Hostname: page-component-78c5997874-dh8gc Total loading time: 0 Render date: 2024-11-10T09:20:32.248Z Has data issue: false hasContentIssue false
  • English
  • Français

Numerical methods for differential algebraic equations

Published online by Cambridge University Press:  07 November 2008

Roswitha März
Roswitha März
Affiliation:
Humboldt-UniversitätFachbereich Mathematik Postfach 1297, D-O-1086 Berlin, Germany, E-mail: ivs@mathematik.hu-berlin.dbp.de
Get access Rights & Permissions [Opens in a new window]

Extract

Differential algebraic equations (DAE) are special implicit ordinary differential equations (ODE)

where the partial Jacobian f′y(y, x, t) is singular for all values of its arguments.

Type
Research Article
Information
Acta Numerica , Volume 1 , January 1992 , pp. 141 - 198
Copyright
Copyright © Cambridge University Press 1992

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

Ascher, U. (1989), ‘On numerical differential algebraic problems with application to semi-conductor device simulations’, SIAM J. Numer. Anal. 26, 517–538.CrossRefGoogle Scholar
Ascher, U. and Petzold, L.R. (1990), Projected implicit Runge-Kutta methods for differential-algebraic equations, Preprint, Lawrence Livermore National LaboratoryGoogle Scholar
Brenan, K.E. and Engquist, B.E. (1988), ‘Backward differentiation approximations of nonlinear differential/algebraic systems’, and Supplement, Math. Comput. 51, 659–676, S7–S16.CrossRefGoogle Scholar
Brenan, K.E., Campbell, S.L. and Petzold, L.R. (1989), Numerical Solution of Initial-Value Problems in Differential-Algebraic Equations, North-Holland (Amsterdam).Google Scholar
Campbell, S.L. (1987), ‘A general form for solvable linear time varying singular systems of differential equations’, SIAM J. Math. Anal. 18, 1101–1115.CrossRefGoogle Scholar
Chua, L.O. and Deng, A. (1989), ‘Impasse points. Part I: Numerical aspects’, Int. J. Circuit Theory Applics 17, 213–235.CrossRefGoogle Scholar
Čistjakov, V.F. (1982), ‘K metodam rešenija singuljarnych linejnych sistem obyknovennych differencial'nych uravnenij’, in Vyroždennyje Sistemy Obyknovennych Differencial'nych Uravnenij (Bojarincev, Ju.E., ed.) Nauka Novosibirsk, 37–65.Google Scholar
Degenhardt, A. (1991), ‘A collocation method for boundary value problems of transferable DAEs’, Numer. Math., to appear.Google Scholar
Deuflhard, P., Hairer, E., Zugck, J. (1987), ‘One-step and extrapolation methods for differential-algebraic systems’ Numer. Math. 51, 501–516.CrossRefGoogle Scholar
Führer, C. (1988), Differential-algebraische Gleichungssysteme in mechanischen Mehrkörpersystemen. Theorie, numerische Ansätze und Anwendungen, Dissertation, Techn. Univ. München, Fak. für Mathematik und Informatik.Google Scholar
Führer, C. and Leimkuhler, B. (1989), Formulation and numerical solution of the equations of constrained mechanical motion, DFVLR-Forschungsbericht 89-08, Deutsche Forschungs- und Versuchsanstalt für Luft- und Raumfahrt, Oberpfaffenhofen.Google Scholar
Führer, C. and Leimkuhler, B.J. (1990), A new class of generalized inverses for the solution of discretized Euler–Lagrange equations, in NATO Advanced Research Workshop on Real-Time Integration Methods for Mechanical System Simulation (Snowbird, Utah 1989) (Haug, E. and Deyo, R., eds.) Springer (Berlin).Google Scholar
Gantmacher, F.R. (1966), Teorija matric, Nauka (Moskva).Google Scholar
Gear, C.W. (1971), ‘The simultaneous numerical solution of differential-algebraic equations’, IEEE Trans. Circuit Theory, CT-18, 89–95.CrossRefGoogle Scholar
Gear, C.W. and Petzold, L.R. (1984), ‘ODE methods for the solution of differential/algebraic systems’, SIAM J. Numer. Anal. 21, 716–728.CrossRefGoogle Scholar
Gear, C.W., Hsu, H.H. and Petzold, L. (1981), Differential-algebraic equations revisited, Proc. ODE Meeting, Oberwolfach, Germany, Institut für Geom. und Praktische Mathematik, Technische Hochschule Aachen, Bericht 9, Germany.Google Scholar
Gear, C.W., Leimkuhler, B. and Gupta, G.K. (1985), ‘Automatic integration of Euler–Lagrange equations with constraints’, J. Comput. Appl. Math. 12 & 13, 77–90.CrossRefGoogle Scholar
Griepentrog, E. (1991), Index reduction methods for differential-algebraic equations, Preprint 91–12, Humboldt-Univ. Berlin, Fachbereich Mathematik.Google Scholar
Griepentrog, E. and März, R. (1986), Differential-Algebraic Equations and their Numerical Treatment (Teubner Texte zur Mathematik 88) Teubner (Leipzig).Google Scholar
Griepentrog, E. and März, R. (1989), ‘Basic properties of some differential-algebraic equations’, Z. Anal. Anwend. 8 (1), 25–40.CrossRefGoogle Scholar
