VSVO formulation of the taylor method for the numerical solution of ODEs

This paper presents an analysis of the Taylor method for the numerical solution of ODEs when a very high precision of the solution is required. Some theoretical properties of the Taylor method are considered. From the practical point of view a variable-stepsize variable-order (VSVO) scheme is presented and its utility is discussed with several examples. To reach the goal of high precision the use of multiprecision libraries is considered. Finally, some numerical tests based on the test problems given in [1] and on a set of important problems in dynamical systems and astrodynamics are presented showing the benefits of the VSVO formulation, especially for high-precision demands, compared with a well established Runge-Kutta code.