Diagnosis of singular points of structured DAEs using automatic differentiation. (English) Zbl 1322.65082
Summary: At a singular point of a differential-algebraic equation (DAE), the initial value problem (IVP) fails to have a unique solution. Hence, numerical integration methods cannot provide reasonable results. Unfortunately, common error control strategies do not always detect these circumstances and arbitrary solutions may be given to the user without warnings of any kind.{
}Automatic (or algorithmic) differentiation (AD) opens new possibilities to realize an analysis of DAEs and to monitor assumptions required for the existence and uniqueness of IVPs. We show how the diagnosis of singular points can be performed for structured quasi-linear DAEs up to index 2. Our approach uses the projector based analysis for DAEs employing AD. The resulting method is illustrated by several examples, with particular emphasis on simple electrical circuits containing controlled sources.
MSC:
| 65L80 | Numerical methods for differential-algebraic equations |
| 65L05 | Numerical methods for initial value problems involving ordinary differential equations |
| 34A09 | Implicit ordinary differential equations, differential-algebraic equations |
Keywords:
differential-algebraic equations; automatid differentiation; initial value problems; singular points; critical points; diagnosis; numerical examples; quasi-linear; electrical circuitReferences:
| [1] | Estévez Schwarz, D.: Consistent initialization for index-2 differential algebraic equations and its application to circuit simulation. Humboldt-Univ., Mathematisch-Naturwissenschaftliche Fakultaẗ II. PhD thesis, Berlin (2000). http://edoc.hu-berlin.de/docviews/abstract.php?id=10218 · Zbl 1021.65042 |
| [2] | Estévez Schwarz, D., Lamour, R.: Monitoring singularities while integrating DAEs. Differential-Algebraic Equations Forum. Springer (To appear) · Zbl 1319.65076 |
| [3] | Estévez Schwarz D., Lamour R.: Projector based integration of DAEs with the Taylor series method using automatic differentiation. J. Comput. Appl. Math. (2014) · Zbl 1301.65087 |
| [4] | Estévez Schwarz, D., Tischendorf, C.: Structural analysis of electric circuits and consequences for the MNA. Int. J. Circuit Theory Appl. 28(2), 131-162 (2000) · Zbl 1054.94529 · doi:10.1002/(SICI)1097-007X(200003/04)28:2<131::AID-CTA100>3.0.CO;2-W |
| [5] | Golub, G.H., van Loan, C.F.: Matrix Computations. John Hopkins University Press, Baltimore and London (1996) · Zbl 0865.65009 |
| [6] | Griepentrog, E., März, R.: Differential-algebraic equations and their numerical treatment., volume 88 of Teubner-Texte zur Mathematik. B.G. Teubner Verlagsgesellschaft, Leipzig (1986) · Zbl 0629.65080 |
| [7] | Hairer, E., Nørsett, S., Wanner, G.: Solving Ordinary Differential Equations I. Springer (1993) · Zbl 0789.65048 |
| [8] | Lamour, R., März, R., Tischendorf, C.: Differential-algebraic equations: A projector based analysis. Differential-Algebraic Equations Forum, vol. 1. Springer, Berlin (2013) · Zbl 1276.65045 · doi:10.1007/978-3-642-27555-5 |
| [9] | Mazzia, F., Magherini, C.: Test set for initial value problems release 2 Technical report, Department of Mathematics, University of Bari and INdAM, vol. 4, Research Unit of Bari (2008) |
| [10] | Rabier, P.J., Rheinboldt, W.C.: On impasse points of quasilinear differential-algebraic equations. J. Math. Anal. Appl. 181(2), 429-454 (1994) · Zbl 0803.34004 · doi:10.1006/jmaa.1994.1033 |
| [11] | Rabier, P.J., Rheinboldt, W.C.: On the computation of impasse points of quasi-linear differential- algebraic equations. Math. Comput. 62(205), 133-154 (1994) · Zbl 0793.65050 |
| [12] | Riaza, R.: Differential-algebraic systems. Analytical aspects and circuit applications. Hackensack. World Scientific, NJ (2008) · Zbl 1184.34004 · doi:10.1142/6746 |
| [13] | Rump. S.M.: INTLAB - INTerval LABoratory. In: Csendes, T. (ed.) Developments in Reliable Computing, pp 77-104. SCAN-98, Kluwer Academic Publishers (1999) · Zbl 0949.65046 |
| [14] | Tuomela, J.: On singular points of quasilinear differential and differential-algebra equations. BIT 37(4), 968-977 (1997) · Zbl 0980.34044 · doi:10.1007/BF02510364 |
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