THE USE OF BUTCHER SERIES IN THE ANALYSIS OF NEWTON-LIKE ITERATIONS IN RUNGE-KUTTA FORMULAS

We consider the numerical solution of initial value problems for both ordinary differential equations and differential-algebraic equations b y Runge-Kutta (RK) formulas. We assume that the internal stage values of. the RK formula are computed by some iterative scheme for solving n onlinear equations, such as Newton's method. Using Butcher series and rooted trees, we give a complete characterization of the local error i n the RK formula after k iterations of the scheme. Results are given f or three specific schemes: simple iteration, the modified Newton itera tion, and the full Newton iteration. The ideas developed in this paper , however, are easily applied to other iterative schemes of this kind.