Differential Algebraic Equations

Differential Algebraic equations (DAEs) are a combination of ordinary differential equations (ODEs) and algebraic constraints. They occur often in space-discrete 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 analysis that differentiation is an unbounded operator, such a process is much more difficult to handle than the integrals used for solving ODEs. Let us consider a DAE with a sinusoidal signal of small amplitude but at high frequency

\[ \begin{aligned}\dot{x}(t) & = y(t)\\0 & = x(t) - \varepsilon\cdot\sin(\omega\cdot t)\end{aligned} \]

This sine wave can be a numerical error at machine precision. Nonetheless the solution

\[ \begin{aligned}x(t) & = \varepsilon\cdot\sin(\omega\cdot t)\\y(t) & = \varepsilon\cdot\omega\cdot\cos(\omega\cdot t)\end{aligned} \]

shows that the the error is magnified by ω and furthermore that initial values cannot be chosen arbitrarily.

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 important to know the index before simulation.

 

Example of a coupled index-2 problem (field/circuit). The error increases although the step size is decreased.

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