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.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsNotes
- 1.
The relation of the work of Weierstrass and Kronecker to Kirchhoff’s circuit laws was pointed out to me by Volker Mehrmann when we met in September 2014 during a Summer School on DAEs in Elgersburg, Germany.
References
Abraham, R., Marsden, J.E., Ratiu, T.: Manifolds, Tensor Analysis, and Applications. Springer, New York (1988)
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)
Altmann, R.: Index reduction for operator differential-algebraic equations in elastodynamics. Z. Angew. Math. Mech. 93 (9), 648–664 (2013)
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)
Arnold, V.I.: Ordinary Differential Equations. MIT Press, Cambridge (1981)
Arnold, M., Murua, A.: Non-stiff integrators for differential–algebraic systems of index 2. Numer. Algorithms 19 (1–4), 25–41 (1998)
Arnold, M., Simeon, B.: Pantograph and catenary dynamics: a benchmark problem and its numerical solution. Appl. Numer. Math. 34, 345–362 (2000)
Ascher, U., Lin, P.: Sequential regularization methods for nonlinear higher index DAEs. SIAM J. Sci. Comput. 18, 160–181 (1997)
Ascher, U.M., Petzold, L.R.: Projected implicit Runge-Kutta methods for differential-algebraic equations. SIAM J. Numer. Anal. 28, 1097–1120 (1991)
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)
Bangerth, W., Rannacher, R.: Adaptive Finite Element Methods for Differential Equations. Birkhäuser, Basel (2013)
Baumgarte, J.: Stabilization of constraints and integrals of motion in dynamical systems. Comput. Methods Appl. Mech. 1, 1–16 (1972)
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)
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)
Brasey, V.: A half-explicit method of order 5 for solving constrained mechanical systems. Computing 48, 191–201 (1992)
Brasey, V., Hairer, E.: Half-explicit Runge–Kutta methods for differential-algebraic systems of index 2. SIAM J. Numer. Anal. 30, 538–552 (1993)
Brenan, K.E., Campbell, S.L., Petzold, L.R.: The Numerical Solution of Initial Value Problems in Ordinary Differential-Algebraic Equations. SIAM, Philadelphia (1996)
Brezzi, F., Fortin, M.: Mixed and Hybrid Finite Element Methods. Springer, New York (1991)
Brizard, A.: An Introduction to Lagrangian Mechanics. World Scientific, Singapore (2008)
Campbell, S.L.: Singular Systems of Differential Equations. Pitman, London (1980)
Campbell, S.L.: Singular Systems of Differential Equations II. Research Notes in Mathematics, vol. 61. Pitman, London (1982)
Campbell, S.L.: Least squares completions for nonlinear differential-algebraic equations. Numer. Math. 65, 77–94 (1993)
Campbell, S., Gear, C.: The index of general nonlinear DAEs. Numer. Math. 72, 173–196 (1995)
Callies, R., Rentrop, P.: Optimal control of rigid-link manipulators by indirect methods. GAMM-Mitteilungen 31 (1), 27–58 (2008)
Deuflhard, P., Hairer, E., Zugck, J.: One-step and extrapolation methods for differential-algebraic systems. Numer. Math. 51 (5), 501–516 (1987)
Drazin, M.: Pseudo inverses in associative rays and semigroups. Am. Math. Mon. 65, 506–514 (1958)
Eich, E.: Convergence results for a coordinate projection method applied to constrained mechanical systems. SIAM J. Numer. Anal. 30 (5), 1467–1482 (1993)
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)
Führer, C., Leimkuhler, B.: Numerical solution of differential-algebraic equations for constrained mechanical motion. Numer. Math. 59, 55–69 (1991)
Gantmacher, F.: Matrizenrechnung, Teil 2. VEB Deutscher Verlag der Wissenschaften, Berlin (1959)
Garcia de Jalón, J., Bayo, E.: Kinematic and Dynamic Simulation of Multibody Systems. Springer, New York (1994)
Gear, C.W.: Numerical Initial Value Problems in Ordinary Differential Equations. Prentice-Hall, Upper Saddle River (1971)
Gear, C.W.: Simultaneous numerical solution of differential-algebraic equations. IEEE Trans. Circuit Theory CT-18 (1), 89–95 (1971)
Gear, C.W.: Differential-algebraic equation index transformation. SIAM J. Sci. Stat. Comput. 9, 39–47 (1988)
Gear, C.W.: Differential-algebraic equations, indices, and integral algebraic equations. SIAM J. Numer. Anal. 27, 1527–1534 (1990)
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)
Griepentrog, E., März, R.: Differential-Algebraic Equations and Their Numerical Treatment. Teubner-Texte zur Mathematik, vol. 88. Teubner Verlagsgesellschaft, Leipzig (1986)
Günther, M.: Partielle differential-algebraische Systeme in der numerischen Zeitbereichsanalyse elektrischer Schaltungen. VDI-Verlag, Reihe 20, Düsseldorf (2001)
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)
Günther, M., Hoschek, M., Rentrop, P.: Differential-algebraic equations in electric circuit simulation. Int. J. Electron. Commun. 54, 101–107 (2000)
Hairer, E., Wanner, G.: Solving Ordinary Differential Equations II: Stiff and Differential-Algebraic Problems. Springer, Berlin (1996)
Hairer, E., Wanner, G.: Stiff differential equations solved by Radau methods. J. Comput. Appl. Math. 111, 93–111 (1999)
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)
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)
Hairer, E., Nørsett, S., Wanner, G.: Solving Ordinary Differential Equations I: Nonstiff Problems. Springer, Berlin (1993)
