Differential algebraic equations

Differential Algebraic equations (DAEs) are a combination of ordinary differential equations (ODEs) and algebraic constraints. They occur often in multiphysics problems due to coupling conditions. The algebraic equations create severe numerical difficulties because the computation necessitates not only integration but also differentiation.

Recalling from calculus that differentiation is an unbounded operation, it is much more difficult to handle than the integration used for solving ordinary differential equations. Let us consider a DAE with a sinusoidal excitation of small amplitude but at high frequency, e.g.

\begin{aligned}\dot{x}(t) & = y(t)\\0 & = x(t)\;-\; \varepsilon\sin(\omega t).\end{aligned}

This sine wave may be numerical noise as small as machine precision but nonetheless its magnitude is amplified by the frequency

\begin{aligned}x(t) & = \varepsilon\sin(\omega t)\\y(t) & = \varepsilon \;\omega \cos(\omega t).\end{aligned}

The more derivatives involved in the exact solution of a DAE, the more problems can occur in the numerical computations. The DAE-index is a measure for this difficultly. That is why it is relevant to know the index before simulation.

References

Cortes Garcia, Idoia ; De Gersem, Herbert ; Schöps, Sebastian (2019):
A Structural Analysis of Field/Circuit Coupled Problems Based on a Generalised Circuit Element.
In: Numerical Algorithms, Springer, ISSN 1017-1398,
DOI: 10.1007/s11075-019-00686-x,
[Online-Edition: https://doi.org/10.1007/s11075-019-00686-x],
[Article]

Cortes Garcia, Idoia ; Schöps, Sebastian ; De Gersem, Herbert ; Baumanns, Sascha
Ilchmann, Achim ; Reis, Timo (Hrsg.) (2018):
Systems of Differential Algebraic Equations in Computational Electromagnetics.
In: Benchmarks in Differential-Algebraic Equations, Heidelberg, Springer, DOI: 10.1007/11221_2018_8,
[Online-Edition: https://doi.org/10.1007/11221_2018_8],
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Schöps, Sebastian ; Bartel, Andreas ; Günther, Michael ; ter Maten, E. Jan W. ; Müller, Peter C. (Hrsg.) (2014):
Progress in Differential-Algebraic Equations. Deskriptor 2013.
Heidelberg, Berlin, Springer, In: Differential-Algebraic Equations Forum, [Book]

Baumanns, Sascha ; Clemens, Markus ; Schöps, Sebastian (2013):
Structural Aspects of Regularized Full Maxwell Electrodynamic Potential Formulations Using FIT.
In: Proceedings of the URSI International Symposium on Electromagnetic Theory, IEEE, S. 1007-1010, [Online-Edition: http://ieeexplore.ieee.org/stamp/stamp.jsp?arnumber=6565916],
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Bartel, Andreas ; Baumanns, Sascha ; Schöps, Sebastian (2011):
Structural Analysis of Electrical Circuits Including Magnetoquasistatic Devices.
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