Examples
In this section, we present a concrete example of how to reduce the index of a differential-algebraic equation (DAE) system and solve it numerically. High-index DAEs often cannot be integrated directly due to hidden constraints or inconsistent derivatives, so index reduction is an essential preprocessing step.
We illustrate the key steps: identifying the structure of the system, performing symbolic or numerical differentiation of constraints if necessary, and rewriting the system in a form suitable for time integration. The example highlights typical issues such as constraint consistency and numerical drift, and demonstrates how basic stabilization or projection methods can help maintain the validity of the solution.
How to use these examples
For each example, first identify the variables and constraints, then verify the required consistent initial data before following the reduction. Finally, use the plotted constraint residuals to compare an unstabilized integration with a stabilized or projected one.