Ascher, U.M., Petzold, L.R.: Projected implicit Runge-Kutta methods for differential-algebraic equations. SIAM J. Numer. Anal. 28, 1097–1120 (1991)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Ascher, U., Chin, H., Petzold, L., Reich, S.: Stabilization of constrained mechanical systems with DAEs and invariant manifolds. J. Mech. Struct. Mach. 23: 135–158 (1995)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Bangerth, W., Rannacher, R.: Adaptive Finite Element Methods for Differential Equations. Birkhäuser, Basel (2013)zbMATHGoogle Scholar
  12. 12.
    Baumgarte, J.: Stabilization of constraints and integrals of motion in dynamical systems. Comput. Methods Appl. Mech. 1, 1–16 (1972)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Benner, P., Losse, P., Mehrmann, V., Voigt, M.: Numerical linear algebra methods for linear differential-algebraic equations. In: Ilchmann, A., Reis, T. (eds.) Surveys in Differential-Algebraic Equations III. DAE-Forum, pp. 117–175. Springer, Cham (2015)CrossRefGoogle Scholar
  14. 14.
    Bornemann, F.A.: An adaptive multilevel approach to parabolic equations: II. Variable-order time discretization based on a multiplicative error correction. IMPACT Comput. Sci. Eng. 3 (2), 93–122 (1991)zbMATHGoogle Scholar
  15. 15.
    Brasey, V.: A half-explicit method of order 5 for solving constrained mechanical systems. Computing 48, 191–201 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Brasey, V., Hairer, E.: Half-explicit Runge–Kutta methods for differential-algebraic systems of index 2. SIAM J. Numer. Anal. 30, 538–552 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Brenan, K.E., Campbell, S.L., Petzold, L.R.: The Numerical Solution of Initial Value Problems in Ordinary Differential-Algebraic Equations. SIAM, Philadelphia (1996)zbMATHGoogle Scholar
  18. 18.
    Brezzi, F., Fortin, M.: Mixed and Hybrid Finite Element Methods. Springer, New York (1991)CrossRefzbMATHGoogle Scholar
  19. 19.
    Brizard, A.: An Introduction to Lagrangian Mechanics. World Scientific, Singapore (2008)CrossRefzbMATHGoogle Scholar
  20. 20.
    Campbell, S.L.: Singular Systems of Differential Equations. Pitman, London (1980)zbMATHGoogle Scholar
  21. 21.
    Campbell, S.L.: Singular Systems of Differential Equations II. Research Notes in Mathematics, vol. 61. Pitman, London (1982)Google Scholar
  22. 22.
    Campbell, S.L.: Least squares completions for nonlinear differential-algebraic equations. Numer. Math. 65, 77–94 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Campbell, S., Gear, C.: The index of general nonlinear DAEs. Numer. Math. 72, 173–196 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Callies, R., Rentrop, P.: Optimal control of rigid-link manipulators by indirect methods. GAMM-Mitteilungen 31 (1), 27–58 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Deuflhard, P., Hairer, E., Zugck, J.: One-step and extrapolation methods for differential-algebraic systems. Numer. Math. 51 (5), 501–516 (1987)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Drazin, M.: Pseudo inverses in associative rays and semigroups. Am. Math. Mon. 65, 506–514 (1958)CrossRefzbMATHGoogle Scholar
  27. 27.
    Eich, E.: Convergence results for a coordinate projection method applied to constrained mechanical systems. SIAM J. Numer. Anal. 30 (5), 1467–1482 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Franzone, P.C., Deuflhard, P., Erdmann, B., Lang, J., Pavarino, L.F.: Adaptivity in space and time for reaction-diffusion systems in electrocardiology. SIAM J. Sci. Comput. 28 (3), 942–962 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Führer, C., Leimkuhler, B.: Numerical solution of differential-algebraic equations for constrained mechanical motion. Numer. Math. 59, 55–69 (1991)MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Gantmacher, F.: Matrizenrechnung, Teil 2. VEB Deutscher Verlag der Wissenschaften, Berlin (1959)zbMATHGoogle Scholar
  31. 31.
