16.3 Index of DAEs

The index along a DAE solution \(y(t)\) is the minimum number of differentiations of some of the components of the system required to solve for \(y'\) uniquely in terms of \(y\) and \(t\), so that the DAE is converted into an ODE for \(y\). Thus an ODE system is of index 0. The above chemical kinetics DAE is of index 1, as we can perform differentiation of the third equation followed by introducing the first two equations in order to obtain the ODE for \(y_3\).

Most DAE solvers, including the one in Stan, support only index-1 DAEs. For a high index DAE problem the user must first convert it to a lower index system. This often can be done by carrying out differentiations analytically (Ascher and Petzold 1998).

References

Ascher, Uri M., and Linda R. Petzold. 1998. Computer Methods for Ordinary Differential Equations and Differential-Algebraic Equations. Philadelphia: SIAM: Society for Industrial; Applied Mathematics.