An Hermite-Obreschkoff method for stiff high-index DAE

We have developed an implicit Hermite-Obreschkoff method for the numerical solution of stiff high-index differential-algebraic equations (DAEs). On each integration step, this method requires the computation of Taylor coefficients and their gradients to construct and solve a nonlinear system for the numerical solution, which is then projected to satisfy the constraints of the problem. We derive this system, show how to compute its Jacobian through automatic differentiation, and present the ingredients of our method, such as predicting an initial guess for Newton’s method, error estimation, and stepsize and order control. We report numerical results on stiff DAEs illustrating the accuracy and performance of our method, and in particular, its ability to take large steps on stiff problems.