A GEneral Nonlinear Differential Algebraic equation solver

GENDA is a Fortran77 sofware package for the numerical solution of nonlinear differential-algebraic equations (DAEs) of arbitrary index
0=F(x,x',t) (1)
on the domain [t0,tf] together with an initial condition
An important invariant in the analysis of DAEs is the so called strangeness index, which generalizes the differentiation index
[2], [3], [5] for systems with undetermined components [B]. It is known that many of the standard integration methods for general DAEs require the system to have differentiation index not higher than one. If this condition is not valid or if the DAE has undetermined components, then the standard methods as implemented in codes like DASSL of Petzold [8] or LIMEX of Deuflhard/Hairer/Zugck [4] may fail.
The implementation of GENDA is based on the construction of the discretization scheme introduced in [A], which transforms the system into a strangeness-free DAE with the same local solution set. The resulting strangeness-free system is allowed to have nonuniqueness in the solution set or inconsistency in the initial values or inhomogeneities. But this information is now available to the user and systems with such properties can be treated in a least squares sense.
In the case that the DAE is found to be uniquely solvable, GENDA is able to compute a consistent initial value and apply an integration scheme for DAEs. In GENDA Runge-Kutta scheme of type RADAU IIa of order 5 [6], [7] is implemented.

General Information

Authors: Peter Kunkel, Universität Leipzig, Mathematisches Institut
Volker Mehrmann, Technische Universität Berlin, Institut für Mathematik
Ingo Seufer, Technische Universität Berlin, Institut für Mathematik
Purpose: GENDA performs the numerical integration of nonlinear DAEs (1) of arbitrary index.
Method: The DAE (1) will be integrated by the implicit Runge-Kutta method of type RADAU IIa of order 5 based on an equivalent strangeness-free formulation of the DAE with the same local solution set.
Documentation: A printed documentation [C] is available.
Support: Technical questions about the proper use of a software package should be directed to the authors of that package.
Disclaimer: Warranty disclaimer: The software is supplied "as is" without warranty of any kind. The copyright holder: (1) disclaim any warranties, express or implied, including but not limited to any implied warranties of merchantability, fitness for a particular purpose, title or non-infringement, (2) do not assume any legal liability or responsibility for the accuracy, completeness, or usefulness of the software, (3) do not represent that use of the software would not infringe privately owned rights, (4) do not warrant that the software will function uninterrupted, that it is error-free or that any errors will be corrected.

Limitation of liability: In no event will the copyright holder: be liable for any indirect, incidental, consequential, special or punitive damages of any kind or nature, including but not limited to loss of profits or loss of data, for any reason whatsoever, whether such liability is asserted on the basis of contract, tort (including negligence or strict liability), or otherwise, even if any of said parties has been warned of the possibility of such loss or damages.


Theoretical and Numerical Results related to GENDA

[A] Peter Kunkel and Volker Mehrmann. A new class of discretization methods for the solution of linear differential-algebraic equations with variable coefficients. SIAM J. Numer. Anal., Vol. 33, No.5, pp. 1941-1961, October 1996.  
[B] Peter Kunkel and Volker Mehrmann. Differential-Algebraic Equations - Analysis and Numerical Solution, EMS Publishing House, Zürich, 2006    
[C] Peter Kunkel, Volker Mehrmann and Ingo Seufer. GENDA: A software package for the solution of General Nonlinear Differential-Algebraic equations, Institut für Mathematik, Technische Universität Berlin, number 730-02. 2002.  


[1] K. E. Brenan, S. L. Campbell and L. R. Petzold. Numerical Solution of Initial-Value Problems in Differential Algebtraic Equations, Elsevier, North Holland, New York, N.Y., 1989.
[2] S. L. Campbell. Comment on controlling generalized state-space (descriptor) systems, Internat. J. Control, 46 (1987), pp. 2229-2230.
[3] S. L. Campbell. Nonregular descriptor systems with delays, in Proc. Symp. Implicit & Nonlinear Systems, Dallas, 1992, pp. 275-281.
[4] P. Deuflhard, E. Hairer amg J. Zugck. One step and extrapolation methods for differential-algebraic systems, Numer. Math., 51 (1987), pp. 501-516.
[5] C. W. Gear. Differential-algebraic equations index transformations, SIAM J. Sci. Statist. Comput., 9 (1988), pp. 39-47.
[6] E. Hairer, C. Lubich and M. Roche. The Numerical Solution of Differential-Algebraic Systems by Runge-Kutta Methods, Lecture Notes in Mathematics No. 1409, Springer-Verlag, Berlin, 1989.
[7] E. Hairer and G. Wanner. Solving Ordinary Differential Equations II , Springer-Verlag, Berlin, 1991.
[8] L. R. Petzold. A description of DASSL: A differential/algebraic system solver, in IMACS Trans. Scientific Computing Vol. 1, R. S. Stepleman et al., eds., North-Holland, Amsterdam, 1993, pp. 65-68.

Impressum Andreas Steinbrecher 16.08.2011