The DAETS solver by Nedialkov and Pryce integrates numerically high-index DAE systems. This solver is based on explicit Taylor series and is efficient on non-stiff to mildly stiff problems, but can have severe stepsize restrictions on (very) stiff problems. Hermite-Obreschkoff (HO) methods can be viewed as a generalization of Taylor series methods. The former have smaller truncation error than the latter and can be A- or L-stable.
We develop an HO method for numerical solution of stiff high-index DAEs. As in DAETS, our method employs Pryce's structural analysis to determine the constraints of the problem and to organize the computations of higher-order derivatives and their gradients. We discuss this method and its ingredients: finding a consistent initial point, computing an initial guess for Newton's method, automatic differentiation for constructing the needed Jacobians, error estimation
and control, and stiffness detection. We report numerical results on stiff DAE and ODE systems illustrating the performance of our method, and in particular, its ability to take large steps on stiff problems.