Ode Solving via Automatic Diierentiation and Rational Prediction

We consider the classical Taylor series approximation to the solution of initial value problems in ordinary diierential equations and examine implicit variants for the numerical solution of stii ODEs. The Taylor coeecients of the state vector are found to be closely related to those of the Jacobian of the right hand side along the solution trajectory. These connections between state and Jacobian coeecients are exploited for their eecient evaluation by automatic diierentiation with a small number of forward and reverse sweeps. It is shown how these coeecients can be utilized in a new rational predictor for the Hermite-Obreshkov-Pad e (HOP) methods, a family of high order numerical integrators, last examined by Wanner in the sixties. The linearly implicit predictor and the full HOP methods yield in the constant coeecient case Pad e approximants of the matrix exponential. A-and L-stability is achieved for the diagonal and rst two subdiagonal choices of the Pad e parameter pair (q; p). Preliminary numerical results demonstrate that on stii and highly oscillatory problems large steps can be realized with a single correction iteration and acceptable discretization error.