F(x(t),x'(t),t)=0, x(t_{0})=x_{0} | (1) |
E(t)x'(t)=A(t)x(t)+f(t), x(t_{0})=x_{0} | (2) |
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.
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. |
Download: |
[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. |
[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 |