zbMATH — the first resource for mathematics

How AD can help solve differential-algebraic equations. (English) Zbl 1409.65050
This paper is concerned with the numerical solution to differential-algebraic equations (DAEs) in their most general form \[ f_i(t, \text{the } x_j(t) \text{ and derivatives of them}) = 0, \;i=1,2,\dots,n, \] where \(x_j(t)\), \(i=1,2,\dots,n\), are unknown variables.
A characteristic feature of differential-algebraic equations is that one needs to find derivatives of some of their equations with respect to time, as part of the so-called index reduction or regularization. The aim of this step is to reformulate a given DAE to a better form for solving it merically. This is commonly done with the help of a computer algebra system. This paper explores two significant cases where it can be done efficiently and simple by pure algorithmic differentiation. The first is the dummy derivatives (DD) method and the second is the Lagrangian formulation of mechanical systems. Both cases are described algorithmically and are illustrated by numerical simulations. The main contribution of this research is to show that: (1) for the DD method, one can combine index and order reduction in one simple framework, so one does not need to reformulate a given system in a first-order form; (2) for Lagrangian formulation of mechanical systems, one can combine all phases by automatic differentiation instead of symbolic algebra.
Reviewer: Phi Ha (Hanoi)
65L80 Numerical methods for differential-algebraic equations
34A09 Implicit ordinary differential equations, differential-algebraic equations
68W30 Symbolic computation and algebraic computation
Full Text: DOI arXiv
[1] Brenan, K. E.; Campbell, S. L.; Petzold, L. R., Numerical Solution of Initial-Value Problems in Differential-Algebraic Equations, (1996), SIAM, Philadelphia, PA · Zbl 0844.65058
[2] Enright, W. H.; Pryce, J. D., two FORTRAN packages for assessing initial value methods, ACM Trans. Math. Softw., 13, 1-27, (1987) · Zbl 0617.65069
[3] vec3d.h, a vector3d.h, accessed March 2017
[4] on the efficient generation of Taylor expansions for DAE solutions by automatic differentiationProceedings of the 7th International Conference on Applied Parallel Computing: State of the Art in Scientific Computing, PARA’04LyngbySpringer200610891098Available at
[5] Griewank, A.; Juedes, D.; Utke, J., ADOL-C, a package for the automatic differentiation of algorithms written in C/C++, ACM Trans. Math. Softw., 22, 131-167, (1996) · Zbl 0884.65015
[6] Hayes, W. B., surfing on the edge: chaos versus near-integrability in the system of Jovian planets, Monthly Not. R. Astron. Soc., 386, 295-306, (2008)
[7] Hindmarsh, A. C.; Brown, P. N.; Grant, K. E.; Lee, S. L.; Serban, R.; Shumaker, D. E.; Woodward, C. S., SUNDIALS: suite of nonlinear and differential/algebraic equation solvers, ACM Trans. Math. Softw., 31, 363-396, (2005) · Zbl 1136.65329
[8] Mattsson, S. E.; Söderlind, G., index reduction in differential-algebraic equations using dummy derivatives, SIAM J. Sci. Comput., 14, 677-692, (1993) · Zbl 0785.65080
[9] Naumann, U., The Art of Differentiating Computer Programs: An Introduction to Algorithmic Differentiation, 24, (2012), SIAM, Philadelphia, PA · Zbl 1275.65015
[10] Nedialkov, N. S.; Pryce, J. D., solving differential-algebraic equations by Taylor series (II): computing the system Jacobian, BIT Numer. Math., 47, 121-135, (2007) · Zbl 1123.65080
[11] Nedialkov, N. S.; Pryce, J. D., solving differential-algebraic equations by Taylor series (III): the DAETS code, JNAIAM J. Numer. Anal. Indust. Appl. Math, 3, 61-80, (2008) · Zbl 1188.65111
[12] DAETS user guide, Tech. Rep. CAS 08-08-NN, Department of Computing and Software, McMaster University, Hamilton, ON, Canada, 2013, 68pp., DAETS is available at
[13] Multi-body Lagrangian simulations, 2017, YouTube channel. Available at
[14] Pantelides, C. C., the consistent initialization of differential-algebraic systems, SIAM J. Sci. Stat. Comput., 9, 213-231, (1988) · Zbl 0643.65039
[15] Pryce, J. D., A simple structural analysis method for DAEs, BIT Numer. Math., 41, 364-394, (2001) · Zbl 0989.34005
[16] Riaza, R.; Tischendorf, C., qualitative features of matrix pencils and DAEs arising in circuit dynamics, Dyn. Syst., 22, 107-131, (2007) · Zbl 1125.37060
[17] A combined structural-algebraic approach for the regularization of coupled systems of DAEs, Tech. Rep. 30, Reihe des Instituts für Mathematik Technische Universität Berlin, Berlin, Germany, 2013
[18] Structural analysis for electric circuits and consequences for MNA, Tech. Rep., Humboldt-Universität zu Berlin, Mathematisch-Naturwissenschaftliche Fakultät II, Institut für Mathematik, 1998
[20] Symbolic-numeric methods for improving structural analysis of differential-algebraic equations, 2015 AMMCS-CAIMS Congress, Wilfrid Laurier University, Waterloo, ON, Canada, 2015
[21] The MathWorks: Teaching Rigid Body Dynamics, Webinar 19 Sep 2017. Available at , accessed Oct 2017
[22] Mathematical equations as executable models of mechanical systems, in Proc. 1st ACM/IEEE Internat. Conf. Cyber-Phys. Sys., Stockholm, Sweden, ACM, New York, 2010, pp. 1–11. Available at
[23] Zonneveld, J., Automatic Numerical Integration, (1970), Mathematisch Centrum, Amsterdam · Zbl 0244.65016
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.