Hairer, E., Lubich, C., Wanner, G.: Geometric Numerical Integration. Springer, Berlin (2002)
Hanke, M.: On the regularization of index 2 differential-algebraic equations. J. Math. Anal. Appl. 151 (1), 236–253 (1990)
Haug, E.: Computer-Aided Kinematics and Dynamics of Mechanical Systems. Allyn and Bacon, Boston (1989)
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)
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)
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)
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)
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)
Kronecker, L.: Algebraische Reduktion der Schaaren bilinearer Formen. Akademie der Wissenschaften Berlin III, 141–155 (1890)
Kunkel, P., Mehrmann, V.: Numerical solution of differential algebraic Riccati equations. Linear Algebra Appl. 137, 39–66 (1990)
Kunkel, P., Mehrmann, V.: Differential-Algebraic Equations – Analysis and Numerical Solution. EMS Publishing House, Zürich (2006)
Lagrange, J.L.: Méchanique analytique. Libraire chez la Veuve Desaint, Paris (1788)
Lamour, R., März, R., Tischendorf, C.: Differential-Algebraic Equations: A Projector Based Analysis. Differential-Algebraic Equations Forum. Springer, Berlin (2013)
Lang, J.: Adaptive Multilevel Solution of Nonlinear Parabolic PDE Systems: Theory, Algorithm, and Applications, vol. 16. Springer, Berlin (2013)
Lötstedt, P., Petzold, L.: Numerical solution of nonlinear differential equations with algebraic constraints i: convergence results for BDF. Math.Comput. 46, 491–516 (1986)
Lubich, C.: h 2 extrapolation methods for differential-algebraic equations of index-2. Impact Comput. Sci. Eng. 1, 260–268 (1989)
Lubich, C.: Integration of stiff mechanical systems by Runge-Kutta methods. ZAMP 44, 1022–1053 (1993)
Lubich, C., Engstler, C., Nowak, U., Pöhle, U.: Numerical integration of constrained mechanical systems using MEXX. Mech. Struct. Mach. 23, 473–495 (1995)
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)
März, R.: Differential algebraic systems anew. Appl. Numer. Math. 42 (1), 315–335 (2002)
März, R., Tischendorf, C.: Recent results in solving index-2 differential-algebraic equations in circuit simulation. SIAM J. Sci. Comput. 18, 139–159 (1997)
Mattson, S., Söderlind, G.: Index reduction in differential-algebraic equations using dummy derivatives. SIAM J. Sci. Comput. 14 (3), 677–692 (1993)
Müller, P.C.: Stability of linear mechanical systems with holonomic constraints. Appl. Mech. Rev. 46 (11S), S160–S164 (1993)
Müller, P.C.: Stability and optimal control of nonlinear descriptor systems: a survey. Appl. Math. Comput. Sci. 8, 269–286 (1998)
Nagel, L.W., Pederson, D.: Spice (simulation program with integrated circuit emphasis). Technical Report UCB/ERL M382, EECS Department, University of California, Berkeley (1973)
O’Malley, R.E.: Introduction to Singular Perturbations. Academic, New York (1974)
Petzold, L.: A description of DASSL: a differential/algebraic system solver. In: Proceedings of 10th IMACS World Congress, Montreal, 8–13 August 1982
Plinninger, T., Simeon, B.: Adaptivity in space and time for solving transient problems in COMSOL. In: Proceedings COMSOL Conference Hannover (2008)
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)
Reich, S.: On a geometric interpretation of DAEs. Circ. Syst. Signal Process. 9, 367–382 (1990)
Reis, T., Stykel, T.: Stability analysis and model order reduction of coupled systems. Math. Comput. Model. Dyn. Syst. 13 (5), 413–436 (2007)
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)
Rheinboldt, W.: Differential - algebraic systems as differential equations on manifolds. Math. Comput. 43 (168), 2473–482 (1984)
Rheinboldt, W.: Manpak: a set of algorithms for computations on implicitly defined manifolds. Comput. Math. Appl. 32, 15–28 (1996)
Schiehlen, W. (ed.): Multibody System Handbook. Springer, Heidelberg (1990)
Schwerin, R.: Multibody System Simulation. Springer, Berlin (1999)
Simeon, B.: Computational Flexible Multibody Dynamics: A Differential-Algebraic Approach. Springer, Heidelberg (2013)
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)
Simeon, B., Führer, C., Rentrop, P.: Differential-algebraic equations in vehicle system dynamics. Surv. Math. Ind. 1, 1–37 (1991)
Simeon, B., Führer, C., Rentrop, P.: The Drazin inverse in multibody system dynamics. Numer. Math. 64, 521–539 (1993)
Weierstrass, K.: Zur Theorie der bilinearen und quadratischen Formen, pp. 310–338. Monatsber. Akad. Wiss., Berlin (1868)
Winkler, R.: Stochastic differential algebraic equations of index 1 and applications in circuit simulation. J. Comput. Appl. Math. 157 (2), 477–505 (2003)
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.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2017 Springer International Publishing AG
About this chapter
Cite this chapter
Simeon, B. (2017). On the History of Differential-Algebraic Equations. In: Ilchmann, A., Reis, T. (eds) Surveys in Differential-Algebraic Equations IV. Differential-Algebraic Equations Forum. Springer, Cham. https://doi.org/10.1007/978-3-319-46618-7_1
Download citation
DOI: https://doi.org/10.1007/978-3-319-46618-7_1
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-46617-0
Online ISBN: 978-3-319-46618-7
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)