The Algorithm

The approach to find a consistent projection of state variables onto a DAE manifold consists of two distinct parts. First, all equations and variables are classified, specified by Algorithm 1. All equations that are a function of rate variables, $\dot{x}$, are assigned to f and the remaining equations constitute h. Equations in h that are no function of y can be differentiated without introducing new variables, and, therefore, are marked $\bar{h}$,otherwise they belong to $\hat{h}$.All state variables x in $\bar{h}$ are marked $\bar{x}$. The variables in $\hat{h}$ that have not been marked are assigned to $\hat{x}$ and $\hat{y}$. Any remaining variables are present in f but not in h and are marked $\tilde{y}$.

 

Next, the algebraic variables $\hat{y}$ are solved from f by partitioning the system of equations f and h such that $\hat{y}$ is isolated. The procedure is described in the previous section and summarized by Algorithm 2.