The use of Butcher series in the analysis of Newton-like iterations in Runge–Kutta formulas

Abstract

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