    Garcia de Jalón, J., Bayo, E.: Kinematic and Dynamic Simulation of Multibody Systems. Springer, New York (1994)Google Scholar
  32. 32.
    Gear, C.W.: Numerical Initial Value Problems in Ordinary Differential Equations. Prentice-Hall, Upper Saddle River (1971)zbMATHGoogle Scholar
  33. 33.
    Gear, C.W.: Simultaneous numerical solution of differential-algebraic equations. IEEE Trans. Circuit Theory CT-18 (1), 89–95 (1971)CrossRefGoogle Scholar
  34. 34.
    Gear, C.W.: Differential-algebraic equation index transformation. SIAM J. Sci. Stat. Comput. 9, 39–47 (1988)MathSciNetCrossRefzbMATHGoogle Scholar
  35. 35.
    Gear, C.W.: Differential-algebraic equations, indices, and integral algebraic equations. SIAM J. Numer. Anal. 27, 1527–1534 (1990)MathSciNetCrossRefzbMATHGoogle Scholar
  36. 36.
    Gear, C.W., Gupta, G., Leimkuhler, B.: Automatic integration of the Euler-Lagrange equations with constraints. J. Comput. Appl. Math. 12 & 13, 77–90 (1985)Google Scholar
  37. 37.
    Griepentrog, E., März, R.: Differential-Algebraic Equations and Their Numerical Treatment. Teubner-Texte zur Mathematik, vol. 88. Teubner Verlagsgesellschaft, Leipzig (1986)Google Scholar
  38. 38.
    Günther, M.: Partielle differential-algebraische Systeme in der numerischen Zeitbereichsanalyse elektrischer Schaltungen. VDI-Verlag, Reihe 20, Düsseldorf (2001)Google Scholar
  39. 39.
    Günther, M., Feldmann, U.: CAD based electric circuit modeling in industry I: mathematical structure and index of network equations. Surv. Math. Ind. 8, 97–129 (1999)MathSciNetzbMATHGoogle Scholar
  40. 40.
    Günther, M., Hoschek, M., Rentrop, P.: Differential-algebraic equations in electric circuit simulation. Int. J. Electron. Commun. 54, 101–107 (2000)Google Scholar
  41. 41.
    Hairer, E., Wanner, G.: Solving Ordinary Differential Equations II: Stiff and Differential-Algebraic Problems. Springer, Berlin (1996)CrossRefzbMATHGoogle Scholar
  42. 42.
    Hairer, E., Wanner, G.: Stiff differential equations solved by Radau methods. J. Comput. Appl. Math. 111, 93–111 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  43. 43.
    Hairer, E., Lubich, C., Roche, M.: Error of Runge–Kutta methods for stiff problems studied via differential algebraic equations. BIT Numer. Math. 28 (3), 678–700 (1988)MathSciNetCrossRefzbMATHGoogle Scholar
  44. 44.
    Hairer, E., Lubich, C., Roche, M.: The Numerical Solution of Differential-Algebraic Equations by Runge-Kutta Methods. Lecture Notes in Mathematics, vol. 1409. Springer, Heidelberg (1989)Google Scholar
  45. 45.
    Hairer, E., Nørsett, S., Wanner, G.: Solving Ordinary Differential Equations I: Nonstiff Problems. Springer, Berlin (1993)zbMATHGoogle Scholar
  46. 46.
    Hairer, E., Lubich, C., Wanner, G.: Geometric Numerical Integration. Springer, Berlin (2002)CrossRefzbMATHGoogle Scholar
  47. 47.
    Hanke, M.: On the regularization of index 2 differential-algebraic equations. J. Math. Anal. Appl. 151 (1), 236–253 (1990)MathSciNetCrossRefzbMATHGoogle Scholar
  48. 48.
    Haug, E.: Computer-Aided Kinematics and Dynamics of Mechanical Systems. Allyn and Bacon, Boston (1989)Google Scholar
  49. 49.
    Higham, D.J., Mao, X., Stuart, A.M.: Strong convergence of Euler-type methods for nonlinear stochastic differential equations. SIAM J. Numer. Anal. 40 (3), 1041–1063 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  50. 50.