Hairer, E. and Wanner, G. (1991), Solving Ordinary Differential Equations II: Stiff and Differential-Algebraic Problems (Springer Series in Computational Mathematics 14) Springer (Berlin).CrossRefGoogle Scholar
Hairer, E., Lubich, Ch. and Roche, M. (1989), The Numerical Solution of Differential-Algebraic Systems by Runge-Kutta Methods (Lecture Notes in Mathematics 1409) Springer (Berlin).CrossRefGoogle Scholar
Hanke, M. (1990) Regularization methods for higher index differential-algebraic equations, Preprint 268, Humboldt-Univ. Berlin, Fachbereich Mathematik.Google Scholar
Hanke, M. (1991), ‘On the asymptotic representation of a regularization approach to nonlinear semiexplicit higher-index differential-algebraic equations’, IMA J. Appl. Math. 46, 225–245.CrossRefGoogle Scholar
Hansen, B. (1989), Comparing different concepts to treat differential-algebraic equations, Preprint 220, Humboldt-Univ. Berlin, Sektion Mathematik.Google Scholar
Hansen, B. (1990), Linear time-varying differential-algebraic equations being tractable with the index k, Preprint 246, Humboldt-Univ. Berlin, Sektion Mathematik.Google Scholar
Keller, H.B. (1975), ‘Approximation methods for nonlinear problems with application to two-point boundary value problems’, Math. Comput. 29 (130), 464–474.CrossRefGoogle Scholar
Keller, H.B. and White, A.B. (1975), ‘Difference methods for boundary value problems in ordinary differential equations’, SIAM J. Numer. Anal. 12, 791–802.CrossRefGoogle Scholar
Lamour, R. (1991a), ‘A well-posed shooting method for transferable DAE's’, Numer. Math. 59.CrossRefGoogle Scholar
Lamour, R. (1991b), Oscillations in differential-algebraic equations, Preprint 272, Humboldt-Universität Berlin, Fachbereich Mathematik.Google Scholar
Lentini, M. and März, R. (1990a), ‘The conditioning of boundary value problems in transferable differential-algebraic equations’, SIAM J. Numer. Anal. 27, 1001–1015.CrossRefGoogle Scholar
Lentini, M. and März, R. (1990b),‘Conditioning and dichotomy in transferable differential-algebraic equations’, SIAM J. Numer. Anal. 27, 1519–1526.CrossRefGoogle Scholar
Lötstedt, P. and Petzold, L. (1986), ‘Numerical solution of nonlinear differential equations with algebraic constraints I: Convergence results for backward differentiation formulas’, Math. Comput. 46, 491–516.Google Scholar
Lubich, Ch. (1990), Extrapolation integrators for constrained multibody systems, Report, Univ. Innsbruck.Google Scholar
März, R. (1984), ‘On difference and shooting methods for boundary value problems in differential-algebraic equations’, ZAMM 64 (11), 463–473.CrossRefGoogle Scholar
März, R. (1985), ‘On initial value problems in differential-algebraic equations and their numerical treatment’, Computing 35, 13–37.CrossRefGoogle Scholar
März, R. (1989), ‘Some new results concerning index-3 differential-algebraic equations’, J. Math. Anal. Applics 140 (1), 177–199.CrossRefGoogle Scholar
März, R. (1990), ‘Higher-index differential-algebraic equations: Analysis and numerical treatment’, Banach Center Publ. 24, 199–222.CrossRefGoogle Scholar
März, R. (1991), On quasilinear index 2 differential algebraic equations, Preprint 269, Humboldt-Universität Berlin, Fachbereich Mathematik.Google Scholar
Mrziglod, T. (1987), Zur Theorie und Numerischen Realisierung von Lösungs-methoden bei Differentialgleichungen mit angekoppelten algebraischen Gleichungen, Diplomarbeit, Universität zu Köln.Google Scholar
Petzold, L.R. (1986), ‘Order results for implicit Runge–Kutta methods applied to differential/algebraic systems’, SIAM J. Numer. Anal. 23, 837–852.CrossRefGoogle Scholar
Petzold, L. and Lötstedt, P. (1986), ‘Numerical solution of nonlinear differential equations with algebraic constraints II: Practical implications’, SIAM J. Sci. Stat. Comput. 7, 720–733.CrossRefGoogle Scholar
Potra, F.A. and Rheinboldt, W.C. (1991), ‘On the numerical solution of Euler–Lagrange equations’, Mech. Struct. Machines 19 (1).CrossRefGoogle Scholar
Rabier, P.J. and Rheinboldt, W.C. (1991), ‘A general existence and uniqueness theory for implicit differential-algebraic equations’, Diff. Int. Eqns 4 (3), 563–582.Google Scholar
Reich, S. (1990), Beitrag zur Theorie der Algebrodifferentialgleichungen, Dissertation (A), Technische Universität Dresden.Google Scholar
Rheinboldt, W.C. (1984), ‘Differential-algebraic systems as differential equations on manifolds’, Math. Comput. 43, 473–482.CrossRefGoogle Scholar
Simeon, B., Führer, C. and Rentrop, P. (1991), ‘Differential-algebraic equations in vehicle system dynamics’, Suru. Math. Ind. 1 (1), 1–37.Google Scholar
Sincovec, R.F., Erisman, A.M, Yip, E.L. and Epton, M.A. (1981), ‘Analysis of descriptor systems using numerical algorithms’, IEEE Trans. Aut. Control, AC-26, 139–147.CrossRefGoogle Scholar