SolveDAE

A MATLAB Toolbox for the Numerical Solution of Differential-Algebraic equations

SolveDAE is a MATLAB toolbox for the numerical solution of systems of linear or nonlinear differential-algebraic equations (DAEs) of arbitrary index given in the form
F(x(t),x'(t),t)=0, x(t0)=x0 (1)
or
E(t)x'(t)=A(t)x(t)+f(t), x(t0)=x0 (2)

on the domain [t0,tf].

The toolbox employs the Fortran77 subroutines GENDA (GEneral Nonlinear Differential-Algebraic equation solver) [B] and GELDA (GEneral Linear Differential-Algebraic equation solver) [C] to solve these kind of problems. Both routines provide solutions for under- and overdetermined systems. The toolbox features symbolic differentiation (the Symbolic Math Toolbox is required). In this way the user only has to supply the functions defining the given DAE and does not need to provide the derivatives or Jacobians of these functions. Other features of the toolbox include the computation of characteristic values and the computation of consistent initial values in the least square sense (the Optimization Toolbox is required). In order to make the utilization of the solvers GELDA and GENDA as easy and comfortable as possible a graphical user interface (GUI) provides the possibility to adjust a variety of parameters that enable the user to customize the solvers to certain problems.

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 [A]. 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 GELDA is based on the construction of the discretization scheme described in
[C], which first determines all the local invariants and then transforms the system (2) into an equivalent strangeness-free DAE with the same 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, GELDA is able to compute a consistent initial value and apply the well-known integration schemes for DAEs. In GELDA the BDF methods [8] and Runge-Kutta schemes [6,7] are implemented.
The implementation of GENDA is based on the construction of the discretization scheme described in [B], which transforms the system into a strangeness-free DAE with the same local solution set. Again, the resulting strangeness-free system is allowed to have nonuniqueness in the solution set or inconsistency in the initial values or inhomogeneities. 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 a 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
Stefan Seidel
Purpose: SolveDAE is a MATLAB Toolbox for the numerical integration of linear or nonlinear DAEs of arbitrary index.
Method: Nonlinear DAEs (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 tha same local solution set. Linear DAEs (2) may be integrated by the implicit Runge-Kutta method of type RADAU IIa of order 5 or by the BDF method of order 1,...,6 based on an equivalent strangeness-free formulation of the DAE with the same solution set.
Documentation: A printed documentation [D] 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.

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Theoretical and Numerical Results related to SolveDAE

[A] Peter Kunkel and Volker Mehrmann. Differential-Algebraic Equations - Analysis and Numerical Solution, EMS Publishing House, Zürich, 2006.    
[B] 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.
[C] Peter Kunkel, Volker Mehrmann, Werner Rath and Jörg Weickert. GELDA: A software package for the solution of General Linear Differential Algebraic equations, SIAM Journal Scientific Computing, Vol. 18, pp. 115-138, 1997.
[D] Peter Kunkel, Volker Mehrmann and Stefan Seidel. A MATLAB package for the numerical solution of General Nonlinear Differential-Algebraic equations, Institut für Mathematik, Technische Universität Berlin, number 16-2005, 2005.

References

[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 10.08.2011