    Hindmarsh, A.C., Brown, P.N., Grant, K.E., Lee, S.L., Serban, R., Shumaker, D.E., Woodward, C.S.: Sundials: suite of nonlinear and differential/algebraic equation solvers. ACM Trans. Math. Softw. 31 (3), 363–396 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  51. 51.
    Hughes, T.J., Franca, L.P., Hulbert, G.M.: A new finite element formulation for computational fluid dynamics: VIII. The Galerkin/least-squares method for advective-diffusive equations. Comput. Methods Appl. Mech. Eng. 73 (2), 173–189 (1989)CrossRefzbMATHGoogle Scholar
  52. 52.
    Kirchhoff, G.: Ueber die Auflösung der Gleichungen, auf welche man bei der Untersuchung der linearen Vertheilung galvanischer Ströme geführt wird. Ann. Phys. 148 (12), 497–508 (1847)CrossRefGoogle Scholar
  53. 53.
    Körkel, S., Kostina, E., Bock, H.G., Schlöder, J.P.: Numerical methods for optimal control problems in design of robust optimal experiments for nonlinear dynamic processes. Optim. Methods Softw. 19 (3–4), 327–338 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  54. 54.
    Kronecker, L.: Algebraische Reduktion der Schaaren bilinearer Formen. Akademie der Wissenschaften Berlin III, 141–155 (1890)Google Scholar
  55. 55.
    Kunkel, P., Mehrmann, V.: Numerical solution of differential algebraic Riccati equations. Linear Algebra Appl. 137, 39–66 (1990)MathSciNetCrossRefzbMATHGoogle Scholar
  56. 56.
    Kunkel, P., Mehrmann, V.: Differential-Algebraic Equations – Analysis and Numerical Solution. EMS Publishing House, Zürich (2006)CrossRefzbMATHGoogle Scholar
  57. 57.
    Lagrange, J.L.: Méchanique analytique. Libraire chez la Veuve Desaint, Paris (1788)Google Scholar
  58. 58.
    Lamour, R., März, R., Tischendorf, C.: Differential-Algebraic Equations: A Projector Based Analysis. Differential-Algebraic Equations Forum. Springer, Berlin (2013)CrossRefzbMATHGoogle Scholar
  59. 59.
    Lang, J.: Adaptive Multilevel Solution of Nonlinear Parabolic PDE Systems: Theory, Algorithm, and Applications, vol. 16. Springer, Berlin (2013)Google Scholar
  60. 60.
    Lötstedt, P., Petzold, L.: Numerical solution of nonlinear differential equations with algebraic constraints i: convergence results for BDF. Math.Comput. 46, 491–516 (1986)CrossRefzbMATHGoogle Scholar
  61. 61.
    Lubich, C.: h 2 extrapolation methods for differential-algebraic equations of index-2. Impact Comput. Sci. Eng. 1, 260–268 (1989)Google Scholar
  62. 62.
    Lubich, C.: Integration of stiff mechanical systems by Runge-Kutta methods. ZAMP 44, 1022–1053 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  63. 63.
    Lubich, C., Engstler, C., Nowak, U., Pöhle, U.: Numerical integration of constrained mechanical systems using MEXX. Mech. Struct. Mach. 23, 473–495 (1995)CrossRefGoogle Scholar
  64. 64.
    Lucht, W., Strehmel, K., Eichler-Liebenow, C.: Indexes and special discretization methods for linear partial differential algebraic equations. BIT Numer. Math. 39 (3), 484–512 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  65. 65.
    März, R.: Differential algebraic systems anew. Appl. Numer. Math. 42 (1), 315–335 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  66. 66.
    März, R., Tischendorf, C.: Recent results in solving index-2 differential-algebraic equations in circuit simulation. SIAM J. Sci. Comput. 18, 139–159 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  67. 67.
    Mattson, S., Söderlind, G.: Index reduction in differential-algebraic equations using dummy derivatives. SIAM J. Sci. Comput. 14 (3), 677–692 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  68. 68.
    Müller, P.C.: Stability of linear mechanical systems with holonomic constraints. Appl. Mech. Rev. 46 (11S), S160–S164 (1993)CrossRefGoogle Scholar
  69. 69.
    Müller, P.C.: Stability and optimal control of nonlinear descriptor systems: a survey. Appl. Math. Comput. Sci. 8, 269–286 (1998)MathSciNetzbMATHGoogle Scholar
  70. 70.
    Nagel, L.W., Pederson, D.: Spice (simulation program with integrated circuit emphasis). Technical Report UCB/ERL M382, EECS Department, University of California, Berkeley (1973)Google Scholar
  71. 71.
    O’Malley, R.E.: Introduction to Singular Perturbations. Academic, New York (1974)zbMATHGoogle Scholar
  72. 72.
    Petzold, L.: A description of DASSL: a differential/algebraic system solver. In: Proceedings of 10th IMACS World Congress, Montreal, 8–13 August 1982Google Scholar
  73. 73.
    Plinninger, T., Simeon, B.: Adaptivity in space and time for solving transient problems in COMSOL. In: Proceedings COMSOL Conference Hannover (2008)Google Scholar
  74. 74.
    Rabier, P., Rheinboldt, W.: Theoretical and numerical analysis of differential-algebraic equations. In: Ciarlet, P., Lions, J. (eds.) Handbook of Numerical Analysis, vol. VIII. Elsevier, Amsterdam (2002)Google Scholar
  75. 75.
    Reich, S.: On a geometric interpretation of DAEs. Circ. Syst. Signal Process. 9, 367–382 (1990)MathSciNetCrossRefzbMATHGoogle Scholar
  76. 76.
    Reis, T., Stykel, T.: Stability analysis and model order reduction of coupled systems. Math. Comput. Model. Dyn. Syst. 13 (5), 413–436 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  77. 77.
    Rentrop, P., Roche, M., Steinebach, G.: The application of Rosenbrock–Wanner type methods with stepsize control in differential-algebraic equations. Numer. Math. 55, 545–563 (1989)MathSciNetCrossRefzbMATHGoogle Scholar
  78. 78.
    Rheinboldt, W.: Differential - algebraic systems as differential equations on manifolds. Math. Comput. 43 (168), 2473–482 (1984)MathSciNetCrossRefGoogle Scholar
  79. 79.
    Rheinboldt, W.: Manpak: a set of algorithms for computations on implicitly defined manifolds. Comput. Math. Appl. 32, 15–28 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  80. 80.
    Schiehlen, W. (ed.): Multibody System Handbook. Springer, Heidelberg (1990)zbMATHGoogle Scholar
  81. 81.
    Schwerin, R.: Multibody System Simulation. Springer, Berlin (1999)CrossRefzbMATHGoogle Scholar
  82. 82.
    Simeon, B.: Computational Flexible Multibody Dynamics: A Differential-Algebraic Approach. Springer, Heidelberg (2013)CrossRefzbMATHGoogle Scholar
  83. 83.
    Simeon, B., Arnold, M.: Coupling DAE’s and PDE’s for simulating the interaction of pantograph and catenary. Math. Comput. Model. Syst. 6, 129–144 (2000)CrossRefzbMATHGoogle Scholar
  84. 84.
    Simeon, B., Führer, C., Rentrop, P.: Differential-algebraic equations in vehicle system dynamics. Surv. Math. Ind. 1, 1–37 (1991)MathSciNetzbMATHGoogle Scholar
  85. 85.
    Simeon, B., Führer, C., Rentrop, P.: The Drazin inverse in multibody system dynamics. Numer. Math. 64, 521–539 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  86. 86.
    Weierstrass, K.: Zur Theorie der bilinearen und quadratischen Formen, pp. 310–338. Monatsber. Akad. Wiss., Berlin (1868)Google Scholar
  87. 87.
    Winkler, R.: Stochastic differential algebraic equations of index 1 and applications in circuit simulation. J. Comput. Appl. Math. 157 (2), 477–505 (2003)MathSciNetCrossRefzbMATHGoogle Scholar

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Authors and Affiliations

  1. 1.Felix-Klein-Zentrum, TU KaiserslauternKaiserslauternGermany

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